diff --git "a/BNE2T4oBgHgl3EQfRgdU/content/tmp_files/load_file.txt" "b/BNE2T4oBgHgl3EQfRgdU/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/BNE2T4oBgHgl3EQfRgdU/content/tmp_files/load_file.txt" @@ -0,0 +1,818 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf,len=817 +page_content='REDUCED CLIQUE GRAPHS: A CORRECTION TO “CHORDAL GRAPHS AND THEIR CLIQUE GRAPHS” DILLON MAYHEW AND ANDREW PROBERT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Galinier, Habib, and Paul introduced the reduced clique graph of a chordal graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The nodes of the reduced clique graph are the maximal cliques of G, and two nodes are joined by an edge if and only if they form a non-disjoint separating pair of cliques in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In this case the weight of the edge is the size of the intersection of the two cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' A clique tree of G is a tree with the maximal cliques of G as its nodes, where for any v ∈ V (G), the subgraph induced by the nodes containing v is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Galinier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' prove that a spanning tree of the reduced clique graph is a clique tree if and only if it has maximum weight, but their proof contains an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We explain and correct this error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In addition, we initiate a study of the structure of reduced clique graphs by proving that they cannot contain any induced cycle of length five (although they may contain induced cycles of length three, four, or six).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We show that no cycle of length four or more is isomorphic to a reduced clique graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We prove that the class of clique graphs of chordal graphs is not comparable to the class of reduced clique graphs of chordal graphs by providing examples that are in each of these classes without being in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Introduction We consider only simple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' A chord of a cycle is an edge that joins two vertices of the cycle without being in the cycle itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' A graph is chordal if any cycle with at least four vertices has a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' A clique is a set of pairwise adjacent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If S is a set of vertices and P is a path, then P is S-avoiding if no internal vertex of P is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assuming that a and b are distinct vertices, an ab-separator is a set S of vertices not containing either a or b such that there is no S-avoiding path from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If, in addition, S does not properly contain an ab-separator then it is a minimal ab-separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If G is a chordal graph, then C(G) is the corresponding clique graph (also known as the clique intersection graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The vertices of C(G) are the maximal cliques of G, and two maximal cliques are adjacent in C(G) if and only if they have a non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The vertices of the reduced clique graph, CR(G), are again the maximal cliques of G, but C and C′ are adjacent in CR(G) if and only if C ∩ C′ ̸= ∅ and C and C′ form a separating pair: that is, there is no (C ∩ C′)-avoiding path from a vertex in C − C′ 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='03781v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='CO] 10 Jan 2023 2 MAYHEW AND PROBERT to a vertex in C′ − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that the vertices of CR(G) are identical to the vertices of C(G), and every edge of CR(G) is an edge of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 234589 234689 235789 123 8910 G C(G) CR(G) 234589 234689 235789 123 8910 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' A chordal graph, its clique graph, and its reduced clique graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The reduced clique graph was introduced in [3] (where it is called a clique graph) and studied further in [5–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a graph, and let T be a tree whose vertices are the maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If, for every v ∈ V (G), the maximal cliques of G that contain v induce a subtree of T, then T is a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Clique trees were introduced by Gavril [4], who proved that a graph has a clique tree exactly when it is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We weight each edge of CR(G) as follows: the edge joining cliques C and C′ is weighted with |C ∩ C′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The following result is [3, Theorem 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a connected chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let T be a spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then T is a clique tree if and only if it is a maximum-weight spanning tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Although the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 is correct, it is not proved in [3, Theorem 6] because of a flaw in the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The issue arises in the proof that a maximum-weight spanning tree must be a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We illustrate the error by using the same argument to prove a false statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Non-theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn be the sequence of maximal cliques in a path of CR(G) where n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that there is a vertex v of G such that v is in C0 ∩ Cn, but in none of the cliques C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then C0 and Cn are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Non-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Consider the subgraph G′ of G induced by C0 ∪ C1 ∪ · · · ∪ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus G′ is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' From [10, Corollary 2] we see that either v is a simplicial vertex (meaning that the neighbours of v in G′ form a clique), or there is a pair, a, b, of vertices such that v belongs to a minimal ab-separator of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In the former case v is in a unique maximal clique of G′ ([1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But C0 and Cn are distinct maximal cliques of G′ that contain v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore REDUCED CLIQUE GRAPHS 3 we can let S be a minimal ab-separator of G′, where v is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The proof of [2, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3] shows that there are two distinct maximal cliques, Da and Db, of G′ such that Da and Db properly contain S, and Da − S is in the same connected component of G′ − S as a, while Db − S is in the same component as b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus Da and Db are maximal cliques of G′ that contain v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But the only maximal cliques of G′ that contain v are C0 and Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore we can assume without loss of generality that Da = C0 and Db = Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Any path from a vertex of C0 − Cn to a vertex of Cn − C0 must contain a vertex in S = C0 ∩ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore C0 and Cn form a non-disjoint separating pair, so C0 and Cn are adjacent in CR(G), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ We can see that this non-theorem is, indeed, not a theorem by examining Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Set C0, C1, and C2 to be the maximal cliques {2, 3, 4, 6, 8, 9}, {1, 2, 3}, and {2, 3, 5, 7, 8, 9}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus C0, C1, C2 is the vertex sequence of a path in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The vertex 8 is in C0 ∩ C2, but not in C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' However C0 and C2 are not adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The error in the “proof” lies in the claim that “the only maximal cliques of G′ that contain v are C0 and Cn”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This need not be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Indeed, {2, 3, 4, 5, 8, 9} is a maximal clique in the subgraph induced by C0 ∪ C1 ∪ C2, and it contains 8, but it is not equal to either C0 or C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Exactly the same error appears in the proof of [3, Theorem 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Nonetheless, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 is true, and we prove it in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Reduced clique graphs and clique trees In [9] we will apply our main theorem to some matroid problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For these purposes we would like to extend its scope somewhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Instead of weighting the edges of CR(G) with sizes of intersections, we consider more general weightings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We consider a function σ which takes {∅} ∪ {C ∩ C′ : C, C are distinct maximal cliques of G} to non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We insist that σ(∅) = 0 and if X and X′ are in the domain of σ and X ⊂ X′, then σ(X) < σ(X′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In such a case the function σ is a legitimate weighting of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a connected chordal graph and let σ be a legitimate weighting of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Every clique tree is a spanning tree of CR(G) and every edge of CR(G) is contained in a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Moreover, a spanning tree of CR(G) is a clique tree if and only if it has maximum weight amongst all spanning trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that the function that takes each intersection C ∩ C′ to |C ∩ C′| is a legitimate weighting, so Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2 does indeed imply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We now start proving the intermediate results required for the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 4 MAYHEW AND PROBERT Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph, and let C and C′ be maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be a set of vertices that contains C∩C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , vk be the vertex sequence of P, a shortest-possible S-avoiding path from a vertex in C − C′ to a vertex in C′ − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then (C ∩ C′) ∪ {vi, vi+1} is a clique for each i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If C ∩ C′ = ∅ then the result holds trivially, so we assume C ∩ C′ is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that every vertex in C ∩ C′ is adjacent to v0, and also to vk, since these vertices are in C − C′ and C′ − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now the result can only fail if there is a vertex x ∈ C ∩ C′ that is not adjacent to vi for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let p be the largest integer such that p < i and x is adjacent to vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Similarly, let q be the smallest integer such that q > i and x is adjacent to vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Consider the cycle obtained by adding the edges vpx and vqx to vp, vp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This cycle contains the distinct vertices vp, vi, vq, and x, so it must contain a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' No chord can join two vertices in the path P, since P is as short as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus any chord is incident with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But x is not adjacent to any of the vertices in vp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , vq−1 by the choice of p and q, so we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph, and let C and C′ be maximal cliques of G where C ∩ C′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If C and C′ are not adjacent in CR(G), then they are joined by a path of CR(G) with vertex sequence C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cs, where each Ci ∩ Ci+1 properly contains C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume this fails for C and C′, and they have been chosen so that C ∩ C′ is as large as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because C and C′ are not adjacent in CR(G), but S ̸= ∅, it follows that there is an S-avoiding path from a vertex in C − C′ to a vertex in C′ − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , vk be the vertex sequence of such a path, where k is as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We assume v0 is in C − C′ while vk is in C′ − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3 and for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k, we let Di be a maximal clique of G that contains S ∪{vi−1, vi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Set D0 to be C and set Dk+1 to be C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that Di ̸= Dj when i < j, because vi−1 is not adjacent to vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For each i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k, the intersection of Di and Di+1 contains S as well as vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If Di and Di+1 are adjacent in CR(G) then we let Pi be the path of CR(G) consisting of Di, Di+1, and the edge between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Otherwise Di and Di+1 are not adjacent in CR(G) and the assumption on the cardinality of S means that there is a path Pi of CR(G) from Di to Di+1 such that every intersection of consecutive cliques in Pi properly contains S ∪ vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We concatenate the paths P0, P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Pk and obtain a walk of CR(G) from C to C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The intersection of any two consecutive cliques in this walk properly contains S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It follows that there is a path of CR(G) from C to C′ with exactly the same property, and now C and C′ fail to provide a counterexample after all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Figure 2 illustrates Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The intersection of cliques C = {1, 2, 3} and C′ = {3, 5, 7, 8} is {3} ̸= ∅, but C and C′ are not adjacent REDUCED CLIQUE GRAPHS 5 in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' However, there is a path between C and C′ in CR(G), and the intersection of any consecutive two cliques in the path properly contains {3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 1 6 4 7 5 2 3 8 123 2345 3567 3456 3578 G CR(G) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a connected chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let T be a clique tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that C and C′ are maximal cliques of G that are adjacent in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then C and C′ are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume C and C′ are adjacent in T, but not in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We partition the maximal cliques of G as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let U be the set of maximal cliques of G such that D is in U if and only if the path of T from D to C does not contain C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Similarly, define U′ so that D′ is in U′ if and only if the path of T from D′ to C′ does not contain C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that every maximal clique of G is in exactly one of U or U′, since T is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Furthermore C is in U and C′ is in U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let U be the union of the cliques in U, and let U ′ be the union of the cliques in U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Every vertex is in at least one maximal clique so U ∪ U ′ = V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that C ⊆ U and C′ ⊆ U ′, so neither U nor U ′ is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If U ∩ U ′ = ∅, then we choose u ∈ U and u′ ∈ U ′ so that u and u′ are adjacent in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' (We are able to do so because G is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=') The edge between u and u′ is contained in a maximal clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If this maximal clique is in U then u′ is in U ∩ U ′, and if it is in U′ then u is in U ∩ U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In either case we have a contradiction, so U ∩ U ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Choose an arbitrary vertex v in U ∩ U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Choose D ∈ U and D′ ∈ U′ such that v is in D ∩ D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because T is a clique tree, it follows that v is contained in all the cliques belonging to the path of T from D to D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In particular, v is contained in C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus U ∩ U ′ ⊆ C ∩ C′ and C ∩ C′ is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since C and C′ are not adjacent in CR(G), we can apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4 and find a path P of CR(G) from C to C′, where the intersection of each pair of consecutive cliques in this path properly contains S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since C is in U and C′ is in U′, there is an edge of P that joins a clique D ∈ U to a clique D′ ∈ U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then D∩D′ properly contains S, so we choose v in (D ∩D′)−S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Again using the fact that T is a clique tree, we see that the path of T from D to D′ consists of cliques that contain v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In particular, v is in C ∩ C′ = S, and we have a contradiction that completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ 6 MAYHEW AND PROBERT It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5 that every clique tree of G is a spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a connected chordal graph and let σ be a legiti- mate weighting of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let T be a clique tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let C and C′ be maximal cliques of G that are adjacent in C(G) and let P be the path of T between C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The weight of any edge in P is at least σ(C ∩ C′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Moreover, if C and C′ are adjacent in CR(G), then at least one edge in P has weight equal to σ(C ∩ C′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let P be the path of T from C to C′, and let the cliques in this path be C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn, where C0 = C and Cn = C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that P is a path of CR(G) by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus any two consecutive cliques in the path have a non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume σ(Ci ∩ Ci+1) < σ(S) for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If S were a subset of Ci ∩ Ci+1, then we would have σ(S) ≤ σ(Ci ∩Ci+1) by the definition of a legitimate weighting, but this is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore we can choose v to be a vertex in S − (Ci ∩ Ci+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now v is a vertex of both C and C′, but the path of T between C and C′ contains at least one maximal clique (either Ci or Ci+1) that does not contain v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This contradicts the fact that T is a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore the weight of any edge in P is at least equal to σ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now assume that C and C′ are adjacent in CR(G), so that they form a separating pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' That is, there are distinct connected components of G − S that contain, respectively, C −S and C′−S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There must be maximal cliques D and D′ that are adjacent in P, where D − S is in the same connected component of G−S as C−S, and D′−S is not in this connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This means that D ∩ D′ is contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Hence σ(D ∩ D′) ≤ σ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The previous paragraph shows that σ(D ∩ D′) ≥ σ(S), so the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ The proof of the next result is a straightforward adaptation of a proof given by Blair and Peyton [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a connected chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let σ be a legitimate weighting of G and let T be a spanning tree of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then T is a clique tree of G if and only if it is a maximum-weight spanning tree of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If T is a clique tree, then for any pair of maximal cliques, C and C′, such that C and C′ are adjacent in C(G), the weight of the edge between C and C′ is no greater than the weight of any edge in the path of T between C and C′ (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It immediately follows that T has maximum weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For the other direction, we assume that T is a maximum-weight spanning tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because every chordal graph has a clique tree, and any clique tree is a spanning tree of CR(G) (and hence of C(G)), we can choose a clique tree T ′ so that T and T ′ have as many edges in common as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We can choose an edge in T that is not in T ′, because otherwise there is nothing left for us to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So let e be such an edge, and assume that e joins maximal cliques C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There are two connected components of T\\e, one containing C REDUCED CLIQUE GRAPHS 7 and the other containing C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let P be the path of T ′ from C to C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We let f be an edge of P which joins two cliques that are not in the same component of T\\e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that f is an edge of T ′, and hence an edge of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If (T − e) ∪ f is not a spanning tree of C(G), then there is a path of T between the end-vertices of f that does not use e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But the end-vertices of f are in different connected components of T\\e, so (T − e) ∪ f is indeed a spanning tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Similarly, if (T ′ − f) ∪ e is not a spanning tree, then there is a path of T ′ between C and C′ that does not contain f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But P is the unique path of T ′ between C and C′, and f is an edge of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So (T − e) ∪ f and (T ′ − f) ∪ e are both spanning trees of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6 to the clique tree T ′ shows that the weight of f is at least the weight of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since T is a maximum-weight spanning tree, and (T − e) ∪ f is a spanning tree it follows that the weights on e and f must be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let D and D′ be the maximal cliques joined by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Any element that is in both C and C′ must be in all the cliques in P, since T ′ is a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This shows that C ∩ C′ ⊆ D ∩ D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If C ∩ C′ were a proper subset of D ∩ D′, then the definition of a legitimate weighting would mean that the weight of e is strictly less than the weight of f, which is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore C ∩ C′ = D ∩ D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We note that (T ′−f)∪e cannot be a clique tree, since it has one more edge in common with T than T ′ does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore we choose a vertex v ∈ V (G) so that the maximal cliques containing v do not induce a subtree of (T ′−f)∪e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let T ′′ be the subtree of T ′ induced by the maximal cliques containing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then f is in T ′′, or else T ′′ would be a subtree of (T ′ − f) ∪ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This means that v is in D ∩ D′ = C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So both C and C′ are in T ′′, but they are not in the same component of T ′′\\f, because in that case (T ′ − f) ∪ e would contain a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So e joins two vertices of T ′′ that are in different components of T ′′\\f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus (T ′′ − f) ∪ e is a subtree of (T ′ − f) ∪ e, and we have a contradiction that completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We have already noted that every clique tree is a spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let T be a clique tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then T is a maximum- weight spanning tree of C(G) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But every edge of T is an edge of CR(G), by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since CR(G) is a subgraph of C(G) it follows that T is a maximum-weight spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For the other direction, we let T be a maximum-weight spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We claim that T is also a maximum-weight spanning tree of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' To prove this claim, let e be an arbitrary edge of C(G) that is not in T, let C and C′ be the maximal cliques of G joined by e, and let P be the path of T that joins C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If e is an edge of CR(G), then the weight of e is no greater than the weight of any edge in P, since T is a maximum-weight spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore we assume that e is not an edge of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now it follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4 and the definition of a legitimate weighting that the edges in P all have weight strictly greater than the weight of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In either case, the weight of e does not exceed the 8 MAYHEW AND PROBERT weight of any edge in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This implies that T is indeed a maximum-weight spanning tree of C(G), and thus T is a clique tree of G by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' To complete the proof, we let e be an arbitrary edge of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We will prove that e is in a maximum-weight spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We let C and C′ be the maximal cliques joined by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let T be an arbitrary maximum- weight spanning tree of CR(G), so that T is a clique tree by the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If e is in T then we have nothing left to prove, so assume that P is the path of T joining C to C′, where P contains more than one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6 shows that P contains an edge, f, with weight equal to the weight of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now (T − f) ∪ e is a maximum-weight spanning tree of CR(G) that contains e, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ From the previous arguments we can deduce further additional facts, both noted in [3]: any edge that is in C(G) but not CR(G) cannot be in any maximum-weight spanning tree of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Secondly, CR(G) is in fact the union of all clique trees of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Although the next fact is incidental to our main results here, we note it for a future application in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a connected chordal graph, and let T be a clique tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let C and C′ be adjacent in T and let S be C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that D and D′ are maximal cliques of G and the path of T from D to D′ contains both C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then D − S and D′ − S are in different connected components of G − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let U be the family of maximal cliques of G such that D is in U if and only if the path of T from D to C does not contain C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Similarly, we let U′ be the family of maximal cliques where D′ is in U′ if and only if the path of T from D′ to C′ does not contain C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that every maximal clique of G belongs to exactly one of U and U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We are asserting that if D ∈ U and D′ ∈ U′, then D − S and D′ − S are in different connected components of G − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that this fails for D and D′, where D ∩ D′ is as large as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let H be the connected component of G − S that contains both D − S and D′ − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let P be the path of T from D to D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore P contains both C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let v be an arbitrary vertex of D ∩ D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then v is in every maximal clique that appears in P, since T is a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In particular, v is in C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus v is in S, and this shows that D ∩ D′ is contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , vk be the vertex sequence of a shortest-possible path of H from a vertex v0 ∈ D − S to a vertex vk ∈ D′ − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This is an S-avoiding path, where S contains D ∩ D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus we can apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k we let Di be a maximal clique of G that contains (D ∩ D′) ∪ {vi−1, vi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let D0 be D and let Dk+1 be D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that each Di − S is contained in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This is true for D0 and Dk+1 by definition, and every other Di contains the edge vi−1vi, which is in the path of H from v0 to vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since D0 is in U and Dk+1 is in U′, we can choose i so that Di is in U and Di+1 is in U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The intersection of Di and Di+1 is larger than D ∩ D′, since it REDUCED CLIQUE GRAPHS 9 contains (D ∩ D′) ∪ vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' As Di − S and Di+1 − S are both contained in H we have a contradiction to the choice of D and D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The structure of reduced clique graphs Habib and Stacho comment on the possibility of investigating the struc- ture of graphs that are isomorphic to reduced clique graphs [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 714].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In this section we make a contribution to this investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We start by answering an obvious question that requires a non-trivial proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then CR(G) is connected if and only if G is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that H and H′ are distinct connected components of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' No maximal clique of H can share a vertex with a maximal clique of H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It follows that there be no path of CR(G) that joins two such cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus CR(G) is not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The other direction is stated without proof in [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 716].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that G is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since G is chordal it has a clique tree [4, Theorem 2], and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5 shows that every edge of the clique tree is an edge of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus CR(G) has a spanning tree, so it is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Next we note a characterisation of clique graphs due to Szwarcfiter and Bornstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2 ([11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The graph H is isomorphic to C(G) for some connected chordal graph G if and only if H has a spanning tree T such that whenever u and v are adjacent in H, the path of T from u to v induces a clique of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Induced cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Next we observe that clique graphs can have induced cycles of any length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We will later show that this is not true for reduced clique graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For an integer n ≥ 3 the wheel graph with n spokes is obtained from a cycle of n vertices by adding a new vertex and making it adjacent to all vertices of the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus the wheel graph with n spokes has an induced cycle of n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For each integer n ≥ 3 the wheel graph with n spokes is isomorphic to the clique graph of a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This is easy to prove using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2, but we will give a direct construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Start with a clique on the n + 1 vertices u0, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , un−1, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For each i ∈ Z/nZ, add a new vertex vi and make it adjacent to ui and ui+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Call the resulting graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It is easy to verify that G is chordal, and its maximal cliques are {u0, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , un−1, x} along with {vi, ui, ui+1} for each i ∈ Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 be a cyclic ordering of the maximal cliques in an induced cycle of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We take the indices to be from Z/nZ, so Ci and Cj are adjacent in CR(G) if and only if 10 MAYHEW AND PROBERT j ∈ {i − 1, i + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If |Ci ∩ Ci+1| ≤ |Cj ∩ Cj+1| for every j ∈ Z/nZ, then we say that the edge between Ci and Ci+1 is a minimal edge of the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 be a cyclic ordering of the maximal cliques in an induced cycle of CR(G), where n ≥ 4 and the indices are from Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that the edge between C0 and C1 is a minimal edge of the induced cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be C0 ∩ C1 and for i = 0, 1 let Hi be the connected component of G−S that contains Ci −S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then H0 and H1 are distinct connected components and Ci − S is contained in H0 or H1 for every i ∈ Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Furthermore, either: (i) H0 contains all of C0 − S, C2 − S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 − S, (ii) H1 contains all of C1 − S, C2 − S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 − S, or (iii) n = 4, and H0 contains C0 −S and C2 −S while H1 contains C1 −S and C3 − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that because C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 are distinct maximal cliques of G, none of them is contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus Ci − S is non-empty for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We consider the connected components of G − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Any set Ci − S is contained in such a component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because C0 and C1 form a separating pair, C0 − S and C1 − S are contained in different connected components of G − S, so H0 and H1 are distinct components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that i and j are distinct indices in Z/nZ such that there are distinct connected components of G−S, call them Hi and Hj, that contain Ci − S and Cj − S respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume also that Ci is adjacent in CR(G) to Cp, where Cp − S is not contained in Hi and that Cj is adjacent to Cq, where Cq − S is not contained in Hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then Ci and Cj are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that because the cycle of CR(G) is induced, p is in {i − 1, i + 1} and q is in {j − 1, j + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note also that Ci ∩ Cp is contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If this containment is proper then |Ci ∩ Cp| < |S| = |C0 ∩ C1| and we have violated our assumption that the edge between C0 and C1 is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore Ci and Cp both contain S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The same argument shows S ⊆ Cj ∩Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now Ci∩Cj is equal to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Moreover Ci − S and Cj − S are in different components of G − S, so Ci and Cj form a separating pair of maximal cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Hence they are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ We colour the cliques of C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For each i ∈ Z/nZ, if Ci − S is contained in H0 we colour Ci red, and if Cj − S is in H1 we colour Ci blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus C0 is red and C1 is blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Any maximal clique Ci is either red or blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If the claim fails then there is some i ∈ Z/nZ−{0, 1} such that Ci−S is contained in neither H0 nor H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let Hi be the connected component of G − S that contains Ci − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We colour any clique Cj in C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 green if Cj −S is contained in Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We know that the collections of red, blue, REDUCED CLIQUE GRAPHS 11 and green cliques are all non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So therefore we can find a red clique, Cred, adjacent to a clique that is not red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We can similarly find Cblue, a blue clique that is adjacent to a non-blue clique, and Cgreen, a green clique that is adjacent to a clique that is not green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 implies that Cred, Cblue, and Cgreen are adjacent to each other in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' As they are three distinct vertices in an induced cycle of CR(G) with at least four vertices, this is an immediate contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ If C1 is the only blue clique, then statement (i) holds and we have nothing left to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Similarly, if C0 is the only red clique, then (ii) holds and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So we assume there are at least two red cliques and at least two blue cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We can choose Cred and C′ red to be distinct red cliques that are adjacent to blue cliques, and we can choose Cblue and C′ blue to be two distinct blue cliques that are adjacent to red cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 implies that Cred and C′ red are adjacent to both Cblue and C′ blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus the four cliques induce a cycle in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This is impossible if n ≥ 5, so we conclude that n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now C0 is a red clique and it is adjacent to two blue cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus C1 and C3 are blue, C2 is red, and we are finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ The example in Figure 1 shows that a reduced clique graph may contain an induced cycle with four vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We will next show that there is no example with an induced cycle of five vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There is no chordal graph G such that CR(G) has an induced cycle with exactly five vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume otherwise and let G be a chordal graph such that CR(G) contains an induced cycle with five vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let C0, C1, C2, C3, C4 be the maximal cliques in this cycle, where the indices are from Z/5Z and Ci is adjacent to Cj if and only if j ∈ {i−1, i+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By adding a constant to these indices as necessary, we may assume that |C0 ∩ C1| ≤ |Ci ∩ Ci+1| for all i ∈ Z/5Z, so that the edge between C0 and C1 is a minimal edge of the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be C0 ∩ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that S is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now we apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By applying the permutation ρ: i �→ 1 − i as necessary, we may assume that statement (ii) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore we let H0 and H1 be connected components of G − S such that H0 contains C0 − S and H1 contains C1 − S, C2 − S, C3 − S, and C4 − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' C0 ∩ C4 = S = C0 ∩ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because C0 − S and C4 − S are contained in different components of G − S, it follows that C0 ∩ C4 ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' All we have left to prove is that this containment is not proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If it were proper, then we would contradict the assumption that the edge between C0 and C1 is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Neither C2 nor C3 contains S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 12 MAYHEW AND PROBERT Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that C0 ∩ C2 ⊆ S because C0 − S and C2 − S are contained in different components of G − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Certainly any path from a vertex of C0 − C2 to a vertex of C2 − C0 must use a vertex of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If C0 ∩ C2 = S, then C0 and C2 form a separating pair, so C0 and C2 are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This contradicts the fact that C0 and C2 are non-consecutive vertices in an induced cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The same argument shows that C3 does not contain S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' C2 ∩ C4 ⊆ C1 and C3 ∩ C1 ⊆ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that x is a vertex of C2 ∩ C4 that is not in C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2 we can let y be a vertex in S − C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus y is in C1 − C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So x is in C2 −C1 and y is in C1 −C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 implies that y is in C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' As x is also in C4 we see that x and y are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because C1 and C2 are adjacent in CR(G) they have a non-empty intersection, but now the edge xy shows that C1 and C2 do not form a separating pair and we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' A symmetric argument shows C3 ∩ C1 ⊆ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' C2 contains a vertex of C1 − C4 and C3 contains a vertex of C4 − C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By symmetry it suffices to prove the first statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that C2 contains no vertex of C1 − C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because C1 and C2 are adjacent in CR(G), they have at least one vertex in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By our assumption, no vertex of C1 ∩ C2 is in C1 − C4, so any such vertex must be in C1 ∩ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore C2 and C4 are not disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since C2 and C4 are not adjacent in CR(G), we can let P be a (C2 ∩C4)-avoiding path from a vertex x ∈ C2 −C4 to y ∈ C4 −C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Our assumption means that x is not in C1 − C4, so it is in C2 − C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Our assumption and Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3 imply that C2 ∩ C4 = C2 ∩ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore P is a (C2 ∩ C1)-avoiding path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2 shows that we can choose a vertex z in S − C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus z is in C1 − C2 and Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 shows that z is in C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assuming that z and y are not equal, they are adjacent, as both are in C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By appending (if necessary) the edge yz to the end of P we obtain a (C1 ∩ C2)-avoiding path from a vertex in C2 − C1 to a vertex in C1 − C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Hence C1 and C2 do not form a separating pair and this contradicts the fact that they are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Either C2 ∩ (C1 ∩ C4) ⊆ C3 or C3 ∩ (C1 ∩ C4) ⊆ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that C2 ∩ C3 is non-empty, since C2 and C3 are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If the claim fails, then we choose x ∈ (C2 ∩ C1 ∩ C4) − C3 and y ∈ (C3 ∩ C1 ∩ C4) − C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now x and y are both in C1 ∩ C4, so they are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Moreover x is in C2 − C3 and y is in C3 − C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus C2 and C3 do not form a separating pair and we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ By using Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5, we will assume that C2 ∩ (C1 ∩ C4) is a subset of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The other outcome from Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5 yields to a symmetric argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Using Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2 we choose a vertex x ∈ S that is not in C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 implies that S is contained in C1 ∩C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So x is in (C1 ∩C4)−C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Our assumption therefore implies that x is not in C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' REDUCED CLIQUE GRAPHS 13 By Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4 we can also choose y in C2 ∩(C1 −C4) and z in C3 ∩(C4 − C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3 implies that y is in C2 −C3 and z is in C3 −C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now x and y are adjacent as they are both in C1, and x and z are adjacent as they are both in C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that C2 ∩ C3 is non-empty as C2 and C3 are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But the path with vertex sequence y, x, z is (C2 ∩ C3)-avoiding, so C2 and C3 do not form a separating pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This final contradiction completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6 shows that the class of reduced clique graphs is contained in the class of graphs with no length-five induced cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We next show that this containment is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let n ≥ 4 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There is no chordal graph G such that either C(G) or CR(G) is a cycle with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Szwarcfiter and Bornstein characterise the clique graphs of chordal graphs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In particular H is isomorphic to C(G) for some chordal graph G if and only if H has a spanning tree T such that whenever u and v are adjacent in H, the path of T from u to v induces a clique of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now assume H is a cycle with at least four vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Any spanning tree of H is a Hamiltonian path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The end vertices of this path are adjacent in H, but the path of the spanning tree between these vertices does not induce a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore H is not isomorphic to C(G) for any chordal graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We turn to reduced chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume for a contradiction that G is a chordal graph with C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 as its list of maximal cliques, where the indices are from Z/nZ, and Ci is adjacent to Cj in CR(G) if and only if j ∈ {i − 1, i + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We can assume without loss of generality that the edge between C0 and C1 is a minimal edge of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be C0 ∩ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that statement (iii) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus n = 4 and there are distinct connected components, H0 and H1, of G − S such that H0 contains C0 − S and C2 − S while H1 contains C1 − S and C3 − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that C0 ∩ C3 ⊆ S, and in fact C0 ∩ C3 is equal to S, or else the minimality of the C0-C1 edge is contradicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Either C0 ∩ C2 is empty, or it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In the latter case, we can apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4 to C0 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We see that either C0 ∩ C1 or C0 ∩ C3 properly contains C0 ∩C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By symmetry, we can assume C0 ∩C2 is a proper subset of C0 ∩ C1 = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus C0 − S and C2 − S are disjoint sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' They are contained in the same connected component of G − S, so we can let P be a shortest-possible path of H0 from a vertex of C0 − S to a vertex of C2 − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' On the other hand, if C0 ∩ C2 is empty, then C0 − S and C2 − S are again disjoint subsets in H0, so we again let P be a shortest-possible path of H0 from C0 − S to C2 − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In either case, P contains exactly one vertex of C0 and exactly one vertex of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then P must contain at least one edge, and this edge is in a maximal clique that is equal to neither C0 nor C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Nor can this maximal clique be C1 or C3, because any edge of P is contained in H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So we have a contradiction in the case that (iii) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 14 MAYHEW AND PROBERT Now we assume that either (i) or (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By applying the permutation ρ: i �→ 1 − i as necessary, we will assume that H0 and H1 are distinct connected components of G − S, and that H0 contains C0 − S while H1 contains Ci − S for i = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By the same argument as earlier, we can see that Cn−1 contains S, or else the choice of the C0 − C1 edge is contradicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now C1 ∩ Cn−1 contains S, and C1 and Cn−1 are non-adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4 and see that there is a path of CR(G) from C1 to Cn−1 such that every intersection of consecutive cliques in the path properly contains C1∩Cn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This path is either C1, C0, Cn−1, or it is C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then C1 ∩ C0 = S properly contains C1 ∩ Cn−1 ⊇ S and we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Hence any intersection of consecutive cliques in C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 properly contains C1 ∩ Cn−1, and hence contains S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It follows that C2 contains S and thus C0 ∩ C2 is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since C0 − S and C2 − S are contained in different components of G − S, any path of a vertex from C0 − C2 to a vertex of C2 − C0 must contain a vertex of S = C0 ∩ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus C0 and C2 form a separating pair in G, and hence they are adjacent in CR(G), which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Clique graphs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' reduced clique graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Consider the classes {C(G)} and {CR(G)}, where G ranges over all chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6 show that the wheel with five spokes is isomorphic to a graph in the former class but not the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Is there a graph that is isomorphic to a graph in the latter class but not the former?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We will show that the answer is, once again, yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Recall that if G and G′ are disjoint graphs, then G ⊠ G′ is obtained from the union of G and G′ by making every vertex of G adjacent to every vertex of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We use Pn to denote the path of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let m, n ≥ 1 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then Pm ⊠ Pn is isomorphic to the reduced clique graph of a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If n ≥ 22, then Pn ⊠ Pn is not isomorphic to the clique graph of a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be the graph obtained from the disjoint union of Pm and Pn and adding a new vertex that is adjacent to every vertex of the disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It is easy to confirm that G is chordal, and that CR(G) is isomorphic to Pm ⊠ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For the second statement, we let H be a graph with disjoint induced paths Pu = u0, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , un−1 and Pv = v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , vn−1, where n ≥ 22 and every ui is adjacent to every vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus H is isomorphic to Pn ⊠ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We will assume for a contradiction that H is isomorphic to C(G) for some chordal graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because C(G) is connected it follows easily that G is connected, so we can apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2 and deduce that H has a spanning tree T, where the path of T from u to v induces a clique of H whenever u and v are adjacent in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' REDUCED CLIQUE GRAPHS 15 Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let i and j be integers satisfying 0 < i, j < n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The path of T from ui to vj is contained in one of: {ui, ui+1, vj, vj+1}, {ui, ui+1, vj−1, vj}, {ui−1, ui, vj, vj+1}, {ui−1, ui, vj−1, vj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let P be the path of T from ui to vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since ui is adjacent to vj it follows that P induces a clique of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' As ui is not adjacent to any of the vertices in u0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , ui−2, ui+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , un−1, it follows that the vertices of P that are in Pu belong to {ui−1, ui, ui+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Similarly, the vertices of P that are in Pv belong to {vj−1, vj, vj+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But ui−1 is not adjacent to ui+1, so P does not contain both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The claim follows by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 implies that the path of T between ui and vj has at most three edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let P be a longest-possible path of T and let p0, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , pk−1 be the ver- tices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k − 1, let Ui be the set of vertices in Pu such that u is in Ui if and only if the shortest path of T from u to a vertex in P contains pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We define Vi to be the analogous set of vertices in Pv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that (U0, U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Uk−1) is a partition of the vertices of Pu, and (V0, V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Vk−1) is a partition of the vertices of Pv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Either max{|Ui|: 0 ≤ i ≤ k − 1} ≤ 3 or max{|Vi|: 0 ≤ i ≤ k − 1} ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume for a contradiction that |Ui| ≥ 4 and |Vj| ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let p and q, respectively, be the smallest (largest) integers such that up, uq ∈ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then q − 1 > p + 1 because |Ui| ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In the same way, let s and t be the smallest (largest) integers such that vs, vt ∈ Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then t−1 > s+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It is simple to see from Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 that the path of T from up to vs has no vertex in common with the path of T from uq to vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But this contradicts the fact that both paths contain pi and pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ By using Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2, we will assume without loss of generality that |Vi| ≤ 3 for each i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since (U0, U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Uk−1) is a partition of the vertices in Pu we can choose i so that Ui contains a vertex x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We claim that if j ≤ i − 4 or j ≥ i + 4, then Vj = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If this fails, then the path of T from a vertex in Vj to x contains at least four edges of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But this contradicts our earlier conclusion that any path of T from a vertex of Pu to a vertex of Pv contains at most three edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So now the vertices of Pv belong to Vi−3 ∪ Vi−2 ∪ · · · ∪ Vi+2 ∪ Vi+3 and this union has cardinality at most 7 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus Pv contains at most 21 vertices and this contradicts n ≥ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Conclusions and open problems Given Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6 it might be natural to believe that reduced clique graphs cannot have any induced cycles with five or more vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But Figure 3 shows a chordal graph G where CR(G) has an induced cycle with six vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 16 MAYHEW AND PROBERT 5 10 235 346 1234 G C(G) CR(G) 2 4 3 6 1 8 9 3479 2378 710 2347 2347 235 346 1234 3479 2378 710 2347 2347 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Nonetheless we believe the following to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There is no chordal graph G such that CR(G) contains an induced cycle with seven or more vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So far as we have been able to tell, every chordal graph is isomorphic to both a clique graph, and to a reduced clique graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We conjecture this holds generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let H be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There are chordal graphs G and G′ such that H is isomorphic to both C(G) and CR(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Szwarcfiter and Bornstein present a polynomial-time algorithm for decid- ing whether a given graph is isomorphic to C(G) for some chordal graph G [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Their techniques do not obviously extend to recognising reduced clique graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Nonetheless, we will make the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There is a polynomial-time algorithm for deciding whether a given graph is isomorphic to CR(G) for some chordal graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' More informally, we ask if there is a structural description for reduced clique graphs that is analogous to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' References [1] Jean R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Blair and Barry Peyton, An introduction to chordal graphs and clique trees, Graph theory and sparse matrix computation, IMA Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 56, Springer, New York, 1993, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 1–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' [2] Peter Buneman, A characterisation of rigid circuit graphs, Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 9 (1974), 205–212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' [3] Philippe Galinier, Michel Habib, and Christophe Paul, Chordal graphs and their clique graphs, Graph-theoretic concepts in computer science (Aachen, 1995), Lecture Notes in Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 1017, Springer, Berlin, 1995, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 358–371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' [4] F˘anic˘a Gavril, The intersection graphs of subtrees in trees are exactly the chordal graphs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Combinatorial Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' B 16 (1974), 47–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' REDUCED CLIQUE GRAPHS 17 [5] Michel Habib and Vincent Limouzy, On some simplicial elimination schemes for chordal graphs, DIMAP Workshop on Algorithmic Graph Theory, Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Notes Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 32, Elsevier Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} 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