diff --git "a/3tAzT4oBgHgl3EQfR_u4/content/tmp_files/load_file.txt" "b/3tAzT4oBgHgl3EQfR_u4/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/3tAzT4oBgHgl3EQfR_u4/content/tmp_files/load_file.txt" @@ -0,0 +1,1004 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf,len=1003 +page_content='k-planar Placement and Packing of ∆-regular Caterpillars Carla Binucci1, Emilio Di Giacomo1, Michael Kaufmann2, Giuseppe Liotta1, and Alessandra Tappini1 1Dipartimento di Ingegneria, Universit`a degli Studi di Perugia, via G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Duranti 93, 06125, Perugia, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' {carla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='binucci, emilio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='digiacomo, giuseppe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='liotta, alessandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='tappini}@unipg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='it 2Wilhelm-Schickard Institut f¨ur Informatik, Universit¨at T¨ubingen, Sand 13, 72076, T¨ubingen, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' mk@informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='uni-tuebingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='de January 4, 2023 Abstract This paper studies a packing problem in the so-called beyond-planar setting, that is when the host graph is “almost-planar” in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Pre- cisely, we consider the case that the host graph is k-planar, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', it admits an embedding with at most k crossings per edge, and focus on families of ∆-regular caterpillars, that are caterpillars whose non-leaf vertices have the same degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We study the dependency of k from the number h of caterpillars that are packed, both in the case that these caterpillars are all isomorphic to one another (in which case the packing is called placement) and when they are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We give necessary and sufficient conditions for the placement of h ∆-regular caterpillars and sufficient conditions for the packing of a set of ∆1-, ∆2-, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ∆h-regular caterpillars such that the degree ∆i and the degree ∆j of the non-leaf vertices can differ from one caterpillar to another, for 1 ≤ i, j ≤ h, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 1 Introduction Graph packing is a classical problem in graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The original formu- lation requires to merge several smaller graphs into a larger graph, called the host graph, without creating multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' More precisely, graphs G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh with Gi = (Vi, Ei) should be combined to a new graph G = (V, E) by injec- tive mappings ηi : Vi → V so that V = V1 ∪ V2 ∪ · · · ∪ Vh and the images of 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='01226v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='CO] 3 Jan 2023 the edge sets Ei do not intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It has been often assumed that |Vi| = n for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' h, and thus the mappings ηi are bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Many combinatorial problems can be regarded as packing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For example, the Hamiltonian cycle problem for a graph G can be stated as the problem of packing an n-vertex cycle with the complement of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' When no restriction is imposed on the host graph, we say that the host graph is Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Some classical results in this setting are those by Bollob´as and Eldridge [6], Teo and Yap [26], Sauer and Spencer [25], while related famous conjectures are by Erd˝os and S´os from 1963 [10] and by Gy´arf´as from 1978 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Within this line of research, Wang and Sauer [27], and Mah´eo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [22] char- acterized triples of trees that admit a packing into Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Haler and Wang [17] extended this result to four copies of a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Further notable work on graph packing into Kn is by Hedetniemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [18], Wozniak and Wojda [28] and Aich- holzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A packing problem with identical copies of a graph is also called a placement problem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', [17, 27, 29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A tighter relation to graph drawing was established when researchers did not consider Kn to be the host graph, but required that the host graph is planar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The main question here is how to pack two trees of size n into a planar graph of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' After a long series of intermediate steps [11, 12, 13, 14, 24] where the class of trees that could be packed has been gradually generalized, Geyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [15] showed that any two non-star trees can be embedded into a planar graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Relaxing the planarity condition allows for packing of more (than two) trees, and restricting the number of crossings for each edge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', in the so-called beyond-planar setting [9, 19, 21], still keeps the host graph sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The study of the packing problem in the beyond planarity setting was started by De Luca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [8], who consider how to pack caterpillars, paths, and cycles into 1-planar graphs (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', [21] for a survey and references on 1-planarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' While two trees can always be packed into a planar graph, it may not be possible to pack three trees into a 1-planar graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In this work we further generalize the problem by allowing the host graph to be k-planar for any k ≥ 1, and we study the dependency of k on the number of caterpillars to be packed and on their vertex degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We consider ∆-regular caterpillars, which are caterpillars whose non-leaf vertices all have the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Our results can be briefly outlined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We consider the packing problem of h copies of the same ∆-regular cater- pillar into a k-planar graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We characterize those families of h ∆-regular caterpillars which admit a placement into a k-planar graph and show that k ∈ O(∆h + h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We extend the study from the placement problem to the packing problem by considering a set of ∆1-, ∆2-, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ∆h-regular caterpillars such that the degree ∆i and the degree ∆j of the non-leaf vertices can differ from one caterpillar to another, with 1 ≤ i, j ≤ h, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By extending the tech- niques of the bullet above, we give sufficient conditions for the existence of a k-planar packing of these caterpillars and show that k ∈ O(∆h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 2 Finally, we prove a general lower bound on k and show that this lower bound can be increased for small values of h and for caterpillars that are not ∆-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Preliminaries are in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The placement of h ∆-regular caterpillars into a k-planar graph is discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Section 4 is devoted to k-planar h-packing, while Section 5 gives lower bounds on the value of k as a function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Concluding remarks and open problems can be found in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 2 Preliminaries We assume familiarity with basic graph drawing and graph theory terminology (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', [5, 20, 23]) and recall here only those concepts and notation that will be used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Given a graph G, we denote by degG(v) the degree of a vertex v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh be h graphs, all having n vertices, an h-packing of G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh is an n-vertex graph G that contains G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh as edge-disjoint spanning subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We also say that G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh can be packed into G and that G is the host graph of G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' An h-packing of h graphs into a host graph G such that the h graphs are all isomorphic to a graph H, is called an h-placement of H into G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We also say that G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Gh can be placed into G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The following property establishes a necessary condition for the existence of an h-packing into any host graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A packing of h connected n-vertex graphs exists only if n ≥ 2h and degGi(v) ≤ n − h, for each i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , h} and for each vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Each Gi has at least n−1 edges (because it is connected);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' thus, if n < 2h the h graphs have more edges in total than the number of edges of any graph with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' But since graphs Gi must be edge-disjoint subgraphs of G, the number of edges of G must be at least the total number of edges of the graphs Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since degGi(v) ≥ 1 for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , h} and for each v (because each Gi is connected) and since �h i=1 degGi(v) ≤ n − 1 (because G cannot have vertex-degree larger that n − 1), it holds that degGi(v) ≤ n − h, for each i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , h} and for each vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A k-planar graph is a graph that admits a drawing in the plane such that each edge is crossed at most k times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If the host graph of an h-packing (h-placement) is k-planar, we will talk about a k-planar h-packing (k-planar h-placement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Sometimes, we shall simply say k-planar packing or k-planar placement, when the value of h is clear from the context or not relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A caterpillar is a tree such that removing all leaves we are left with a path, called spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A caterpillar T is ∆-regular, for ∆ ≥ 2, if degT (v) = ∆ for every vertex v of the spine of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The number of vertices of a ∆-regular caterpillar is n = σ(∆ − 1) + 2 for some positive integer σ, which is the number of vertices of the spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 3 3 h-placement of ∆-regular Caterpillars into k- planar Graphs Given h copies of a same ∆-regular caterpillar, we want to study under which conditions they admit a placement into a k-planar graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We start by showing that the necessary condition stated in Property 1 is, in general, not sufficient to guarantee a placement even for ∆-regular caterpillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For every h ≥ 2, let ∆ be a positive integer such that h−1 ∆−1 is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A set of h ∆-regular caterpillars with n = 2h vertices does not admit a placement into any graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since each caterpillar has n − 1 edges and the number of caterpillars is h = n 2 , the total number of edges is n(n−1) 2 and thus, if a placement exists, the host graph can only be Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now prove that this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Denote by C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch the h caterpillars and suppose that a packing into Kn exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let v be a vertex of Kn and let v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , vh be the h vertices that are mapped to v, with vi being a vertex of Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Each vertex vi has degree in Ci that is either ∆ or 1 (because each Ci is ∆-regular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Denote by c the number of vertices among v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , vh that have degree ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the degree of v in the packing is c∆ + (h − c) and since the degree of v in Kn is n − 1, it must be c∆ + (h − c) = n − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', c∆ + (h − c) = 2h − 1, which can be rewritten as c = h−1 ∆−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' But this is not possible because c is integer, while h−1 ∆−1 is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In the rest of this section we shall establish necessary and sufficient condi- tions that characterize when a set of h isomorphic ∆-regular caterpillars admit a k-planar h-placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Concerning the sufficiency, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='1 we describe a constructive argument that computes a set of so-called zig-zag drawings and study the properties of such drawings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='2, we complete the charac- terization by also giving necessary conditions for an h-placement of ∆-regular caterpillars into a k-planar graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' in the same section, we give an upper bound on k as a function of h and ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We recall that a ∆-regular caterpillar has a number of vertices n that is equal to σ(∆ − 1) + 2 for some natural number σ, which is the number of vertices of the spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' While ∆-regular caterpillars are defined for any value of σ ≥ 1, when we want to pack a set of h ≥ 2 caterpillars, Property 1 requires that each caterpillar has at least two spine vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', that σ ≥ 2 for each caterpillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Otherwise, the unique spine vertex would have degree n − 1 and Property 1 would not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='1 Zig-zag Drawings of ∆-regular caterpillars Let C be a ∆-regular caterpillar with n vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we construct a drawing Γ of C as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The number of vertices of the spine of C is σ = n−2 ∆−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' consider a set of σ points on a circle γ and denote by u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , uσ these points according to the circular clockwise order they appear along γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Draw the spine 4 u1 u2 u3 u4 u5 (a) v17 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 upper part lower part hole short edge (b) v17 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 (c) Figure 1: (a) A zig-zag drawing of a 4-regular caterpillar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (b) the upper and the lower part are highlighted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' c) a 2-packing obtained by the drawing of (b) with a copy of it rotated by one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' of C by connecting, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ⌊ σ 2 ⌋, the points ui and ui+1 to the point uσ−i+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If σ is even and i = σ 2 , the points ui+1 and uσ−i+1 coincide and therefore the point u σ 2 is connected only to u σ 2 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that all points ui have two incident edges, except u1 and u⌊ σ 2 ⌋+1 which have only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We add the leaves adjacent to each vertex ui ̸∈ {u1, u⌊ σ 2 ⌋+1} by connecting uσ−i+1 to ∆−2 points between ui and ui+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we then add the leaves adjacent to u1 by connecting it to ∆−1 points between uσ and u1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we finally add the leaves adjacent to u⌊ σ 2 ⌋+1 by connecting it to ∆ − 1 points between u σ 2 and u σ 2 +1 if σ is even, or to ∆−1 points between u⌊ σ 2 ⌋+1 and u⌊ σ 2 ⌋+2 if σ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The resulting drawing is called a zig-zag drawing of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' From now on, we assume that in a zig-zag drawing the points that represent vertices are equally spaced on the circle γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let χ be the convex hull of the points representing the vertices of C in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A zig-zag drawing has exactly two sides of χ that coincide with two edges of C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we call these two edges short edges of Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' each other side of χ is called a hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Denote by v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , vn the vertices of Γ according to the circular clockwise order they appear along χ with v1 ≡ u1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that (v1, vn) is a short edge and vn is the degree-1 vertex of this edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Consider a straight line s that intersects both short edges of Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' line s inter- sects all the edges of the zig-zag drawing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Without loss of generality, assume that s is horizontal and denote by U the set of vertices that are above s and by L the set of vertices that are below s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The vertices in U form the upper part of Γ and those in L form the lower part of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Without loss of generality assume that v1 is in the upper part (and therefore vn is in the lower part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It follows that each edge has the end-vertex with lower index in the upper part, and the end-vertex with higher index in the lower part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence the short edge different from (v1, vn), which we denote as (vr−1, vr), is such that vr−1 is in the upper part and vr is in the lower part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The first vertex of the upper part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', vertex v1, is called starting point of Γ, while the first vertex of the lower part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', 5 vertex vr, is called ending point of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We observe that r = n 2 + 1 if the number of vertices of the spine σ = n−2 ∆−1 is even, while r = 1 + n−(∆−1) 2 if σ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' This can be written with a single formula as r = 1+ n−(∆−1)(σ mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The two short edges separate two sets of consecutive holes, one completely contained in the upper part and one completely contained in the lower part;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' if σ is even, these two sets have the same number of holes equal to n−2 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' if σ is odd, then one of the two sets has n−∆−1 2 holes, while the other has n+∆−3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Note that the smaller set is in the upper part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let ℓ be a positive integer and let Γ′ be the drawing obtained by re-mapping vertex vi to the point1 representing vi+ℓ in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We say that Γ′ is the drawing obtained by rotating Γ by ℓ steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Note that the starting point of Γ′ is vj with j = 1 + ℓ and the ending point is vr with r = 1 + ℓ + n−(∆−1)(σ mod 2) 2 = j + n−(∆−1)(σ mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The drawing in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 1(c) is the union of two zig-zag drawings Γ1 and Γ2, where Γ2 is obtained by rotating Γ1 by one step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the starting point of Γ1 is v1 while its ending point is v8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the starting point of Γ2 is v2, while its ending point is v9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let Γ1 be a zig-zag drawing of a ∆-regular caterpillar C with starting point j1 and ending point r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' let Γ2 be a zig-zag drawing of C with starting point j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If 0 < j2 − j1 < n−(∆−1)(σ mod 2) 2 , where σ is the number of spine vertices of C, then Γ1 ∪ Γ2 has no multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We first observe that Γ2 is obtained by rotating Γ1 by ℓ steps, where ℓ = j2 − j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Suppose that a multiple edge (vi, vg), with i < g exists in Γ1 ∪ Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' This implies that in the drawing Γ1 there must be an edge (vi′, vg′) that, when rotated by ℓ steps, coincides with (vi, vg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In other words, the two edges (vi, vg) and (vi′, vg′) must be such that: (i) i′ < i < r1 ≤ g < g′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (ii) g = i′ + ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (iii) the number α of vertices encountered between vi and vg when going clockwise from vi to vg is the same as the number of vertices encountered when going clockwise from vg′ to vi′ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Denote by β the number of vertices encountered when going clockwise from vi′ to vi, and by ζ the number of vertices encountered when going clockwise from vg to vg′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We have 2α + β + ζ + 4 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If σ is even, then β = ζ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 2(a)), which implies α + β + 2 = n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that g = i′ + ℓ implies that ℓ = β + α + 2 (ℓ is equal to the number of vertices encountered clockwise between vi′ and vg plus one) and therefore (vi′, vg′) can coincide with (vi, vg) after a rotation of ℓ steps only if ℓ = n 2 but, when σ is even, we have ℓ = j2 − j1 < n 2 and therefore a multiple edge cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If σ is odd, then β − (∆ − 1) ≤ ζ ≤ β + (∆ − 1) (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 2(b) and 2(c)) and therefore 2α + 2β − (∆ − 1) + 4 ≤ 2α + β + ζ + 4 = n ≤ 2α + 2β + (∆ − 1) + 4, which can be rewritten as n−(∆−1) 2 ≤ α + β + 2 ≤ n+(∆−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It follows that, in order to have (vi′, vg′) and (vi, vg) coincident after a rotation of ℓ steps, the value of ℓ must be such that n−(∆−1) 2 ≤ ℓ ≤ n+(∆−1) 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' but, when σ is odd, we have ℓ = j2 − j1 < n−(∆−1) 2 and therefore a multiple edge cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 1In a drawing in convex position the indices of the vertices are taken modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 6 vj1 vr1 vi′ vh′ vi vh β ζ α α ℓ (a) vj1≡vi′ vr1 vh′ vi vh β ζ α α ℓ (b) vj1 vr1 vi′ vh′ vi vh β ζ α α ℓ (c) Figure 2: Illustration for the proof of Lemma 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (a) σ even;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (b)-(c) σ odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We conclude this section by computing the maximum number of crossings per edge in the union of two zig-zag drawings without overlapping edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We state this lemma in general terms assuming that the two ∆-regular caterpillars can have different vertex degrees, as we are going to use the lemma to establish upper bounds on k both for k-planar h-placements and for k-planar h-packings (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let Γ be a union of a set of zig-zag drawings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To ease the description that follows, we regard Γ as a sub-drawing of a straight-line drawing of Kn whose vertices coincide with those of Γ (and therefore are equally spaced along a circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In particular, for each vertex vj, we denote by ej,0, ej,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ej,n−2 the edges incident to vj in Kn according to the circular counterclockwise order around vj starting from ej,0 = (vj, vj−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Each of the zig-zag drawings that form Γ contains a subset of these edges and Γ is a valid packing if there is no edge that belongs to two different zig-zag drawings in the set whose union is Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We denote by Sn the (circular) sequence of slopes si = i · π n, for i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that, without loss of generality, we can assume that the convex hull of Γ has a side with slope s0 and, as a consequence, every edge of Γ has a slope in the set Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let vj be a vertex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' if the slope of ej,0 is sij, then the slope of ej,p is sij+p (with indices taken modulo n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' in other words, the edges incident to each vertex have slopes that form a sub-sequence of n − 1 consecutive elements of Sn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we denote such a sequence as ψ(ij), where ij indicates that the first element of ψ(ij) is sij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We say that vj uses the sequence ψ(ij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If we consider two different vertices vj and vj+p and vj uses the sequence ψ(ij), then vj+p uses the sequence ψ(ij − 2p) (with indices taken modulo n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' in other words, the sequence used by a vertex shifts clockwise by two elements moving to the next vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1 be an n-vertex ∆1-regular caterpillar and let C2 be an n- vertex ∆2-regular caterpillar with ∆i ≤ n − 2 (for i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let Γ1 be a zig-zag drawing of C1 with starting point vj1 and let Γ2 be a zig-zag drawing of C2 with starting point vj2 with 0 < j2 − j1 < n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If Γ1 ∪ Γ2 has no multiple edges, then any edge of Γ1 ∪ Γ2 is crossed at most 2(∆1 + ∆2) + 4(j2 − j1) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 7 s0 s1 sn−1 sn−2 vj ej,0 ej,n−2 ej+1,0 vj+1 ψ(j) ψ(j + 1) Sn ej+1,n−2 ≡ Figure 3: Illustration for the definition of slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We first observe that the edges of a zig-zag drawing of a ∆-regular cater- pillar are all drawn as segments whose slope belongs to a set of ∆ slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In particular, for every spine vertex v, the edges incident to v are drawn using all these ∆ slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Consider the starting vertex vj1 of Γ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the edges incident to vj1 are drawn with the first ∆1 slopes of ψ(ij1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Analogously, the edges incident to the starting vertex vj2 of Γ2 are drawn with the first ∆2 slopes of ψ(ij2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The sequence ψ(ij2) is shifted clockwise by 2(j2−j1) units with respect to ψ(ij1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, since j2−j1 < n 2 , the first slope of ψ(ij2) is distinct from the first slope of ψ(ij1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let e = (vi, vg) be an edge of Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now prove that the number of crossings along e is at most the one given in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let e1 = (vi, va) be the edge of Γ2 incident to vi that forms the smallest angle with e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' analogously, let e2 = (vg, vb) be the edge of Γ2 incident to vg that forms the smallest angle with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that, in principle there are four possible clockwise orders of vi, va, vg, and vb (see cases (a)–(d) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 4 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' However the case (b) cannot happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Namely, in case (b) the slopes used to draw the edges of Γ2 would be shifted counterclockwise with respect to those used to represent the edges of Γ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' but, as observed above, the slopes used by Γ2 are shifted clockwise with respect to those used by Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let α1 be the angle between e and e1 and let α2 be the angle between e and e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let V1 be the set of vertices seen by the angle α1 including va and excluding vg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' analogously let V2 be the set of vertices seen by the angle α2 including vb and excluding vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In each of the three cases (a), (c), and (d), at least one of α1 and α2 is such that e sweeps the angle moving clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let αl with l ∈ {1, 2} be the angle that satisfies this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In particular, for case (a) αl can be both α1 or α2, in case (c) αl is α2 and in case (d) αl is α1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Every edge that crosses e has an end-vertex in V1 and one end-vertex in V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To count the number of such edges (and therefore the number of crossings along e), we evaluate |Vl|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The value of |Vl| is at most the number of slopes of Sn that are encountered in counterclockwise order between the slope s ∈ Sn of el and the slope s′ ∈ Sn of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In particular, in case (a) |Vl| is exactly this number, while in case (c) and (d) |Vl| is less than this number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The slope s′ is at most the last 8 vi vg va vb α1 α2 V1 V2 (a) vi vg va vb α1 α2 (b) vi vg va vb α1 α2 V2 (c) vi vg va vb α1 α2 V1 (d) Figure 4: Illustration for the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The edges of Γ2 are dashed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 9 slope used by Γ1, which is sp with p = j1 + ∆1, while the slope s is at least the first slope used by Γ2, which is sq with q = j1 − 2(j2 − j1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Thus, the number of slopes between s′ (included) and s (excluded) is at most p−q = ∆1 +2(j2 −j1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence |Vl| ≤ ∆1 + 2(j2 − j1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We call a block a subset of consecutive vertices of Vl starting with a spine vertex and containing all the leaves that follow that spine vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The number of edges of Γ2 incident to the vertices of a block is 2(∆2 − 1) (since ∆2 edges are incident to the spine vertex and ∆2 − 2 is the number of leaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The number of blocks in Vl is � |Vl| (∆2−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It follows that the number of crossings χe along e is at most � |Vl| (∆2−1) � 2(∆2 − 1) which is less than 2(|Vl| + ∆2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since |Vl| ≤ ∆1 + 2(j2 − j1), we have χe ≤ 2(∆1 + ∆2) + 4(j2 − j1), which concludes the proof in the case when e belongs to Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The case when the edge e belongs to Γ2 is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In particular, when e belongs to Γ2, the cases (b), (c), and (d) apply, while case (a) does not happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='2 Characterization We are now ready to characterize the ∆-regular caterpillars that admit an h- placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C be a ∆-regular caterpillar with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' An h-placement of C exists if and only if: (i) ∆ ≤ n − h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' and (ii) n ≥ 2h + (∆ − 1) · (σ mod 2), where σ is the number of spine vertices of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Further, if an h-placement exists, there exists one that is k-planar for k ∈ O(∆h + h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We first prove the sufficient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch be the h cater- pillars and assume that n ≥ 2h+(∆−1)(σ mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We compute an h-placement of C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch starting from a zig-zag drawing Γ1 of C1 and obtaining the drawing Γi of Ci by rotating Γ1 by i − 1 steps, for i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that, when the number of spine vertices σ of each Ci is even, h ≤ n 2 and therefore each Γi is rotated by less than n 2 steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' when σ is odd h ≤ n−(∆−1) 2 and each Γi is rotated by less than n−(∆−1) 2 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In both cases, each pair of drawings Γi and Γj satisfies the conditions of Lemma 1 and therefore there are no multiple edges, that is, the union of all Γi is a valid h-placement of C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now prove the necessary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If σ is even, then conditions (i) and (ii) are necessary by Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence, consider the case when σ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Condition (i) is necessary by Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Assume, by contradiction, that (ii) is not necessary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', there exists an h-placement of h caterpillars C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch such that n < 2h + (∆ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch admit an h-placement, by Property 1 n must be at least 2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Thus, it would be 2h ≤ n < 2h + (∆ − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' in other words, n = 2h + α with 0 ≤ α ≤ ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let G be the host graph of the h-placement and let v be the vertex of G to which the largest number of spine vertices of C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch is mapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let β be the number of spine vertices that are mapped to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' There are other h − β 10 leaf vertices that are mapped to v (because one vertex per caterpillar has to be mapped on each vertex of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The degree of v in G is at most n − 1 and each of the spine vertices mapped to v has degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence, the β spine vertices mapped to v have degree β∆ in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Vertex v can have at most other n−1−β∆ edges and therefore it must be n − 1 − β∆ ≥ h − β, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', β ≤ n−1−h ∆−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, there are σh spine vertices in total and, since G has n vertices, there are at least ⌈ σh n ⌉ spine vertices mapped to v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', β ≥ ⌈ σh n ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Putting together the two conditions on β we obtain: �σh n � ≤ β ≤ n − 1 − h ∆ − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since n = 2h + α, we have h = n−α 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing h in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='2, we obtain: �σ 2 − σα 2n � ≤ β ≤ n + α − 2 2(∆ − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since n = σ(∆ − 1) + 2, we have: �σ 2 − σα 2(σ(∆ − 1) + 2) � ≤ β ≤ σ(∆ − 1) + α 2(∆ − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (1) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (1) implies that: �σ 2 − α 2(∆ − 1) + 4 σ � ≤ σ 2 + α 2(∆ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (2) We now prove that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (2) cannot be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since σ is odd, it is σ = 2i+1 for some i ∈ N, and thus: � i + 1 2 − ζ � ≤ k + 1 2 + ζ′, (3) with ζ = α 2(∆−1)+ 4 σ and ζ′ = α 2(∆−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We have ζ < ζ′ and we prove that ζ′ < 1 2: ζ′ = α 2(∆ − 1) ≤ ∆ − 2 2(∆ − 1) < ∆ − 1 2(∆ − 1) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (3) is i + 1 because 0 < 1 2 − ζ < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the second term is less than i + 1 because 0 < 1 2 + ζ′ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It follows that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (3) does not hold and therefore Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (2) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now prove the bound on the number of crossings along an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We consider an edge of Γ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the number of crossings along an edge of the drawing of another caterpillar is bounded by the same number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let e be an edge of the drawing Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By Lemma 2, the number of crossings χe along e due to the edges of another drawing Γl (with 2 ≤ l ≤ h) is at most 2(∆1+∆l)+4(jl−j1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Summing 11 u1 u2 u3 u4 u5 (a) v17 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 (b) Figure 5: (a) An inner zig-zag drawing and (d) an outer zig-zag drawing of a 4-regular caterpillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' over all drawings distinct from Γ1, we obtain χe ≤ �h l=2(2(∆1+∆l)+4(jl−j1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Considering that ∆l = ∆ for every l and that jl − j1 = l − 1, we have χe ≤ h � l=2 (4∆ + 4(l − 1)) ≤ (4∆ − 2)h + 2h2 − 4∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (4) We conclude by observing that the number of crossings given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (4) can be reduced, although not asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A zig-zag drawing can be embedded inside the circle (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 5(a)) or outside the circle (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Thus, the number given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (4) can be halved by embedding half of the caterpillars inside the circle and the other half outside the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 4 h-packing of ∆-regular Caterpillars in k-planar Graphs In this section we study h-packings of h ∆1-, ∆2-, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ∆h-regular caterpillars such that the degree ∆i and the degree ∆j of the spine vertices can differ from one caterpillar to another, for 1 ≤ i, j ≤ h, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1 be an n-vertex ∆1-regular caterpillar and let C2 be an n- vertex ∆2-regular caterpillar such that ∆1 > ∆2 and ∆1 ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let Γ1 be a zig-zag drawing of C1 with starting point vj1 and ending point vr1, and let Γ2 be a zig-zag drawing of C2 with starting point vj2 and ending point vr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If ∆2 2 ≤ j2 − j1 < n−(∆1−1) 2 , then Γ1 ∪ Γ2 has no multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' As described in the proof of Lemma 2, the edges of a zig-zag drawing of a ∆-regular caterpillar are all drawn as segments whose slope belongs to a set of ∆ slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In particular, for every spine vertex v, the edges incident to v are drawn using all these ∆ slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Based on this observation, we show that the ∆1 slopes used to represent the edges of Γ1 are distinct from the ∆2 slopes used to represent the edges of Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We use the same notation used in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Consider the staring vertex vj1 of Γ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the edges incident to vj1 are drawn with the first ∆1 slopes of ψ(ij1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Analogously, the edges incident to the starting vertex vj2 of Γ2 are drawn with the first ∆2 slopes of ψ(ij2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since j2 −j1 ≥ ∆2 2 , the sequence ψ(ij2) is shifted clockwise by ∆2 units with respect to ψ(ij1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, since j2 − j1 ≤ n−(∆1−1) 2 , the sequence of the first ∆2 slopes of ψ(ij2) does not overlap with the first ∆1 slopes of ψ(ij1), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch be h caterpillars such that Ci is ∆i-regular, for 1 ≤ i ≤ h, and ∆h ≤ ∆h−1 ≤ · · · ≤ ∆1 ≤ n − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If �h i=1 ∆i ≤ n − 1 and �h i=2 � ∆i 2 � < n−(∆1−1) 2 , then there exists a k-planar packing with k ∈ O(∆1h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We compute a zig-zag drawing of C1 with starting point vj1, with j1 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' for each Ci, with 2 ≤ i ≤ h, we compute a zig-zag drawing Γi with starting vertex vji where ji = ji−1 + � ∆i 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that, each vertex v of Γ1 ∪Γ2 ∪· · ·∪Γh has degree at most n−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' namely �h i=1 degCi(v) ≤ �h i=1 ∆i ≤ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Moreover, given two caterpillars Ci and Ci′ with 1 ≤ i < i′ ≤ h, we have that: (i) ji′ − ji ≥ ji′ − ji′−1 = � ∆i′ 2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' and (ii) ji′ − ji ≤ jh − j1 = �h i=2⌈ ∆i 2 ⌉, which gives ji′ − ji < n−(∆1−1) 2 < n−(∆i−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Putting together (i) and (ii), we obtain ∆i′ 2 < ji′ − ji < n−(∆i−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence, Lemma 3 holds for every pair of caterpillars and the union of all the zig-zag drawings Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Γh is a valid packing of C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now prove the bound on the number of crossings along an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We consider an edge of Γ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the number of crossings along an edge of another drawing is bounded by the same number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let e be an edge of the drawing Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By Lemma 2, the number of crossings χe along e due to the edges of another drawing Γl (with 2 ≤ l ≤ h) is at most 2(∆1 + ∆l) + 4(jl − j1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Summing over all drawings distinct form Γ1, we obtain χe ≤ �h l=2(2(∆1 + ∆l) + 4(jl − j1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Considering that jl ≥ jl−1 + � ∆i 2 � , we obtain that jl − j1 = �l i=2 � ∆i 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since ∆l ≤ ∆1 for every 2 ≤ l ≤ h, we have jl − j1 ≤ (l − 1)( ∆1 2 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Therefore, we obtain χe ≤ �h l=2(4∆1 + 4(l − 1)( ∆1 2 + 1)) ≤ (∆1 + 2)h2 + 4∆1(h − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now consider a special case of packing a set of h ∆1-, ∆2-, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ∆h- regular caterpillars where, for each ∆i (1 ≤ i ≤ h), we have that ∆i − 1 is a multiple of ∆i+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In this case, we show that the sufficient conditions of Theorem 3 can be relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For example, consider the packing of a 17-regular caterpillar and two 9-regular caterpillars, each having 34 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' These three caterpillars do not satisfy the sufficient condition of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' However, a k- planar packing of these caterpillars is possible, as proven in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We start 13 vj vi vr vg upper part lower part c = 0 c = 1 c = 2 d = 0 d = 1 d = 2 Figure 6: Illustration for Property 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' σ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For each spine vertex, c and d are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Considering adjacent spine vertices, the sum of c and d is 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with the following property, which immediately follows from the construction of a zig-zag drawing (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 6 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let Γ be a zig-zag drawing of a ∆-regular caterpillar with starting vertex vj and ending vertex vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If vi is a spine vertex in the upper part of Γ, then i = j + c(∆ − 1) for some c ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' if vg is a spine vertex in the lower part of Γ, then g = r + d(∆ − 1) for some d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Moreover, if vi and vg are adjacent then either c + d = � σ 2 � − 1 or c + d = � σ 2 � , where σ is the number of spine vertices of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Property 2 is extensively used in the proof of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1 be an n-vertex ∆1-regular caterpillar and let C2 be an n- vertex ∆2-regular caterpillar such that ∆1−1 = q(∆2−1), for some q ∈ N+ and ∆i ≤ n − 2 (for i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let Γ1 be a zig-zag drawing of C1 with starting point vj1 and ending point vr1, and let Γ2 be a zig-zag drawing of C2 with starting point vj2 and ending point vr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If 0 < j2 − j1 < n−(∆1−1) 2 , then Γ1 ∪ Γ2 has no multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let (vi1, vg1) be an edge of Γ1 and (vi2, vg2) be an edge of Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Assume that vi1 belongs to the upper part of Γ1 and vi2 belongs to the upper part of Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Note that this implies that vg1 belongs to the lower part of Γ1 and vg2 belongs to the lower part of Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that (vi1, vg1) and (vi2, vg2) do not coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We first show that it does not happen that vi1 coincides with vi2 and vg1 coincides with vg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We then show that it does not happen that vi1 coincides with vg2 and vg1 coincides with vi2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In the rest of the proof we will express the four indices i1, i2, g1 and g2 in terms of the values j1, j2, r1 and r2, according to Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Without loss of generality, we can assume that r2 ≤ n and j1 ≥ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', that the vertices vr2, vn, v1, and vj1 appear in this clockwise order, with vr2 and vn possibly coincident and with v1 and vj1 possibly coincident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' With these assumptions, we have j1 < j2 < r1 < r2 and vi1 can coincide with vg2 14 only if i1 = g2 − n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', only if the value of g2 is greater than n and coincides with i1 modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Thus, while assuming that vi1 coincides with vi2 implies that i1 = i2, assuming that vi1 coincides with vg2 implies that i1 = g2 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 1: It does not happen that vi1 coincides with vi2 and vg1 coincides with vg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' At least one vertex per edge is a spine vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We distinguish four sub-cases depending on which vertex is a spine vertex for each edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since all the cases are very similar, we give here only the first case and the others can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='a: vi1 and vi2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By Property 2 we have, for some c1, c2 ∈ N: i1 = j1 + c1(∆1 − 1) = j1 + qc1(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (5) and i2 = j2 + c2(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (6) If vi1 coincides with vi2, we have i1 = i2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (6) we obtain: j2 − j1 = (qc1 − c2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (7) Concerning vg1 and vg2, we have: gm 1 ≤ g1 ≤ gM 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' gm 2 ≤ g2 ≤ gM 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with gm l = rl + dl(∆l − 1), gM l = rl + (dl + 1)(∆l − 1) for some dl ∈ N such that cl + dl = � σl 2 � − 1, where σl is the number of spine vertices of Cl, for l = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that gM 1 < gm 2 , which implies g1 ̸= g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have gM 1 < gm 2 it must be: r1 + (d1 + 1)(∆1 − 1) < r2 + d2(∆2 − 1) r1 + q(d1 + 1)(∆2 − 1) < r2 + d2(∆2 − 1) r2 − r1 > (qd1 + q − d2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (8) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (8) can be rewritten as: j2 − j1 > � (qd1 + q − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 � (∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (9) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (9) we obtain: qc1 − c2 > (qd1 + q − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qc1 − c2 > qσ1 2 + q(σ1 mod 2) 2 − qc1 − q + q − σ2 2 − 1(σ2 mod 2) 2 + c2+ + 1 + σ2 mod 2 2 − q(σ1 mod 2) 2 15 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: qc1 − c2 > 1 2 (10) In summary, to have gM 1 < gm 2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (10) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (7) and from the hypothesis that j2−j1 > 0 we obtain (qc1−c2)(∆2−1) > 0 which, since (∆2 − 1) > 0, implies qc1 − c2 > 0 and, since qc1 − c2 is integer, can be rewritten as qc1 − c2 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' This implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (10) holds and therefore that gM 1 < gm 2 and g1 ̸= g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 2: It does not happen that vi1 coincides with vg2 and vg1 coincides with vi2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Also in this case we distinguish four sub-cases depending on which vertex is a spine vertex for each edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' As in Case 1, we give here only the first sub-case, while the others can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='a: vi1 and vi2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since vg2 is a vertex in the lower part of Γ2, it must be g2 = r2 + d2(∆2 − 1) + α2, for some α2 such that 0 ≤ α2 < ∆2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If vg2 coincides with vi1, as explained above, it must be i1 = g2 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Combining the expression of g2 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (5) we obtain: r2 − j1 = (qc1 − d2)(∆2 − 1) − α2 + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (11) Concerning vg1, we have: gm 1 ≤ g1 ≤ gM 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with gm 1 = r1 + d1(∆1 − 1), gM 1 = r1 + (d1 + 1)(∆1 − 1) for some d1 ∈ N such that c1 + d1 = � σ1 2 � − 1, where σ1 is the number of spine vertices of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that i2 < gm 1 , which implies i2 ̸= g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have i2 < gm 1 it must be: j2 + c2(∆2 − 1) < r1 + d1(∆1 − 1) j2 + c2(∆2 − 1) < r1 + qd1(∆2 − 1) j2 − r1 < (qd1 − c2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (12) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (11) can be rewritten as: j2 − j1 = (qc1 − d2)(∆2 − 1) − α2 + n 2 + (∆2 − 1)(σ2 mod 2) 2 , (13) while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (12) can be rewritten as: j2 − j1 < � (qd1 − c2) − q(σ1 mod 2) 2 � (∆2 − 1) + n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (14) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (13) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (14) we obtain: qc1 − d2 < (qd1 − c2) − (σ2 mod 2) 2 − q(σ1 mod 2) 2 + α2 ∆2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 16 Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qc1 − σ2 2 − σ2 mod 2 2 + c2 + 1 < qσ1 2 + q(σ1 mod 2) 2 − qc1 − q − c2− − σ2 mod 2 2 − q(σ1 mod 2) 2 + α2 ∆2 − 1 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: qc1 − qσ1 2 + c2 < −q + 1 2 + α2 2(∆2 − 1) (15) In summary, to have iM 2 < gm 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (15) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (13) and from the hypothesis that j2 − j1 < n−(∆1−1) 2 = n−q(∆2−1) 2 we obtain: qc1 − d2 + 1 2(σ2 mod 2) < −q 2 + α2 ∆2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Replacing again d2 with σ2+1(σ2 mod 2) 2 − c2 − 1, we obtain: qc1 − qσ1 2 + c2 < −q + 2 2 + α2 ∆2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (16) We have that − q 2 − 1 + α2 ∆2−1 < − q 2 − 1 2 + α2 2(∆2−1), since α2 2(∆2−1) < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In other words, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (16) implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (15) holds and therefore that i2 < gm 1 and i2 ̸= g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch be h caterpillars such that Ci is ∆i-regular, ∆i − 1 is a multiple of ∆i+1 − 1, with 1 ≤ i < h, and ∆i ≤ n − h (for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If n ≥ 2h + (∆1 − 1), then there exists a k-planar packing with k ∈ O(∆1h + h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For each Ci, with 1 ≤ i ≤ h, we compute a zig-zag drawing Γi with starting vertex vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that, given two caterpillars Cj1 and Cj2 with 1 ≤ j1 < j2 ≤ h, we have that ∆j1 − 1 is a multiple of ∆j2 − 1, and the zig-zag drawings Γj1 and Γj2 have starting vertices vj1 and vj2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence, 0 < j2 − j1 < h and by hypothesis h ≤ n−(∆1−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hence, Lemma 4 holds for every pair of caterpillars and the union of all zig-zag drawings Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Γh is a valid packing of C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The proof of the bound on the number of crossings along an edge is the same as the one of Theorem 2, considering that ∆l ��� ∆1 and that jl − j1 = l − 1 for every 2 ≤ l ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 17 5 Lower bounds In this section we first give a general lower bound on the value of k for k-planar h-packings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we then increase this lower bound for some small values of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Every k-planar h-packing of h graphs with n vertices and m edges is such that k ≥ h2m2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='6n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The number of edges of a k-planar graph with n vertices is at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='81 √ k ·n, for k ≥ 2 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since the h graphs have h·m edges in total, a k-planar packing of these graphs can exist only if h ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='81 √ k n m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', if k ≥ h2m2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='6n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since for a tree m = n − 1, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Every k-planar h-packing of h trees is such that k ≥ h2 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We now refine the lower bound above for small values of h in an h-placement of caterpillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Specifically we show that for values of h equal to 3, 4, and 5 the corresponding lower bounds are 2, 3, and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Note that for all these cases the lower bound implied by Corollary 1 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For h = 3, 4 there exists a caterpillar C with at least h+7 vertices for which every k-planar h-placement of C is such that k ≥ h − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' For h = 5 there exists a caterpillar C with at least 24 vertices for which every k-planar 5-placement of C is such that k ≥ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case h = 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let n be an integer such that n ≥ h + 7, and let Cn,h be the n-vertex caterpillar shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that the vertex of Cn,h denoted as v in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 7 has degree n − h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we call it the center of Cn,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Consider any h-placement of Cn,h into a graph G and denote as vi the vertex of G which the center of Ci is mapped to (i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The vertices v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , vh must be distinct because, if two centers were mapped to the same vertex of G then this vertex would have degree larger than n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Namely, if two centers are mapped to the same vertex, this vertex has degree 2n − 2h which is larger than n − 1 if n > 2h − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', if h + 7 > 2h − 1, which is true for h < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since each vi (1 ≤ i ≤ h) has degree n − h in Ci and degree 1 in each of the h − 1 other caterpillars, its degree in G is n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Thus, G contains Kh,n−h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Thus, for h = 3, G contains K3,7 (n ≥ 10 in this case), which is not 1-planar [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' for h = 4, G contains K4,7 (n ≥ 11 in this case), which is not 2-planar [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The case h = 5 is analogous with K5,19, which is not 4-planar [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 6 Concluding Remarks and Open Problems This paper studied the placement and the packing of caterpillars into k-planar graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It proved necessary and sufficient conditions for the h-placement of ∆- regular caterpillars in a k-planar graph and sufficient conditions for the packing of a set of ∆1-, ∆2-, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , ∆h-regular caterpillars with k ∈ O(∆1h2) (∆1 is the 18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' v h + 2 h n − h − 2 Cn,h Figure 7: A caterpillar as described in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' maximum vertex degree in the set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' The work in this paper contributes to the rich literature concerning the placement and the packing problem in planar and non-planar host graphs and it specifically relates with a recent re-visitation of these questions in the beyond-planar context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Many open problems naturally arise from the research in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We conclude the paper by listing some of those that, in our opinion, are among the most interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Extend the characterization of Theorem 2 to the placement of caterpillars that are not ∆-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorems 4 and 3 give sufficient conditions for the k-planar packing of some families of caterpillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It would be interesting to give a complete characterization of the packability of these families into k-planar graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 6 improves the lower bound of Theorem 5 for caterpillars that are not ∆-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' It would be interesting to find a similar result with ∆-regular caterpillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Finally, we point out that one could investigate what graphs can be packed/placed into a k-planar graph for a given value of k, instead of studying how k varies with the number h and the vertex degree of the caterpillars that are packed/placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' While the interested reader can refer to [3] for results with k = 1, the following theorem gives a preliminary result for k = 2 (see the appendix for a proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (4) in the proof of Theorem 2 would give upper bounds in the range [86, 137] for the caterpillars considered by the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A ∆-regular caterpillar with 4 ≤ ∆ ≤ 7 admits a 2-planar 3- placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' References [1] Eyal Ackerman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On topological graphs with at most four crossings per edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Computational Geometry, 85:101574, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [2] Oswin Aichholzer, Thomas Hackl, Matias Korman, Marc van Kreveld, Maarten L¨offler, Alexander Pilz, Bettina Speckmann, and Emo 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', Keszthely, 1976), volume 1, pages 463–469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' North-Holland New York, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [17] Sean P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Haler and Hong Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Packing four copies of a tree into a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Australas.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Hedetniemi, Stephen Hedetniemi, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Slater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A note on packing two trees into Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Ars Combinatoria, 11, 01 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [19] Seok-Hee Hong and Takeshi Tokuyama, editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Beyond Planar Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='1007/978-981-15-6533-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [20] Michael Kaufmann and Dorothea Wagner, editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Drawing Graphs, Meth- ods and Models, volume 2025 of Lecture Notes in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Springer, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='1007/3-540-44969-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [21] Stephen 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European Workshop on Computational Geometry, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [25] Norbert Sauer and Joel Spencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Edge disjoint placement of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Theory, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' B, 25(3):295–302, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Teo and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Yap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Packing two graphs of order n having total size at most 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Graphs Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', 6(2):197–205, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [27] Hong Wang and Norbert Sauer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Packing three copies of a tree into a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' European Journal of Combinatorics, 14(2):137 – 142, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [28] Mariusz Wozniak and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Pawel Wojda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Triple placement of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Graphs and Combinatorics, 9(1):85–91, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' [29] Andrzej Zak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A note on k-placeable graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Discret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', 311(22):2634– 2636, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 21 A Missing cases for the proof of Lemma 4 Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='b: vg1 and vg2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By Property 2 we have, for some d1, d2 ∈ N: g1 = r1 + d1(∆1 − 1) = r1 + qd1(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (17) and g2 = r2 + d2(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (18) If vg1 coincides with vg2, we have g1 = g2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (17) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (18) we obtain: r2 − r1 = (qd1 − d2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (19) Concerning vi1 and vi2, we have: im 1 ≤ i1 ≤ iM 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' im 2 ≤ i2 ≤ iM 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with im l = il + cl(∆l − 1), iM l = il + (cl + 1)(∆l − 1) for some cl ∈ N such that cl + dl = � σl 2 � − 1, where σl is the number of spine vertices of Cl, for l = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that iM 1 < im 2 , which implies i1 ̸= i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have iM 1 < im 2 it must be: j1 + (c1 + 1)(∆1 − 1) < j2 + c2(∆2 − 1) j1 + q(c1 + 1)(∆2 − 1) < j2 + c2(∆2 − 1) j2 − j1 > (qc1 + q − c2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (20) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (19) can be rewritten as: j2 − j1 = � (qd1 − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 � (∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (21) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (21) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (20) we obtain: c2 − qc1 − q > −1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (22) In summary, to have iM 1 < im 2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (22) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (21) and from the hypothesis that j2 − j1 > 0 we obtain c2 − qc1 − q > −1 which, since c2 − qc1 − q is integer, can be rewritten as c2 − qc1 − q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' This implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (22) holds and therefore that iM 1 < im 2 and i1 ̸= i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='c: vi1 and vg2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By Property 2 we have, for some c1 ∈ N: i1 = j1 + c1(∆1 − 1) = j1 + qc1(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (23) We also have, for some c2 ∈ N and 0 ≤ α2 < ∆2 − 1: i2 = j2 + c2(∆2 − 1) + α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (24) 22 If vi1 coincides with vi2, we have i1 = i2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (23) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (24) we obtain: j2 − j1 = (qc1 − c2)(∆2 − 1) − α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (25) Concerning vg1, we have: gm 1 ≤ g1 ≤ gM 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with gm 1 = r1 + d1(∆1 − 1), gM 1 = r1 + (d1 + 1)(∆1 − 1) for some d1 ∈ N such that c1 + d1 = � σ1 2 � − 1, where σ1 is the number of spine vertices of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since vg2 is a vertex in the lower part of Γ2, it must be g2 = r2 + d2(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that gM 1 < g2, which implies g1 ̸= g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have gM 1 < g2 it must be: r1 + (d1 + 1)(∆1 − 1) < r2 + d2(∆2 − 1) r1 + q(d1 + 1)(∆2 − 1) < r2 + d2(∆2 − 1) r2 − r1 > (qd1 + q − d2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (26) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (26) can be rewritten as: j2 − j1 > � (qd1 + q − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 � (∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (27) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (25) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (27) we obtain: qc1 − c2 − α2 ∆2 − 1 > (qd1 + q − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qc1 − c2 − α2 ∆2 − 1 > qσ1 2 + q(σ1 mod 2) 2 − qc1 − q + q − σ2 2 − 1(σ2 mod 2) 2 + c2+ + 1 + σ2 mod 2 2 − q(σ1 mod 2) 2 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: qc1 − c2 > 1 2 + α2 2(∆2 − 1) (28) In summary, to have gM 1 < g2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (28) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (25) and from the hypothesis that j2 −j1 > 0 we obtain (qc1 −c2)(∆2 −1)− α2 > 0 which, since (∆2 − 1) > 0, implies qc1 − c2 > α2 ∆2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since 0 ≤ α2 ∆2−1 < 1 and qc1 − c2 is integer, we have qc1 − c2 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' This implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (28) holds and therefore that gM 1 < g2 and g1 ̸= g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='d: vg1 and vi2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' By Property 2 we have, for some c2 ∈ N: i2 = j2 + c2(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (29) 23 We also have, for some c1 ∈ N and 0 ≤ α1 < ∆2 − 1: i1 = j1 + c1(∆1 − 1) + α1 = j1 + qc1(∆2 − 1) + α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (30) If vi1 coincides with vi2, we have i1 = i2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (30) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (29) we obtain: j2 − j1 = (qc1 − c2)(∆2 − 1) + α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (31) Concerning vg2, we have: gm 2 ≤ g2 ≤ gM 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with gm 2 = r2 + d2(∆2 − 1), gM 2 = r2 + (d2 + 1)(∆2 − 1) for some d2 ∈ N such that c2 + d2 = � σ2 2 � − 1, where σ2 is the number of spine vertices of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since vg1 is a vertex in the lower part of Γ1, it must be g1 = r1 + d1(∆1 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that g1 < gm 2 , which implies g1 ̸= g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have g1 < gm 2 it must be: r1 + d1(∆1 − 1) < r2 + d2(∆2 − 1) r1 + qd1(∆2 − 1) < r2 + d2(∆2 − 1) r2 − r1 > (qd1 − d2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (32) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (32) can be rewritten as: j2 − j1 > � (qd1 − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 � (∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (33) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (31) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (33) we obtain: qc1 − c2 + α1 ∆2 − 1 > (qd1 − d2) + (σ2 mod 2) 2 − q(σ1 mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qc1 − c2 + α1 ∆2 − 1 > qσ1 2 + q(σ1 mod 2) 2 − qc1 − q − σ2 2 − 1(σ2 mod 2) 2 + c2+ + 1 + σ2 mod 2 2 − q(σ1 mod 2) 2 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: qc1 − c2 > 1 − q 2 − α1 2(∆2 − 1) (34) In summary, to have g1 < gm 2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (34) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (31) and from the hypothesis that j2 −j1 > 0 we obtain (qc1 −c2)(∆2 −1)+ α1 > 0 which, since (∆2−1) > 0, implies qc1−c2 > − α1 ∆2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since 0 ≤ α1 ∆2−1 < 1 and qc1 − c2 is integer, we have qc1 − c2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since q is a positive integer, this implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (34) holds and therefore that g1 < gm 2 and g1 ̸= g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 24 Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='b: vg1 and vg2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since vg2 is a vertex in the lower part of Γ2, it must be g2 = r2+d2(∆2−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If vg2 coincides with vi1, as explained above, it must be i1 = g2 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Combining the expression of g2 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (30) we obtain: r2 − j1 = (qc1 − d2)(∆2 − 1) + α1 + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (35) Concerning vi2, we have: im 2 ≤ i2 ≤ iM 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with iM 2 = j2 + (c2 + 1)(∆2 − 1) for some c2 ∈ N such that c2 + d2 = � σ2 2 � − 1, where σ2 is the number of spine vertices of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that iM 2 < g1, which implies i2 ̸= g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have iM 2 < g1 it must be: j2 + (c2 + 1)(∆2 − 1) < r1 + d1(∆1 − 1) j2 + (c2 + 1)(∆2 − 1) < r1 + qd1(∆2 − 1) j2 − r1 < (qd1 − c2 − 1)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (36) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (35) can be rewritten as: j2 − j1 = (qc1 − d2)(∆2 − 1)/α1 + n 2 + (∆2 − 1)(σ2 mod 2) 2 , (37) while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (36) can be rewritten as: j2 − j1 < � (qd1 − c2 − 1) − q(σ1 mod 2) 2 � (∆2 − 1) + n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (38) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (37) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (38) we obtain: qc1 − d2 < (qd1 − c2 − 1) − (σ2 mod 2) 2 − q(σ1 mod 2) 2 − α1 ∆2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qc1 − σ2 2 − σ2 mod 2 2 + c2 + 1 + σ2 mod 2 2 + α1 ∆2 − 1 < qσ1 2 + q(σ1 mod 2) 2 − − qc1 − q − c2 − q(σ1 mod 2) 2 − 1 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: qc1 − qσ1 2 + c2 + 1 < −q 2 − α1 2(∆2 − 1) (39) In summary, to have iM 2 < gm 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (39) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (37) and from the hypothesis that j2 − j1 < n−(∆1−1) 2 = n−q(∆2−1) 2 we obtain: qc1 − qσ1 2 + c2 + 1 < −q 2 − α1 ∆2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (40) 25 We have that − q 2 − α1 ∆2−1 < − q 2 − α1 2(∆2−1), since α1 2(∆2−1) < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In other words, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (40) implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (39) holds and therefore that iM 2 < g1 and i2 ̸= g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='c: vi1 and vg2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since vg2 is a vertex in the lower part of Γ2, it must be g2 = r2+d2(∆2−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If vg2 coincides with vi1, as explained above, it must be i1 = g2 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Combining the expression of g2 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (23) we obtain: r2 − j1 = (qc1 − d2)(∆2 − 1) + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (41) Concerning vg1, we have: gm 1 ≤ g1 ≤ gM 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with gm 1 = r1 + d1(∆1 − 1), gM 1 = r1 + (d1 + 1)(∆1 − 1) for some d1 ∈ N such that c1 + d1 = � σ1 2 � − 1, where σ1 is the number of spine vertices of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that iM 2 < gm 1 , which implies i2 ̸= g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have iM 2 < gm 1 it must be: j2 + (c2 + 1)(∆2 − 1) < r1 + d1(∆1 − 1) j2 + (c2 + 1)(∆2 − 1) < r1 + qd1(∆2 − 1) j2 − r1 < (qd1 − c2 − 1)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (42) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (41) can be rewritten as: j2 − j1 = (qc1 − d2)(∆2 − 1) + n 2 + (∆2 − 1)(σ2 mod 2) 2 , (43) while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (42) can be rewritten as: j2 − j1 < � (qd1 − c2 − 1) − q(σ1 mod 2) 2 � (∆2 − 1) + n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (44) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (43) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (44) we obtain: qc1 − d2 < (qd1 − c2 − 1) − (σ2 mod 2) 2 − q(σ1 mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qc1 − σ2 2 − σ2 mod 2 2 + c2 + 1 < qσ1 2 + q(σ1 mod 2) 2 − qc1 − q − c2 − 1− − σ2 mod 2 2 − q(σ1 mod 2) 2 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: qc1 − qσ1 2 + c2 + 1 < −q 2 (45) 26 In summary, to have iM 2 < gm 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (45) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (43) and from the hypothesis that j2 − j1 < n−(∆1−1) 2 = n−q(∆2−1) 2 we obtain: qc1 − d2 + 1 2(σ2 mod 2) < −q 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Replacing again d2 with σ2+1(σ2 mod 2) 2 − c2 − 1, we obtain: qc1 − qσ1 2 + c2 + 1 < −q 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (46) Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (45) is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (46), we can conclude that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (45) holds and therefore that iM 2 < gm 1 and i2 ̸= g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='d: vg1 and vi2 are spine vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Since vg1 is a vertex in the lower part of Γ1, it must be g1 = r1 + d1(∆1 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' If vg1 coincides with vi2, combining the expression of g1 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (6) we obtain: j2 − r1 = (qd1 − c2)(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (47) Concerning vi1 vg2, we have: im 1 ≤ i1 ≤ iM 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' and gm 2 ≤ g2 ≤ gM 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' with im 1 = j1 + c1(∆1 − 1) = j1 + qc1(∆2 − 1), gM 2 = r2 + (d2 + 1)(∆2 − 1) − n for some d2 ∈ N such that c2 + d2 = � σ2 2 � − 1, where σ2 is the number of spine vertices of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We prove that gM 2 < im 1 , which implies g2 ̸= i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To have gM 2 < im 1 it must be: r2 + (d2 + 1)(∆2 − 1) − n < j1 + c1(∆1 − 1) r2 + (d2 + 1)(∆2 − 1) − n < j1 + qc1(∆2 − 1) r2 − j1 < (qc1 − d2 − 1)(∆2 − 1) + n(∆2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (48) Since rl = jl + n−(∆l−1)(σl mod 2) 2 , for l = 1, 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (47) can be rewritten as: j2 − j1 = (qd1 − c2)(∆2 − 1) − q(∆2 − 1)(σ1 mod 2) 2 + n 2 , (49) while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (48) can be rewritten as: j2 − j1 < (qc1 − d2 − 1)(∆2 − 1) + (∆2 − 1)(σ2 mod 2) 2 + n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (50) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (49) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (50) we obtain: qd1 − c2 − q(σ1 mod 2) 2 < (qc1 − d2 − 1) + (σ2 mod 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 27 Since cl + dl = � σl 2 � − 1, we have dl = σl+1(σl mod 2) 2 − cl − 1, for l = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' replacing d1 and d2 in the previous equation, we obtain: qσ1 2 + q(σ1 mod 2) 2 − qc1 − q − c2 − q(σ1 mod 2) 2 < qc1 − σ2 2 − σ2 mod 2 2 + + c2 + 1 − 1 + σ2 mod 2 2 which, considering that σ2 = n−2 ∆2−1 = q(n−2) ∆1−1 = qσ1, implies: 2 � qc1 − qσ1 2 + c2 � > −q (51) In summary, to have gM 2 < im 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (51) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (49) and from the hypothesis that j2 − j1 < n−(∆1−1) 2 = n−q(∆2−1) 2 we obtain: qd1 − c2 − q 2(σ1 mod 2) < −q 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Replacing again d1 with σ1+1(σ1 mod 2) 2 − c1 − 1, we obtain: 2 � qc1 − qσ1 2 + c2 � > −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (52) Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (51) is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (52), we can conclude that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' (51) holds and therefore that gM 2 < im 1 and g2 ̸= i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' B Proof of Theorem 7 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' A ∆-regular caterpillar with 4 ≤ ∆ ≤ 7 admits a 2-planar 3- placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let C1, C2, and C3 be three copies (shown in red, blue and green, respec- tively, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 8) of a ∆-regular caterpillar C with 4 ≤ ∆ ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We denote the vertices of caterpillar Cj for j = 1, 2, 3 as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the spine vertices are denoted as vj 0, vj 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , vj c−1 in the order they appear along the spine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the leaves adjacent to vertex vj i (for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , c−1) are denoted as uj i,l with l = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , d, where d = ∆ − 2 if i = 0 or i = c − 1 and d = ∆ − 3 if 0 < i < c − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Let p0, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , pn−1 be n points on a circle in clockwise order (with indices taken modulo n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To construct the packing, we compute a drawing for each caterpillar such that the vertices are mapped to points p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , pn and the union of the three drawings is a 2-planar drawing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' We describe the construction for ∆ = 4, 5, 6 (see also Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 8(a) to 8(c));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the construction in the case ∆ = 7 is slightly different and it is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 8(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Caterpillar C1 is drawn outside the circle so that vertex v1 0 is mapped to point p0, each vertex v1 i , for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , c − 1 is mapped to pi(∆−1)+1, each leaf u1 0,l is mapped to the point pl+1, and each leaf u1 i,l is mapped to the point pi(∆−1)+2+l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' In other words, each vertex of the spine is followed clockwise by 28 ∆ = 4 (a) ∆ = 5 (b) ∆ = 6 (c) ∆ = 7 (d) Figure 8: 2-planar 3-placements of ∆-regular caterpillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' its leaves and the last of these leaves is followed by the next vertex of the spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Caterpillar C2 is drawn inside the circle so that vertex v2 i is mapped to the point immediately following clockwise the point hosting v1 i and each leaf u2 i,l is mapped to the point immediately following clockwise u1 i,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Clearly, the drawings of the first two caterpillars do not cross each other because they are on different sides of the circle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' also, their union has no multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Concerning C3, the vertex v3 i , for i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' , c−2 is mapped to the point that hosts u1 i,d and u2 i,d−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', the last leaf of v1 i and the second last leaf of v2 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' the vertex v3 c−1 is mapped to the point that hosts u1 i,d−1 and u2 i,d−2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=', the second last leaf of v1 c−1 and the third last leaf of v2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' About this mapping, observe that if we draw the edges of the spine of C3 outside the circle, each edge of the spine of C3 intersects two consecutive edges of the spine of C1 and each edge of the spine of C1 intersects at most two consecutive edges of the spine of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' To complete the drawing, we need to draw the leaves of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Consider two consecutive spine vertices v3 i and v3 i+1, with 0 ≤ i ≤ c − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' between these two vertices there are ∆ − 2 points not yet used by C3, we connect the first two of these vertices in clockwise order to vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Depending on the value of ∆, there remain 0, 1, or 2 points between vi and vi+1 not yet used by C3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we connect these points to vi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' Notice that, there remain to map ∆ − 3 leaves adjacent to v3 0 and 3 leaves adjacent to v3 c−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' On the other hand, there are ∆ points not yet used by C3 that are between v3 c−1 and v3 0 clockwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' we connect the three vertices following clockwise v3 c−1 to v3 c−1, and the remaining ones to v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' All the edges of C3 that are incident to leaves are drawn inside the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' This mapping of C3 does not create multiple edges and gives rise to at most two crossings along the edges of C2 and C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfR_u4/content/2301.01226v1.pdf'}