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+Parameterised and Fine-grained Subgraph
+Counting, modulo 2∗
+Leslie Ann Goldberg
+University of Oxford
+Marc Roth
+University of Oxford
+Abstract
+Given a class of graphs H, the problem ⊕Sub(H) is defined as follows. The input is a graph H ∈ H
+together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs
+of G that are isomorphic to H. The goal of this research is to determine for which classes H the
+problem ⊕Sub(H) is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|) · |G|O(1).
+Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that ⊕Sub(H) is FPT if and only if the
+class of allowed patterns H is matching splittable, which means that for some fixed B, every H ∈ H
+can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at
+most B vertices.
+Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all
+hereditary pattern classes H, and (II) all tree pattern classes, i.e., all classes H such that every
+H ∈ H is a tree.
+We also establish almost tight fine-grained upper and lower bounds for the case of hereditary
+patterns (I).
+2012 ACM Subject Classification Theory of computation → Problems, reductions and completeness;
+Mathematics of computing → Discrete mathematics
+Keywords and phrases modular counting, parameterised complexity, fine-grained complexity, sub-
+graph counting
+Acknowledgements We want to thank Radu Curticapean, Holger Dell and Thore Husfeldt for
+insightful discussions on an early draft of this work.
+1
+Introduction
+The last two decades have seen remarkable progress in the classification of subgraph counting
+problems: Given a small pattern graph H and a large host graph G, how often does H occur
+as a subgraph if G? Since it was discovered that subgraph counts from small patterns reveal
+global properties of complex networks [25, 26], subgraph counting has also found several
+applications in fields such as biology [1, 32] genetics [34], phylogeny [24], and data mining [35].
+Moreover, the theoretical study of subgraph counting and related problems has led to many
+deep structural insights, establishing both new algorithmic techniques and tight lower bounds
+under the lenses of fine-grained and parameterised complexity theory [18, 15, 9, 13, 12, 5, 3].
+Without any additional restrictions, the subgraph counting problem is infeasible. The
+complexity class #W[1] is the parameterised complexity class analgous to NP (see Section 2
+for more detail). Under standard assumptions, problems that are #W[1]-hard are not fixed-
+parameter tractable (FPT). However, the canonical complete problem for #W[1], the problem
+of counting k-cliques, corresponds to the special case of the subgraph counting problem
+∗ For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any
+Author Accepted Manuscript version arising from this submission. All data is provided in full in the
+results section of this paper.
+arXiv:2301.01696v1  [cs.CC]  4 Jan 2023
+
+2
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+where H is a clique of size k. This problem cannot be solved in time f(k) · no(k) for any
+function f unless the Exponential Time Hypothesis (ETH) fails [7, 8]. Due to this hardness
+result, the research focus in this area shifted to the question: Under which restrictions on the
+patterns H and the hosts G is algorithmic progress possible? More precisely, under which
+restrictions can the problem be solved in time f(|H|) · |G|O(1), for some computable function
+f? Instances that can be solved within such a run time bound are called fixed-parameter
+tractable (FPT); allowing a potential super-polynomial overhead in the size of the pattern
+|H| formalises the assumption that H is assumed to be (significantly) smaller than G.
+If only the patterns are restricted, then the situation if fully understood. Formally, given
+a class H of patterns, the problem #Sub(H) asks, given as input a graph H ∈ H and an
+arbitrary graph G, to compute the number of subgraphs of G that are isomorphic to H.
+Following initial work by Flum and Grohe [18] and by Curticapean [10], Curticapean and
+Marx [13] proved that, under standard assumptions, #Sub(H) is FPT if and only if H has
+bounded matching number, that is, if there is a positive integer B such that the size of
+any matching in any graph in H is at most B. They also proved that all FPT cases are
+polynomial-time solvable.
+In stark contrast, almost nothing is known for the decision version Sub(H). Here, the
+task is to correctly decide whether there is a copy of H ∈ H in G, rather than to count
+the copies. It is known that Sub(H) is FPT whenever H has bounded treewidth (see e.g.
+[19, Chapter 13]), and it is conjectured that those are all FPT cases. However, resolving
+this conjecture belongs to the “most infamous” open problems in parameterised complexity
+theory [17, Chapter 33.1]
+1.1
+Counting Modulo 2
+To interpolate between the fully understood realm of (exact) counting and the barely
+understood realm of decision, Curticapean, Dell and Husfeldt proposed the study of counting
+subgraphs, modulo 2 [11]. Formally, they introduced the problem ⊕Sub(H), which expects
+as input a graph H ∈ H and an arbitrary graph G, and the goal is to compute modulo 2 the
+number of subgraphs of G isomorphic to H.
+The study of counting modulo 2 received significant attention from the viewpoint of
+classical and structural complexity theory. For example, one way to state Toda’s Theorem [33]
+is PH ⊆ P⊕P, implying that counting satisfying assignments of a CNF, modulo 2, is at least
+as hard as the polynomial hierarchy. Another example is the quest to classify the complexity
+of counting modulo 2 the homomorphisms to a fixed graph, which was very recently resolved
+by Bulatov and Kazeminia [6].
+In their work [11], Curticapean, Dell and Husfeldt proved that the problem of counting
+k-matchings modulo 2, that is, the problem ⊕Sub(H) where H is the class of all 1-regular
+graphs, is fixed-parameter tractable, where the parameter k is |H|. Since the exact counting
+version of this problem is #W[1]-hard [10], their result provides an example where counting
+modulo 2 is strictly easier than exact counting (subject to complexity assumptions). The
+complexity class ⊕W[1] can be defined via the complete problem of counting k-cliques
+modulo 2. Crucially, ⊕W[1]-hard problems are not fixed-parameter tractable, unless the
+randomised ETH (rETH) fails. Curticapean et al. [11] proved that counting k-paths modulo
+2 is ⊕W[1]-hard. Since finding a k-path in a graph G is fixed-parameter tractable via colour-
+coding [2], this hardness result provides an example where counting modulo 2 is strictly
+harder than decision (subject to complexity assumptions). Combining those observations,
+it appears that counting subgraphs modulo 2 may lie strictly in between the complexity of
+decision and the complexity of exact counting.
+
+L. A. Goldberg and M. Roth
+3
+A matching is a graph whose degree is at most 1. The matching-split number of a graph
+H is the minimum size of a set S ⊆ V (H) such that H \ S is a matching. A class of graphs
+H is called matching splittable if there is a positive integer B such that the matching-split
+number of any H ∈ H is at most B. For example, the class of all matchings is matching
+splittable while the class of all cycles is not. Curticapean, Dell and Husfeldt extended their
+FTP algorithm for counting k-matchings modulo 2 to obtain an FPT algorithm for ⊕Sub(H)
+for any matching-splittable class H. On this basis, they then made the following conjecture.
+▶ Conjecture 1 ([11]). ⊕Sub(H) is FPT if and only if H is matching splittable.
+A class H of graphs is called hereditary if it is closed under vertex removal. Intriguingly,
+if Conjecture 1 is true, then the FPT criterion for counting subgraphs modulo 2 (⊕Sub(H))
+would coincide with the polynomial-time criterion for finding subgraphs (Sub(H)) for hered-
+itary pattern classes H as established by Jansen and Marx.
+▶ Theorem 2 ([23]). Let H be a hereditary class of graphs and assume P ̸= NP. Then
+Sub(H) is solvable in polynomial time if and only if H is matching splittable.
+Jansen and Marx also conjecture that the condition of H being hereditary can be removed.
+▶ Conjecture 3 ([23]). Sub(H) is solvable in polynomial time if and only if H is matching
+splittable.
+Conjectures 1 and 3 have the remarkable consequence that ⊕Sub(H) is FPT if and only
+if Sub(H) is solvable in polynomial time. In the current work we establish this consequence
+for all hereditary pattern classes.
+1.2
+Our Contributions
+We resolve Conjecture 1 for all hereditary classes H, as well as for every class H consisting
+only of trees.
+▶ Theorem 4. Let H be a hereditary class of graphs. If H is matching splittable, then
+⊕Sub(H) is fixed-parameter tractable.
+Otherwise, the problem is ⊕W[1]-complete and,
+assuming rETH, cannot be solved in time f(|H|) · |G|o(|V (H)|/ log |V (H)|) for any function f.
+▶ Theorem 5. Let T be a recursively enumerable class of trees. If T is matching splittable,
+then ⊕Sub(T ) is fixed-parameter tractable. Otherwise ⊕Sub(T ) is ⊕W[1]-complete.
+In order to prove our classifications, we adapt the by-now-standard technique for ana-
+lysing subgraph counting problems established by Curticapean, Dell and Marx [12]. Let
+#Sub(H → G) denote the number of subgraphs of a graph G that are isomorphic to a
+graph H and let #Hom(F → G) denotes the number of homomorphisms (edge-preserving
+mappings) from a graph F to a graph G. Given a graph H, there is a function aH from
+graphs to rationals with finite support such that the following holds for any graph G:
+#Sub(H → G) =
+�
+F
+aH(F) · #Hom(F → G) ,
+(1)
+where the sum is over all (isomorphism types of) graphs. Since aH has finite support,
+aH(F) = 0 for all but finitely-many graphs F. Thus, equation (1) allows us to express the
+solution to the exact counting problem as a finite linear combination of homomorphism counts.
+In a nutshell, the framework of [12] states that computing the function G �→ #Sub(H → G)
+
+4
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+is hard to compute if and only if there is a graph F of high treewidth with aH(F) ̸= 0.
+This translates the complexity of (exact) subgraph counting to the purely combinatorial
+problem of understanding the coefficients aH. One might hope that this strategy transfers
+to counting modulo 2 as well. Unfortunately, this is not possible as Equation (1) might
+not be well-defined if arithmetic is done modulo 2. The reason for this is the fact that the
+coefficients aH(F) are of the form µ(F, H) × |Aut(H)|−1, where µ(F, H) is an integer, and
+Aut(H) is the automorphism group of the graph H [12]. Thus there is, a priori, no hope
+to extend the framework to counting modulo 2 for pattern graphs with an even number of
+automorphisms. In fact, according to Curticapean, Dell and Husfeldt [11], the absence of a
+comparable framework for counting modulo 2 is one of the main challenges for establishing
+the hardness part of Conjecture 1, and it is the main reason why the reductions in [11] use
+more classical, gadget-based reductions.
+In this work, we solve the problem of patterns with an even number of automorphisms
+by considering a colourful intermediate problem. More concretely, we will equip each edge
+of the pattern H with a distinct colour and show that it will be sufficient to consider only
+automorphisms that preserve the colours. If H has no isolated vertices, then this is only
+the trivial automorphism. Formally, the coloured approach will be based on the notion of
+so-called fractured graphs introduced by Peyerimhoff et al. [29].
+In what follows (Section 2), we will first introduced all required notions and previous
+results. In Section 3, we will prove the classification for hereditary pattern classes (Theorem 4).
+On a technical level, this proof can be considered a warm-up for the significantly harder
+challenge of establishing the classification for trees (Theorem 5), which we prove in Section 4.
+2
+Preliminaries
+Let f : A1 × A2 → B be a function. For each a1 ∈ A1 we write f(a1, ⋆) : A2 → B for the
+function that maps a2 ∈ A2 to f(a1, a2).
+Graphs in this work are undirected and without self loops. A homomorphism from a
+graph H to a graph G is a mapping ϕ from the vertices V (H) of H to the vertices V (G)
+of G such that for each edge e = {u, v} ∈ E(H) of H, the image ϕ(e) = {ϕ(u), ϕ(v)} is an
+edge of G. A homomorphism is called an embedding if it is injective. We write Hom(H → G)
+and Emb(H → G) for the sets of homomorphisms and embeddings, respectively, from H
+to G. An embedding ϕ ∈ Emb(H → G) is called an isomorphism if it is bijective and
+{u, v} ∈ E(H) ⇔ {ϕ(u), ϕ(v)} ∈ E(G). We say that H and G are isomorphic, denoted by
+H ∼= G, if an isomorphism from H to G exists. A graph invariant ι is a function from graphs
+to rationals such that ι(H) = ι(G) for each pair of isomorphic graphs H and G.
+A subgraph of G is a graph G′ with V (G′) ⊆ V (G) and E(G′) ⊆ E(G). We write
+Sub(H → G) for the set of all subgraphs of G that are isomorphic to H. Given a subset of
+vertices S ⊆ V (G) of a graph G, we write G[S] for the graph induced by S, that is, G[S] has
+vertices S and edges {{u, v} ⊆ S | {u, v} ∈ E(G)}.
+We denote by tw(G) the treewidth of the graph G. Since we will rely on treewidth purely
+in a black-box manner, we omit the technical definition and refer the reader to [14, Chapter
+7].
+Given any graph invariant ι (such as treewidth) and a class of graphs G, we say that
+ι is bounded in G if there is a non-negative integer B such that, for all G ∈ G, ι(G) ≤ B.
+Otherwise we say that ι is unbounded in G.
+Given a graph H = (V, E), a splitting set of H is a subset of vertices S such that every
+vertex in H[V \S] has degree at most 1. The matching-split number of H is the minimum size
+
+L. A. Goldberg and M. Roth
+5
+v
+vB1 vB2
+Figure 1 Illustration of the construction of a fractured graph from [29]. The left picture shows
+a vertex v of a graph Q with incident edges EQ(v) = { , , , , , }. The right picture shows the
+splitting of v in the construction of the fractured graph Q
+♯
+σ for a fracture σ satisfying that the
+partition σv contains two blocks B1 = { , , }, and B2 = { , , }.
+of a splitting set of H. A class of graphs H is called matching splittable the matching-split
+number of H is bounded.
+2.1
+Colour-Preserving Homomorphisms and Embeddings
+A homomorphism c from a graph G to a graph Q is sometimes called a “Q-colouring” of G.
+A Q-coloured graph is a pair consisting of a graph G and a homomorphism c from G to Q.
+Note that the identity function idQ on V (Q) is a Q-colouring of Q. If a homomorphism c
+from G to Q is vertex surjective, then we call (G, c) a surjectively Q-coloured graph.
+▶ Definition 6 (cE). A Q-colouring c of a graph G induces a (not necessarily proper)
+edge-colouring cE : E(G) → E(Q) given by cE({u, v}) = {c(u), c(v)}.
+Notation: Given a Q-coloured graph (G, c) and a vertex u ∈ V (Q), we will use the
+capitalised letter U to denote the subset of vertices of G that are coloured by c with u, that
+is, U := c−1(u) ⊆ V (G).
+Given two Q-coloured graphs (H, cH) and (G, cG), we call a homomorphism ϕ from H
+to G colour-preserving if for each v ∈ V (H) we have cG(ϕ(v)) = cH(v). We note the
+special case in which Q = H and cH is the identity idQ; then the condition simplifies to
+cG(ϕ(v)) = v. A colour-preserving embedding of (H, cH) in (G, cG) is a vertex injective colour-
+preserving homomorphism from (H, cH) to (G, cG). We write Hom((H, cH) → (G, cG)) and
+Emb((H, cH) → (G, cG)) for the sets of all colour-preserving homomorphisms and embeddings,
+respectively, from (H, cH) to (G, cG).
+Let k be a positive integer, let H be a graph with k edges, and let (G, γ) be a pair
+consisting of a graph G and a function that maps each edge of G to one of k distinct colours.
+We refer to γ as a “k-edge colouring” of G. For example, in most of our applications we will fix
+a graph Q with k edges and a Q-colouring c of G and we will take γ to be the edge-colouring
+cE from Definition 6. We write ColSub(H → (G, γ)) for the set of all subgraphs of G that
+are isomorphic to H and that contain each of the k edge colours precisely once.
+2.2
+Fractures and Fractured Graphs
+In this work, we will crucially rely on and extend the framework of fractured graphs as
+introduced in [29].
+▶ Definition 7 (Fractures). Let Q be a graph. For each vertex v of Q, let EQ(v) be the set
+of edges of Q that are incident to v. A fracture of Q is a tuple ρ = (ρv)v∈V (Q), where for
+each vertex v of Q, ρv is a partition of EQ(v).
+
+6
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+Note that a fracture describes how to split (or how to fracture) each vertex of a given
+graph: for each vertex v, create a vertex vB for each block B in the partition ρv; edges
+originally incident to v are made incident to vB if and only if they are contained in B. We
+call the resulting graph the fractured graph H
+♯
+ρ; a formal definition is given in Definition 8,
+a visualisation is given in Figure 1.
+▶ Definition 8 (Fractured Graph Q
+♯
+ρ). Given a graph Q, we consider the matching MQ
+containing one edge for each edge of Q; formally,
+V (MQ) :=
+�
+e={u,v}∈E(Q)
+{ue, ve}
+and
+E(MQ) := {{ue, ve} | e = {u, v} ∈ E(Q)}.
+For a fracture ρ of Q, we define the graph Q
+♯
+ρ to be the quotient graph of MQ under
+the equivalence relation on V (MQ) which identifies two vertices ve, wf of MQ if and only if
+v = w and e, f are in the same block B of the partition ρv of EQ(v). We write vB for the
+vertex of Q
+♯
+ρ given by the equivalence class of the vertices ve (for which e ∈ B) of MQ.
+▶ Definition 9 (Canonical Q-colouring cρ). Let Q be a graph and let ρ be a fracture of Q.
+The canonical Q-colouring of the fractured graph Q
+♯
+ρ maps vB to v for each v ∈ V (Q) and
+block B ∈ ρv, and is denoted by cρ.
+Observe that cρ is the identity in V (Q) if ρ is the coarsest fracture (that is, each partition
+ρv only contains one block, in which case Q
+♯
+ρ = Q).
+2.3
+Parameterised and Fine-grained Computation
+A parameterised computational problem is a pair consisting of a function P : Σ∗ → {0, 1} and
+a computable parameterisation κ : Σ∗ → N. A fixed-parameter tractable (FPT) algorithm for
+(P, κ) is an algorithm that computes P and runs, on input x ∈ Σ∗, in time f(κ(x)) · |x|O(1)
+for some computable function f. We call (P, κ) fixed-parameter tractable (FPT) if an FPT
+algorithm for (P, κ) exists.
+A parameterised Turing-reduction from (P, κ) to (P ′, κ′) is an FPT algorithm for (P, κ)
+that is equipped with oracle access to P ′ and for which there is a computable function g such
+that, on input x, each oracle query y satisfies κ′(y) ≤ g(κ(x)). We write (P, κ) ≤fpt
+T (P ′, κ′)
+if a parameterised Turing-reduction from (P, κ) to (P ′, κ′) exists. This guarantees that
+fixed-parameter tractability of (P ′, κ′) implies fixed-parameter tractability of (P, κ). For a
+more comprehensive introduction, we refer the reader the standard textbooks [14] and [19].
+Counting modulo 2 and the rETH
+The lower bounds in this work will rely on the hardness of the parameterised complexity
+class ⊕W[1], which can be considered a parameterised equivalent of ⊕P. Following [11], we
+define ⊕W[1] via the complete problem ⊕Clique: Given as input a graph G and a positive
+integer k, the goal is to compute the number of k-cliques in G modulo 2, i.e., to compute
+⊕Sub(Kk → G). The problem is parameterised by k. A parameterised problem (P, κ) is
+called ⊕W[1]-hard if ⊕Clique ≤fpt
+T (P, κ), and it is called ⊕W[1]-complete if, additionally,
+(P, κ) ≤fpt
+T ⊕Clique.
+Modifications of the classical Isolation Lemma (see e.g. [4] and [36]) yield a randomised
+parameterised Turing reduction from finding a k-clique to computing the parity of the
+number of k-cliques. In combination with existing fine-grained lower bounds for finding a
+k-clique [7, 8], it can then be shown that ⊕Clique cannot be solved in time f(k) · |G|o(k)
+for any function f, unless the randomised Exponential Time Hypothesis fails:
+
+L. A. Goldberg and M. Roth
+7
+▶ Definition 10 (rETH, [22]). The randomised Exponential Time Hypothesis (rETH) asserts
+that 3-SAT cannot be solved by a randomised algorithm in time exp o(n), where n is the
+number of variables of the input formula.
+As an immediate consequence, the rETH implies that ⊕W[1]-hard problems are not fixed-
+parameter tractable.
+For the lower bounds in this work, we won’t reduce from ⊕Clique directly, but instead
+from the following, more general problem:
+▶ Definition 11 (⊕cp-Hom). Let H be a class of graphs. The problem ⊕cp-Hom(H) has
+as input a graph H ∈ H and a surjectively H-coloured graph (G, c). The goal is to compute
+⊕Hom((H, idH) → (G, c)). The problem is parameterised by |H|.
+The following lower bound was proved independently in [27, 29] and [11].
+▶ Theorem 12. Let H be a recursively enumerable class of graphs. If the treewidth of H is
+unbounded then ⊕cp-Hom(H) is ⊕W[1]-hard and, assuming the rETH, it cannot be solved
+in time f(|H|) · |G|o(tw(H)/ log tw(H)) for any function f.
+Next is the central problem in this work.
+▶ Definition 13 (⊕Sub). Let H be a class of graphs. The problem ⊕Sub(H) has as input
+a graph H ∈ H and a graph G. The goal is to compute ⊕Sub(H → G). The problem is
+parameterised by |H|.
+For example, writing K for the set of all complete graphs, the problem ⊕Sub(K) is
+equivalent to ⊕Clique.
+Complexity Monotonicity and Inclusion-Exclusion
+Throughout this work, we will rely on two important tools introduced in [29]. For the sake
+of being self-contained, we encapsulate them below in individual lemmas.
+The first tool is an adaptation of the so-called Complexity Monotonicity principle to
+the realm of fractured graphs and modular counting (see [29, Sections 4.1 and 6.3] for a
+detailed treatment and for a proof). Intuitively, the subsequent lemma states that evaluating,
+modulo 2, a linear combination of colour-prescribed homomorphism counts from fractured
+graphs, is as hard as evaluating its hardest term with an odd coefficient.
+▶ Lemma 14 ([29]). There is a deterministic algorithm A and a computable function f such
+that the following conditions are satisfied:
+1. A expects as input a graph Q and a Q-coloured graph (G, c).
+2. A is equipped with oracle access to a function
+(G′, c′) �→
+�
+ρ
+a(ρ) · ⊕Hom((Q
+♯
+ρ, cρ) → (G′, c′))
+mod 2 ,
+where the sum is over all fractures of Q and a is a function from fractures of Q to integers.
+3. Each oracle query (G′, c′) is of size at most f(|Q|) · |G|.
+4. A computes ⊕Hom((Q
+♯
+ρ, cρ) → (G, c)) for each fracture ρ with a(ρ) ̸= 0 mod 2.
+5. The running time of A is bounded by f(|Q|) · |G|O(1).
+The second tool is a standard application of the inclusion-exclusion principle (see e.g. [29,
+Sections 4.2 and 6.3]). It will be used in the final steps of our reductions to remove the
+colourings.
+
+8
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+▶ Lemma 15 ([29]). There is a deterministic algorithm A that satisfies the following condi-
+tions:
+1. A expects as input a graph H with k edges, a graph G and a k-edge colouring γ of G.
+2. A is equipped with oracle access to the function ⊕Sub(H → ⋆), and each oracle query G′
+satisfies |G′| ≤ |G|.
+3. A computes ⊕ColSub(H → (G, γ)).
+4. The running time of A is bounded by 2|H| · |G|O(1).
+3
+Classification for Hereditary Graph Classes
+In this section, we will completely classify the complexity of ⊕Sub(H) for hereditary classes.
+Let us start by restating the classification theorem.
+▶ Theorem 4. Let H be a hereditary class of graphs. If H is matching splittable, then
+⊕Sub(H) is fixed-parameter tractable.
+Otherwise, the problem is ⊕W[1]-complete and,
+assuming rETH, cannot be solved in time f(|H|) · |G|o(|V (H)|/ log |V (H)|) for any function f.
+The proof of Theorem 4 is split in four cases, which stem from a structural property of
+non matching splittable hereditary graph classes H due to Jansen and Marx [23]. For the
+statement, we need to consider the following classes:
+Fω is the class of all complete graphs.
+Fβ is the class of all complete bipartite graphs.
+FP2 is the class of all P2-packings, that is, disjoint unions of paths with two edges.1
+FK3 is the class of all triangle packings, that is, disjoint unions of the complete graph of
+size 3.
+▶ Theorem 16 (Theorem 3.5 in [23]). Let H be a hereditary class of graphs. If H is not
+matching splittable then at least one of the following are true: (1.) Fω ⊆ H, (2.) Fβ ⊆ H,
+(3.) FP2 ⊆ H, or (4.) FK3 ⊆ H.
+Thus, it suffices to consider cases 1. - 4. to prove Theorem 4. We start with the easy
+cases of cliques and bicliques; they follow implicitly from previous works [11, 16, 27] and we
+only include a proof for completeness. Note that a tight bound under rETH is known for
+those cases:
+▶ Lemma 17. Let H be a hereditary class of graphs. If Fω ⊆ H or Fβ ⊆ H then ⊕Sub(H)
+is ⊕W[1]-hard and, assuming rETH, cannot be solved in time f(|H|) · |G|o(|V (H)|) for any
+function f.
+Proof. If Fω ⊆ H then ⊕W[1]-hardness follows immediately from the fact that ⊕Clique
+is the canonical ⊕W[1]-complete problem [11]. For the rETH lower bound, we can reduce
+from the problem of deciding the existence of a k-clique via a (randomised) reduction using a
+version of the Isolation Lemma due to Williams et al. [36, Lemma 2.1]. This reduction does
+not increase k or the size of the host graph and is thus tight with respect to the well-known
+lower bound for the clique problem due to Chen et al. [7, 8]: Deciding the existence of a
+k-clique in an n-vertex graph cannot be done in time f(k) · no(k) for any function f, unless
+ETH fails. Our lower bound under rETH follows since the reduction is randomised.
+1 To avoid confusion, we remark that [23] uses P3 to denote the path of two edges (and three vertices).
+In the current work, it will be more convenient to use the number of edges of a path as index.
+
+L. A. Goldberg and M. Roth
+9
+If Fβ ⊆ H, then the claim holds by [16, Theorem 5], which established the problem of
+counting, modulo 2, the induced copies of a k-by-k-biclique in an n-vertex bipartite graph
+to be ⊕W[1]-hard and not solvable in time f(k) · no(k) for any function f, unless rETH
+fails. Since a copy of a biclique (with at least one edge) in a bipartite graph must always be
+induced, the claim follows. This concludes the proof of Lemma 17.
+◀
+The more interesting cases are FP2 ⊆ H and FK3 ⊆ H. One reason for this is that, in
+contrast to cliques and bicliques, the decision version of those instances are fixed-parameter
+tractable. Hence a reduction from the decision version via e.g. an isolation lemma does not
+help. In other words, establishing hardness for those cases requires us to rely on the full
+power of counting modulo 2. More precisely, we will rely on the framework of fractures
+graphs (see Section 2). Both cases can be considered simpler applications of the machinery
+used in the later sections, so we will present all steps in great detail. While this might seem
+unnecessary given the simplicity of the constructions, we hope that it enables the reader to
+make themselves familiar with the general reduction strategies which will be used throughout
+the later sections of this work.
+3.1
+Triangle Packings
+The goal of this subsection is to establish hardness of ⊕Sub(FK3). To this end, let ∆ be an
+infinite computable class of cubic bipartite expander graphs, and let Q = {L(H) | H ∈ ∆}
+where L(H) is constructed as follows: Each v ∈ V (H) becomes a triangle with vertices vx,
+vy, and vz corresponding to the three neighbours x, y, and z of v. Finally, for every edge
+{u, v} ∈ E(H) we identify vu and uv. In fact, L(H) is just the line graph of H: Every edge of
+H becomes a vertex in L(H), and two vertices of L(H) are made adjacent if and only if the
+corresponding edges in H are incident. Since all H ∈ ∆ are bipartite (and thus triangle-free),
+we can easily observe the following.2
+▶ Observation 18. The mapping v �→ (vx, vy, vz) is a bijection from vertices of H to triangles
+in L(H).
+We also consider the fracture of L(H) that splits L(H) back into |V (H)| triangles; consider
+Figure 2 for an illustration.
+▶ Definition 19 (τ(H)). Let H ∈ ∆ and recall that each vertex w of L(H) is obtained by
+identifying vu and uv for some edge {u, v} ∈ E(H). Moreover, w has four incident edges
+ex, ey, ea, eb, to vx, vy, ua, ub, respectively, where x, y, u are the neighbours of v in H and
+v, a, b are the neighbours of u in H. We define τ(H)w := {{ex, ey}, {ea, eb}}, and we proceed
+similar for all vertices of L(H).
+Next, we use that tw(L(H)) = Ω(tw(H)) (see e.g. [21]). Moreover, tw(L(H)) ≤ |V (L(H))|
+since the treewidth of a graph is always bounded by the number of its vertices. Additionally,
+|V (L(H))| = |E(H)| by construction. Since the graphs in ∆ are cubic, we further have that
+|E(H)| = Θ(|V (H)|) for H ∈ ∆. We combine those bounds with the fact that expander
+graphs have treewidth linear in the number of vertices (see e.g. [20]); therefore ∆ and thus
+Q have unbounded treewidth. Putting these facts together, we obtain the following.
+▶ Fact 20. Q has unbounded treewidth and tw(L(H)) = Θ(|V (L(H))|) = Θ(|V (H)|) for
+H ∈ ∆.
+2 Observation 18 is also an immediate consequence of Whitney’s Isomorphism Theorem implying that a
+triangle of a line graph corresponds to either a claw or to a triangle in its primal graph.
+
+10
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+We are now able to establish hardness of ⊕Sub(FK3). The proof will heavily rely on the
+transformation from edge-coloured subgraphs to homomorphisms established in [29].
+▶ Lemma 21. The problem ⊕Sub(FK3) is ⊕W[1]-hard. Furthermore, on input kK3 and G,
+the problem cannot be solved in time f(k) · |G|o(k/ log k) for any function f, unless rETH fails.
+Proof. We reduce from ⊕cp-Hom(Q), which, by Fact 20 and Theorem 12, is ⊕W[1]-hard
+and for L(H) ∈ Q, it cannot be solved in time f(|L(H)|) · |G|o(|V (L(H))|/ log |V (L(H))|), unless
+rETH fails.
+Let L and (G, c) be an input instance to ⊕cp-Hom(Q). Recall that ∆ is computable —
+that is, there is an algorithm that takes a graph H and determines whether it is in ∆. Thus,
+there is an algorithm that takes input L ∈ Q and finds a graph H ∈ ∆ with L = L(H). The
+run time of this algorithm depends on |L| but clearly not on (G, c). Let k = |V (H)| and
+note that |E(L(H))| = 3k, since, by construction, each vertex v of H becomes a triangle of
+L(H). We consider the graph G as a 3k-edge-coloured graph, coloured by cE. That is, each
+edge e = {x, y} of G is assigned the colour cE(e) = {c(x), c(y)} which is an edge of L (see
+Figure 2 for an illustration).
+Now, for any L-coloured graph (G′, c′) recall that ColSub(kK3 → (G′, c′
+E)) is the set of
+subgraphs of G′ that are isomorphic to kK3 and that include each edge colour (each edge of
+L) precisely once. We will see later that ⊕ColSub(kK3 → (G′, c′
+E)) can be computed using
+our oracle for ⊕Sub(FK3) using the principle of inclusion and exclusion.
+It was shown in [29, Lemma 4.1] that there is a unique function a such that for every
+L-coloured graph (G′, c′) we have3
+#ColSub(kK3 → (G′, c′
+E)) =
+�
+ρ
+a(ρ) · Hom(L
+♯
+ρ → (G′, c′)) .
+(2)
+where the sum is over all fractures of L. Additionally, it was shown in [29, Corollary 4.3]
+that
+a(⊤) =
+�
+ρ∈F(kK3,L)
+�
+w∈V (L)
+(−1)|ρw|−1 · (|ρw| − 1)! ,
+(3)
+where ⊤ is the fracture in which each partition consists only of one block (that is, L
+♯
+⊤ = L),
+and F(kK3, L) is the set of all fractures ρ of L such that L
+♯
+ρ ∼= kK3. However, note that,
+by Observation 18, there is only way to fracture L into k disjoint triangles, and this fracture
+is given by τ(H). Thus, (3) simplifies to
+a(⊤) =
+�
+w∈V (L)
+(−1)|τ(H)w|−1 · (|τ(H)w| − 1)! ,
+(4)
+which is odd since each partition of τ(H) consists of precisely two blocks (so in fact the
+expression in (4) is (−1)|V (L)|).
+Note that the algorithm for ⊕cp-Hom(Q) is supposed to compute ⊕Hom((L, idL) → (G, c))
+which is equal to ⊕Hom(L
+♯
+⊤ → (G, c⊤)). Since a(⊤) is odd, we can invoke Lemma 14 to
+recover this term by evaluating the entire linear combination (2), that is, by evaluating
+the function ⊕ColSub(kK3 → ⋆). More concretely, this means that we need to compute
+⊕ColSub(kK3 → (G′, c′
+E)) for some L-coloured graphs (G′, c′) of size at most f(|L|) · |G| for
+3 In the language of [29], Equation (2) is obtained by choosing Φ as the property of being isomorphic
+to kK3.
+
+L. A. Goldberg and M. Roth
+11
+Figure 2 (Top:) A cubic bipartite graph H ∈ ∆, its line graph L(H), and the fractured graph
+induced by τ(H). (Below:) An L(H)-coloured graph (G, c); emphasised in distinct colours is the
+edge-colouring cE of G induced by the mapping {u, v} �→ {c(u), c(v)}. Additionally we depict an
+element S ∈ ColSub(kK3 → (G, cE)), that is, a subgraph of G isomorphic to kK3 that contains each
+edge colour of G precisely once.
+
+12
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+some computable function f (see 3. in Lemma 14). This can easily be done using Lemma 15
+since we have oracle access to the function ⊕Sub(kK3 → ⋆). We emphasise that, by condition
+2. of Lemma 15, each oracle query ˆG satisfies | ˆG| ≤ |G′|, where (G′, c′) is the L-coloured
+graph for which we wish to compute ⊕ColSub(kK3 → (G′, c′
+E)). Since |(G′, c′)| ≤ f(|L|) · |G|,
+we obtain that | ˆG| ≤ f(|L|) · |G| as well.
+Since, by Fact 20, k = Θ(|kK3|) = Θ(|V (L)|) = Θ(tw(L)), our reduction yields ⊕W[1]-
+hardness and transfers the conditional lower bound under rETH as desired.
+◀
+3.2
+P2-packings
+Next we establish hardness for the case of P2-packings. The strategy will be similar in spirit
+to the construction for triangle packings; however, rather then identifying a unique fracture
+for which the technique applies, we will encounter an odd number of possible fractures in the
+current section.
+Let ∆ be a computable infinite class of 4-regular expander graphs, and let Q be the class
+of all subdivisions of graphs in ∆, that is Q = {H2 | H ∈ ∆}, where H2 is obtained from H
+by subdividing each edge once.
+We start by establishing an easy but convenient fact on the treewidth of the graphs in Q.
+▶ Lemma 22. Q has unbounded treewidth and tw(H2) = Θ(|V (H)|) for H ∈ ∆.
+Proof. As in Section 3.1, tw(H) = Θ(|V (H)|) for H ∈ ∆, since expanders have treewidth
+linear in the number of vertices. Since H is a minor of H2, and since taking minors cannot
+increase treewidth (see [14, Exercise 7.7]), we thus have that tw(H2) = Ω(|V (H)|)). Finally,
+we have tw(H2) ≤ |V (H2)| since the treewidth is at most the number of vertices, and
+|V (H2)| = O(|V (H)|) since H is 4-regular. In combination, we obtain tw(H2) = Θ(|V (H)|)
+for H ∈ ∆. Note that this also implies that Q has unbounded treewidth (as ∆ is infinite).
+◀
+For what follows, given a subdivision H2 of a graph H, it will be convenient to assume
+that V (H2) = V (H) ∪ SE, where SE = {se | e ∈ E(H}) is the set of the subdivision vertices.
+▶ Definition 23 (Odd Fractures). Let H ∈ ∆ and let τ be a fracture of H2. We say that τ is
+odd if the following two conditions are satisfied:
+1. For each s ∈ SE the partition τs consists of two singleton blocks.
+2. For each v ∈ V (H) the partition τv consists of two blocks of size 2.
+Consider Figure 3 for a depiction of an odd fracture.
+The following two lemmas are crucial for our construction.
+▶ Lemma 24. Let H ∈ ∆. The number of odd fractures of H2 is odd.
+Proof. The first condition in Definition 23 leaves only one choice for subdivision vertices.
+Let us thus consider a vertex v ∈ V (H) = V (H2) \ SE. Since H is 4-regular, there are 4
+incident edges to v. Now note that there are precisely 3 partitions of a 4-element set with two
+blocks of size 2. Thus the total number of odd fractures of H2 is 3|V (H)|, which is odd.
+◀
+▶ Lemma 25. Let H ∈ ∆, let k = 2|V (H)| and let τ be a fracture of H2 such that τv consists
+of at most 2 blocks for each v ∈ V (H2). Then H2
+♯
+τ ∼= kP2 if and only if τ is odd.
+Proof. First observe that |E(H2)| = 2|E(H)| = 4|V (H)| = 2k. Thus the number of edges of
+H2
+♯
+τ is equal to 2k (for each fracture τ of H2), which is also equal to the number of edges
+of kP2.
+
+L. A. Goldberg and M. Roth
+13
+Figure 3 (Top:) Subdividing a 4-regular expander in ∆ depicted by the neighbourhood of an
+individual vertex. (Centre:) Illustrations of odd fractures (Definition 23). For each non-subdivision
+vertex, there are only three ways to satisfy 2. in Definition 23. This observation is used in Lemma 24 to
+show that the number of odd fractures is a power of 3. (Bottom:) Elements of ColSub(kP2 → (G, cE))
+inducing fractures of H2 such that each partition has at most two blocks. Lemma 25 shows that
+those are precisely the odd fractures of H2.
+
+14
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+Thus, H2
+♯
+τ is isomorphic to kP2 if and only if each connected component of H2
+♯
+τ is
+a path of length 2. It follows immediately by Definition 23 that τ being odd implies that
+H2
+♯
+τ consists only of disjoint P2. It thus remains to show the other direction.
+Assume for contradiction that there is a subdivision vertex s ∈ SE of H2 such that τs
+consists of only one block (recall that s has degree 2, thus τs either consists of two singleton
+blocks, or of one block of size 2). Let e = {u, v} ∈ E(H) be the edge corresponding to s, that
+is, s was created by subdividing e. Since H2
+♯
+τ is a union of P2, we can infer that τv and τu
+contain a singleton block (otherwise we would have created a connected component which is
+not isomorphic to P2). Now recall that both u and v have degree 4, since H is 4-regular. We
+obtain a contradiction as follows: By assumption of the lemma, we know that τv and τu can
+have at most two blocks. Since we have just shown that both contain a singleton block, it
+follows that both τv and τu contain one further block of size 3. However, a block of size 3
+yields a vertex of degree 3 in the fractured graph H2
+♯
+τ, contradicting the fact that H2
+♯
+τ
+consists only of disjoint P2.
+Thus we have established that, for each s ∈ SE, the partition τs consists of two singleton
+blocks. Given this fact, the only way for H2
+♯
+τ being a disjoint union of P2 is that each
+partition τv, for v ∈ V (H) = V (H2) \ SE, consists of two blocks of size 2.
+◀
+We are now able to prove our hardness result.
+▶ Lemma 26. The problem ⊕Sub(FP2) is ⊕W[1]-hard. Furthermore, on input kP2 and G,
+the problem cannot be solved in time f(k) · |G|o(k/ log k) for any function f, unless rETH fails.
+Proof. We reduce from ⊕cp-Hom(Q), which, by Lemma 22 and Theorem 12, is ⊕W[1]-hard
+and for H′ ∈ Q, it cannot be solved in time f(|H′|) · |G|o(|V (H′)|/ log |V (H′)|), unless rETH
+fails.
+Let H′ and (G, c) be an input instance to ⊕cp-Hom(Q). There is an algorithm that
+takes as input a graph H′ ∈ Q and finds a graph H ∈ ∆ with H′ = H2 — this is basically
+2-colouring. The run time of this algorithm depends on |H′| but clearly not on (G, c). Let
+k = 2|V (H)| and note that |E(H2)| = 2|E(H)| = 4|V (H)| = 2k. We consider the graph G
+as a 2k-edge-coloured graph, coloured by cE. That is, each edge e = {x, y} of G is assigned
+the colour cE(e) = {c(x), c(y)} which is an edge of H′ = H2.
+Now, for any H2-coloured graph (G′, c′) recall that ColSub(kP2 → (G′, c′
+E)) is the set of
+subgraphs of G′ that are isomorphic to kP2 and that include each edge colour (each edge of
+H2) precisely once. We will see later that ⊕ColSub(kP2 → (G′, c′
+E)) can be computed using
+our oracle for ⊕Sub(FP2) using the principle of inclusion and exclusion.
+It was shown in [29, Lemma 4.1] that there is a unique function a such that, for every
+H2-coloured graph (G′, c′),
+#ColSub(kP2 → (G′, c′
+E)) =
+�
+ρ
+a(ρ) · Hom(H2
+♯
+ρ → (G′, c′)) .
+(5)
+where the sum is over all fractures of H2. As in Section 3.1 from [29, Corollary 4.3] we know
+that
+a(⊤) =
+�
+ρ∈F(kP2,H2)
+�
+w∈V (H2)
+(−1)|ρw|−1 · (|ρw| − 1)! ,
+(6)
+where ⊤ is the fracture in which each partition consists only of one block and F(kP2, H2) is
+the set of all fractures ρ of H2 such that H2
+♯
+ρ ∼= kP2.
+Our next goal is to show that a(⊤) = 1 mod 2. First, suppose that a fracture ρ contains
+a partition ρw with at least three blocks. Then (|ρw| − 1)! = 0 mod 2. Thus such fractures
+
+L. A. Goldberg and M. Roth
+15
+do not contribute to a(⊤) if arithmetic is done modulo 2. Next, note that if, for each w, the
+partition ρw contains at most 2 blocks, then
+�
+w∈V (H2)
+(−1)|ρw|−1 · (|ρw| − 1)! = 1
+mod 2.
+Let Odd(kP2, H2) be the set of all fractures ρ of H2 such that H2
+♯
+ρ ∼= kP2 and each
+partition of ρ consists of at most 2 blocks. Our analysis then yields a(⊤) = |Odd(kP2, H2)|
+mod 2. Finally, Lemma 25 states that Odd(kP2, H2) is precisely the set of odd fractures, and
+Lemma 24 thus implies that |Odd(kP2, H2)| = 1 mod 2. Consequently, a(⊤) = 1 mod 2 as
+well, and we have achieved the goal.
+Next we can proceed similarly to the case of triangle packings. As in that case, the goal
+is to compute ⊕Hom((H2, idH2) → (G, c))) which is equal to ⊕Hom((H2
+♯
+⊤, c⊤) → (G, c)).
+Since a(⊤) is odd, we can invoke Lemma 14 to recover this term by evaluating the entire
+linear combination (5), that is, if we can evaluate the function ⊕ColSub(kP2 → ⋆). This can
+be done by using Lemma 15. Each call to the oracle is of the form ⊕Sub(kP2 → ˆG) where
+| ˆG| is bounded by f(k) · |G|.
+Now recall that k ∈ Θ(|V (H)|). By Lemma 22, we thus have k = Θ(tw(H2)). Hence our
+reduction yields ⊕W[1]-hardness and transfers the conditional lower bound under rETH as
+desired.
+◀
+We can now conclude the treatment of hereditary pattern classes by proving Theorem 4,
+which we restate for convenience.
+▶ Theorem 4. Let H be a hereditary class of graphs. If H is matching splittable, then
+⊕Sub(H) is fixed-parameter tractable.
+Otherwise, the problem is ⊕W[1]-complete and,
+assuming rETH, cannot be solved in time f(|H|) · |G|o(|V (H)|/ log |V (H)|) for any function f.
+Proof. The fixed-parameter tractability result was shown in [11]. For the hardness result,
+using the fact that H is not matching splittable and Theorem 16 we obtain four cases.
+If H contains all cliques or all bicliques, then hardness follows from Lemma 17.
+If H contains all triangle packings, then hardness follows from Lemma 21.
+If H contains all P2-packings, then hardness follows from Lemma 26.
+Since the case distinction is exhaustive, the proof is concluded.
+◀
+4
+Classification for Trees
+Our overall goal is to prove Theorem 5, which we restate for convenience:
+▶ Theorem 5. Let T be a recursively enumerable class of trees. If T is matching splittable,
+then ⊕Sub(T ) is fixed-parameter tractable. Otherwise ⊕Sub(T ) is ⊕W[1]-complete.
+We start by introducing some terminology for trees which will be used in the remainder
+of this section.
+▶ Definition 27 (2-paths). A 2-path of length a of a tree T is a path x0, x1, . . . , xa such that
+deg(x0) ̸= 2, deg(x1) = · · · = deg(xa−1) = 2 and deg(xa) ̸= 2.
+Next we introduce rays, which are restricted 2-paths that will be crucial in our analysis.
+
+16
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+▶ Definition 28 (source, ray, degL,a, degL, and degNL). Let T be a tree. A source of T is any
+vertex with degree greater than 2. A ray of length a of T is a 2-path x0, x1, . . . , xa such that
+deg(x0) > 2 and deg(xa) = 1. We call x0 the source of the ray. Given a vertex s of degree
+at least 3, we write degL,a(s) for the number of rays of length a with source s. We set
+degL(s) :=
+�
+a
+degL,a(s) .
+Finally, we set degNL(s) := deg(s) − degL(s).
+Next, we introduce parameters Fa,b, Sc and Cd. Our goal is then to show that, for every
+non-matching-splittable class of trees, at least one of those two parameters is unbounded.
+▶ Definition 29 (Forks and Fa,b). Let a, b be positive integers. A source s of a tree T is
+called an a-b-fork if degNL(s) = 1 and one of the following is true
+a ̸= b and degL,a(s), degL,b(s) > 0.
+a = b and degL,a(s) > 1.
+The a-b-fork number of T, denoted by Fa,b(T) is the maximum size of an independent set
+containing only a-b-forks. Finally, we say that a class of trees T has unbounded fork number
+if for every positive integer B there are positive integers a and b and a tree T ∈ T such that
+Fa,b(T) ≥ B.
+▶ Definition 30 (Stars and Sc). A star of size k > 1 in a tree T is a collection of k distinct
+rays that have a common source s. For a positive integer c ≥ 3, a c-star of size k in a tree T
+is a collection of k distinct rays of length c that have a common source s.
+The c-star number of a tree T, denoted by Sc(T) is the maximum size of a c-star in
+T. Finally, we say that a class of trees T has unbounded star number if for every positive
+integer B there exists c ≥ 3, and a tree T ∈ T such that Sc(T) ≥ B.
+▶ Definition 31 (C-gadgets and Cd). A C-gadget4 of order d and length k in a tree T is a
+path x0, . . . , xk such that one of the following is true for each inner vertex xi ∈ {1, . . . , k −1}:
+(i)
+deg(xi) = 2, that is N(xi) = {xi−1, xi+1}, or
+(ii)
+xi is a source and every neighbour v ∈ N(xi)\{xi−1, xi+1} is contained in a ray of length
+at most d from xi to a leaf.
+The Cd-number of a tree T, denoted by Cd(T) is the length of the longest C-gadget of order d.
+Finally, we say that a class of trees T has unbounded C-number if there exists d > 0 such
+that for every positive integer B, and a tree T ∈ T such that Cd(T) ≥ B.
+Note that the ordering of the quantifiers in the definition of the Cd-number is different from
+the ordering in the definition of the c-star-number. This is due to technical reasons which
+are important for the proof of Lemma 32.
+▶ Lemma 32. Let T be a class of trees. If T is not matching splittable, then T has either
+unbounded fork number, unbounded star number, or unbounded C-number.
+Proof. We can assume that there is an overall bound d on the length of 2-paths in trees in
+T : Otherwise, T already has unbounded C-number (see (i) in Definition 31)). Hence the
+length of every ray in any tree in T is bounded by d as well. Thus
+T has unbounded fork number if and only if for every positive integer B there are
+a, b ∈ {1, . . . , d} and a tree T ∈ T such that Fa,b(T) ≥ B.
+4 C stands for caterpillar, the shape of which resembles the structure of a C-gadget.
+
+L. A. Goldberg and M. Roth
+17
+T has unbounded C-number if and only if Cd is unbounded in T (see Definition 31)).
+T has unbounded star number if and only if for every positive integer s there is a
+c ∈ {3, . . . , d} and a tree T ∈ T such that Sc(T) ≥ s.
+We split the proof into two cases.
+Case 1. T has unbounded diameter.
+In Case 1, we show that T has unbounded fork number or unbounded C-number. If Cd
+is unbounded in T then T has unbounded C-number and we are done so assume that there
+is a constant h such that Cd(T) ≤ h for every T ∈ T .
+Now let B be a positive integer. We show that there are a, b ∈ {1, . . . , d} and T ∈ T
+with Fa,b(T) ≥ B. To this end, we use the premise that T has unbounded diameter. Let
+k > (h + 2)(Bd2 + 1) be a positive integer, and let T ∈ T be such that there is a path
+P = s, p0, . . . , pk, t in T. Observe that the deletion of all edges in P decomposes T into a
+family of disjoint subtrees. We write Ti for the subtree that contains pi. Now decompose P
+into segments P1, P2, . . . of length h + 2. Note that a segment Pj = pj0, . . . , pjh+2 yields a
+C-gadget of order d and length > h if and only if Tji is either a star or an isolated vertex for
+each i ∈ {1, . . . , h + 1}.
+Since no such C-gadgets exist by assumption, we obtain that each segment Pj of the path
+P contains a vertex pij such that Tij is neither a star nor an isolated vertex.
+Assume that Tij is rooted at pij. Since Tij is neither an isolated vertex nor a star, there
+must be a (proper) descendant vij of pij (in Tij) such that vij is an (aij, bij)-fork for some
+aij, bij ∈ {1, . . . , d}. Now note that there are at most d2 pairs of integers in {1, . . . , d}. Since
+we have at least one fork for every segment and since there are at least ⌊k/(h + 2)⌋ > Bd2 + 1
+segments, we thus obtain by the pigeon-hole principle that there is a pair a, b ∈ {1, . . . , d}
+such that, for at least B segments Pij, the node vij is an (a, b)-fork in Tij and thus also in
+T. Since those forks are pairwise non-adjacent, we obtain, as desired, that the (a, b)-fork
+number of T is at least B, concluding Case 1.
+Case 2. T has bounded diameter.
+Let D be the assumed upper bound on the diameter of trees in T . If T has unbounded
+star number then we are finished. Assume instead that T has bounded star number. Then
+there is a positive integer s such that for all c ∈ {3, . . . , d} and every tree T ∈ T , Sc(T) < s.
+We will show that T has unbounded fork number. Consider any positive integer B. We will
+show that there are a, b ∈ {1, . . . , d} and T ∈ T with Fa,b(T) ≥ B.
+Let k > (D+1)(Bd2 +1)(d2s+1) be a positive integer. Since T is not matching splittable,
+there is a tree T ∈ T whose matching-split number is at least k. Note that T is not a
+path since every path with matching-split number at least k has length greater than k > D,
+contradicting the bound on the diameter.
+Now fix any vertex r of T as the root. Given a vertex v of T, we write Tv for the subtree
+rooted at v (assuming that r is the overall root). We call v a rooted fork if Tv is a star —
+observe that each rooted fork must indeed be a fork. Let f be the number of rooted forks.
+Similar to the argument in Case 1, if f > Bd2 + 1, then by the pigeon-hole principle there
+are a, b ∈ {1, . . . , d} such that Fa,b(T) ≥ B.
+Hence assume for contradiction that f ≤ Bd2 + 1. Let R be the set of all rays of T and
+recall that each ray in R is, by definition, a 2-path of the form R = x0, . . . , xd′ for d′ ≤ d,
+where deg(x0) > 2 and xd′ is a leaf. We call a ray R long if d′ ≥ 3. Note that the source of
+every ray must either be a rooted fork, or it must lie on a path from the root r to one of the
+rooted forks.
+Let T ′ be the subtree of T induced by all vertices that lie on paths between r and a
+rooted fork (including r and all rooted forks). Since there are f rooted forks and the depth
+
+18
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+of T is bounded by D, |V (T ′)| ≤ (D + 1)f ≤ (D + 1)(Bd2 + 1).
+Consider a vertex v of T ′. Assume for contradiction that v is the source of > ds long
+rays (in T). Recall that for all c ∈ {3, . . . , d} we have that Sc(T) < s. Recall further that
+each long ray has length d′ for some 3 ≤ d′ ≤ d. Thus we obtain a contradiction by the
+pigeon-hole principle.
+Now let S be the set containing all vertices of T ′ and all vertices of long rays. Noting
+that each long ray has length at most d, and that the source of each long ray must be a
+vertex of T ′ by construction, we can use the observation that each vertex of T ′ is the source
+of at most ds long rays to (generously) bound
+|S| ≤ |V (T ′)| + |V (T ′)| · d · ds .
+Note further that T[V (T) \ S] consists only of isolated edges and vertices: The only vertices
+in V (T) \ S are non-source vertices of rays of length < 3, the sources of which are in T ′.
+Thus, S is a splitting set. Finally, recalling that |V (T ′)| ≤ (D + 1)f ≤ (D + 1)(Bd2 + 1), we
+have
+|S| ≤ |V (T ′)| + |V (T ′)| · d · ds ≤ (D + 1)(Bd2 + 1)(d2s + 1) ,
+contradicting the fact that the matching-split number of T is strictly larger than (D +
+1)(Bd2 + 1)(d2s + 1). This concludes Case 2, and hence the proof.
+◀
+In the next three subsections, we will prove hardness of ⊕Sub(T ) for non-matching-
+splittable T in each of the three cases given by Lemma 32.
+4.1
+Unbounded C-number
+For our hardness proof, it will be useful to find a proper sub-gadget of a C-gadget in a tree.
+▶ Definition 33 (Strong C-gadgets, junctions, and closedness). Let C = x0, . . . , xL be a
+C-gadget of order d and length L in a tree T. We call C a strong C-gadget with k junctions
+if there are integers 0 = i0 < i1 < · · · < ik < ik+1 = L such that
+(I)
+for all j ∈ {0, . . . , k}, ij+1 − ij > 2d, and
+(II)
+for all j ∈ {1, . . . , k}, xij is the source of a ray Rj of length d that does not contain one of
+the neighbours xij−1 and xij+1 of xij. The vertices xi1, . . . , xik are called the junctions.
+Finally, a strong C-gadget is called closed if neither xi1 nor xik are forks.5
+Consider the bottom part of Figure 4 for a visualisation. We start with the following lemma
+which establishes the existence of a strong C-gadget with many junctions inside a long enough
+C-gadget.
+▶ Lemma 34. Let T be a tree such that the longest 2-path in T has length d ≥ 1, and let
+k be a positive integer. Then there exists L > 0 (only depending on k and d) such that
+the following is true: If T contains an C-gadget of order d and length L, then there exists
+1 ≤ d′ ≤ d such that T contains a strong C-gadget of order d′ with at least k junctions.
+Proof. Let f(x) = x/(k + 1) − 2d − 1 and let L be large enough such that f d(L) > d. Let
+Hd = x0, . . . , xL be a C-gadget of order d and length L in T.
+Let d′ = d. Note that Hd′ is a C-gadget of order d′ and length at least L = f d−d′(L) in
+T. For each graph Hd′ with d′ ≥ 1 we will either
+5 The condition of being closed rules out the special case in which x0 or xL are leaves of T. More generally
+it rules out the case where there is a ray from x1 including x0 or from xk including xL.
+
+L. A. Goldberg and M. Roth
+19
+(1) construct a strong C-gadget with k junctions with order d′, or
+(2) find a subsequence Hd′−1 of Hd′ that is an C-gadget of order d′ − 1 of length at least
+f d−(d′−1)(L).
+If we ever do (1) we are finished. If from d′ = 1 we do (2) then we find a 2-path of length at
+least f d(L) > d, which is a contradiction.
+Here is how we proceed from Hd′ = y0, . . . , yℓ. We set i0 = 0. Then iteratively, for each
+j ∈ {1, . . . , k} we will either construct Hd′−1 as in (2) or we find ij ∈ {ij−1 + 2d + 1, . . . , ℓ}
+such that yij is the source of a length-d′ ray that does not contain yij − 1 or yij + 1. If we
+succeed in defining i1, . . . , ik, ik+1 in this way then y0, . . . , yik+1 is a strong C-gadget with k
+junctions of order d′ so (1) is satisfied.
+Let us now make this argument rigorous; again, assume that Hd′ = y0, . . . , yℓ is a C-
+gadget of order d′ and length ℓ ≥ f d−d′(L). Set i0 = 0 and, starting with j = 0, proceed
+iteratively as follows:
+1. Let Sj be the set of all indices i ∈ {ij−1 + 2d + 1, . . . , ℓ} such that yi is the source of a
+length-d′ ray that does not contain yi−1 and yi+1.
+2. If Sj = ∅ then set stop = j and terminate. Otherwise, set ij = min Sj and j ← j + 1, and
+go back to 1.
+We now distinguish two cases: If stop ≥ k + 1, then we found indices i0, . . . , ik+1 such
+that ˆHd′ := y0, . . . , yik+1 is a strong hardness gadget of order d′ with k junctions; hence we
+achieved (1) and we are done. Otherwise we have stop < k + 1. Let Ij := {ij, . . . , ij+1 − 1}
+for all 0 ≤ j < stop, and let Istop = {istop, . . . , ℓ}. By the pigeon-hole principle, at least one
+of those intervals, say Ij′, has size at least ℓ/(stop + 1) ≥ ℓ/(k + 1). Now, by construction of
+our iterative procedure above, we find that the sub-interval {ij′ + 2d + 1, . . . , ij′+1 − 1} ⊆ Ij′
+contains no index i such that yi is the source of a length-d′ ray that does not contain yi−1
+and yi+1. Thus, the subsequence Hd′−1 := yij′+2d+1, . . . , yij′+1−1 constitutes a C-gadget
+of order d′ − 1. Furthermore, Hd′−1 has length at least ℓ/(k + 1) − 2d − 1 = f(ℓ). Since
+ℓ ≥ f d−d′(L), and since f is monotonically increasing, we find that f(ℓ) ≥ f d−(d′−1)(L).
+Hence we achieved (2) and we can conclude this case as well.
+◀
+Now, by removing the first and the last junction, we can also ensure the existence of a
+closed strong C-gadget
+▶ Corollary 35. Let T be a tree such that the longest 2-path in T has length d ≥ 1, and
+let k be a positive integer. Then there exists L > 0 (only depending on k and d) such that
+the following is true: If T contains an C-gadget of order d and length L, then there exists
+1 ≤ d′ ≤ d such that T contains a closed strong C-gadget of order d′ with at least k junctions.
+Proof. Use Lemma 34 with k + 2 rather than k and observe that every strong C-gadget with
+k + 2 junctions also yields a closed strong C-gadget with k junctions by removing i1 and
+ik+2 from the list of indices. Since xi1 and xik+2 must have degree at least 3 (they are inner
+vertices of a C-gadget and they are junctions), we obtain that neither xi2 and xik+1 can be
+forks of T.
+◀
+4.1.1
+Constructions of Q and ˆG
+For the scope of this subsection, to avoid notational clutter, we assume the following are
+given:
+Positive integers k and d.
+
+20
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+A tree T that contains a closed strong C-gadget H = x0, . . . , xℓ of order d with k junctions
+xi1, . . . , xik. Additionally, for each j ∈ [k], we fix a ray Rj = xij, r1
+j, . . . , rd
+j of length d,
+the source of which is xij and which does not contain one of the neighbours xij−1 and
+xij+1 — note that the Rj must exist as the xij are junctions.
+A k-vertex cubic graph ∆ containing a Hamiltonian cycle v1, . . . , vk, v1.
+We emphasise that the set of edges of ∆ not contained in the Hamilton cycle must
+constitute a perfect matching, that is, a set of k/2 pairwise non-incident edges. This must
+be satisfied since ∆ is cubic.
+▶ Definition 36. The core of H, denoted by C(H), contains the subsequence xi1, xi1+1, . . . , xik−1, xik
+and the vertices of the rays Rj, that is
+C(H) := {xi1, xi1+1, . . . , xik−1, xik} ∪
+k�
+j=1
+V (Rj) .
+▶ Definition 37 (Q(∆, T, H) and τQ). Set ℓj := ij+1 − ij. The graph Q = Q(∆, T, H) is
+obtained from ∆ as follows:
+1. The edge {vk, v1} is deleted.
+2. For each j ∈ {1, . . . , k − 1} the edge {vj, vj+1} is replaced by a path of length ℓj:
+Pj = vj, u1
+j, . . . , uℓj−1
+j
+, vj+1 ,
+where the ut
+j are fresh vertices.
+3. Each edge e = {vi, vj} not contained on the Hamilton cycle, i.e., j /∈ {i − 1, i + 1}, is
+replaced by a path Pi,j of length 2d:
+Pi,j = vi, w1
+i , . . . , wd−1
+i
+, m(e), wd−1
+j
+, . . . , w1
+j, vj ,
+where the wt
+i and wt
+j are fresh vertices.
+Finally τ = τ(∆, T, H) is a fracture of Q defined as follows: For each m(e), the partition
+τm(e) contains two singleton blocks, and for all remaining vertices v of Q the partition τv
+only contains one block.
+Since ∆, T and H are fixed in this subsection, to avoid notational clutter, we just write Q
+and τ, rather than Q(∆, T, H) and τ(∆, T, H).
+It turns out that Q is isomorphic to a quotient graph of T[C(H)] obtained by identifying
+the endpoints of the rays Ri and Rj for every {vi, vj} ∈ E(∆) with j /∈ {i − 1, i + 1}. This
+induces a homomorphism from T[C(H)] to Q that will be useful in the construction of ˆG;
+hence we explicitly define this mapping below:
+▶ Definition 38 (γ). We define a function γ : C(H) → V (Q) as follows.
+1. We map the sequence xi1, xi1+1, . . . , xik−1, xik in C(H) to the sequence v1, . . . , vk in Q.
+More precisely, for each j ∈ {1, . . . , k − 1} and t ∈ {1, . . . , ℓj − 1}, we set γ(xij) := vj
+and γ(xij+t) := ut
+j.
+2. For each edge e = {vi, vj} of ∆ with j /∈ {i − 1, i + 1}, we map V (Ri) and V (Rj) to the
+path Pi,j. More precisely, for each t ∈ {1, . . . , d − 1} we set γ(rt
+i) := wt
+i and γ(rt
+j) = wt
+j.
+Furthermore, we set γ(rd
+i ) := m(e) =: γ(rd
+j ). (Note that the images of the sources of the
+rays Ri and Rj are already set in 1.)
+▶ Observation 39. The function γ is an edge-bijective homomorphism from T[C(H)] to Q.
+Let us provide the induced egde-bijection explicitly:
+
+L. A. Goldberg and M. Roth
+21
+▶ Definition 40. (E′, γE) Define E′ := E(T[C(H)]), that is, E′ ⊆ E(T) contains all
+edges on the sub-path xi1, . . . , xik of H and all edges of the rays R1, . . . , Rk. We write
+γE : E′ → E(Q) for the edge-bijection from E′ to E(Q) induced by the homomorphism γ.
+Now let (G, c) be a Q-coloured graph. We state the following fact explicitly, since it will
+be crucial in our construction:
+▶ Observation 41. Let (G, c) be a Q-coloured graph. The mapping cE ◦ γ−1
+E
+is a map from
+E(G) to E′.
+Our goal is to construct a graph ˆG = ˆG(G, c, T, H) from G, and an edge-colouring ˆγ :
+E( ˆG) �→ E(T) whose range is E(T) such that
+⊕Emb((Q
+♯
+τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)),
+that is, the number of colour-preserving embeddings from the fractured graph Q
+♯
+τ to (G, c)
+is equal, modulo 2, to the number of subgraphs of ˆG that are isomorphic to T and that
+contain each edge-colour in E(T) precisely once.
+For what follows, let V (R) := ∪k
+j=1V (Rj) be the set of all vertices of the rays R1, . . . , Rk.
+We are now able to define ˆG = ˆG(G, c, T, H); the construction is illustrated in Figure 4.
+The definition uses the function cE introduced in Definition 6 and the functions γ and γE
+introduced in Definitions 38 and 40, respectively. It also uses the mapping cE ◦ γ−1
+E
+from
+E(G) to E′ (see Observation 41).
+▶ Definition 42 ( ˆG(G, c, T, H), ˆγ(G, c, T, H)). Let (G, c) be a Q-coloured graph. The pair
+( ˆG, ˆγ) = ( ˆG(G, c, T, H), ˆγ(G, c, T, H)) is an edge-coloured graph constructed as follows, where
+the co-domain of ˆγ is E(T):
+(A)
+The graph ˆG contains G as a subgraph. For each e ∈ E(G), define ˆγ(e) = γ−1
+E (cE(e)).
+(B)
+The vertex set of ˆG is the union of V (G) and V (T) \ C(H).
+(C)
+Pairs of vertices in V (T) \ C(H) are connected by an edge in ˆG if and only if they are
+adjacent in T. For each such edge e, ˆγ(e) = e.
+(D)
+The remaining edges of ˆG are defined as follows. For each edge e ∈ E(T) that connects a
+vertex z ∈ V (T) \ C(H) to a vertex y ∈ C(H) there are corresponding edges in ˆG. These
+edges connect z to all vertices g ∈ V (G) such that c(g) = γ(y) For each such edge e′ in ˆG,
+ˆγ(e′) = e.
+Observe that for each element Tcol ∈ ColSub(T → ( ˆG, ˆγ)) the induced subgraph
+Tcol[G] := Tcol[V (Tcol) ∩ V (G)]
+of Tcol is an edge-colourful subgraph in G, that is, Tcol[G] contains precisely one edge per
+edge-colour of G under the edge colouring ˆγ hence it contains precisely one edge per edge-
+colour of G under cE. As shown in Section 3 in the full version [30] of [31], Tcol[G] thus
+induces a fracture ρ = ρ(Tcol) of Q: Two edges {v, w} and {v, y} of Q are in the same block
+in the partition ρv corresponding to vertex v of Q if and only if the edges of Tcol[G] that are
+coloured γ−1
+E ({v, w}) and γ−1
+E ({v, y}) are adjacent. In what follows, we show that ρ must
+always be equal to τ(∆, T, H) (see Definition 37).
+▶ Lemma 43. For every Tcol ∈ ColSub(T → ( ˆG, ˆγ)) we have that ρ(Tcol) = τ(∆, T, H).
+Proof. To avoid notational clutter, we set ρ := ρ(Tcol) and τ := τ(∆, T, H). Let T1 and T2
+be the subtrees of T attached to the ends of the C-gadget H as shown in the bottom part of
+Figure 4.
+
+22
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+Figure 4 (Below): The tree T containing a closed strong C-gadget of order d; the green dashed
+lines are rays of length d. (Left): The construction of ˆG = ˆG(G, c, T, H); note that the removal of
+the vertices and edges coloured blue yields G (see Definition 42), and note that G is Q-coloured as
+depicted. (Right): The graphs ∆ and Q = Q(∆, T, H); we assume in the picture that {v2, vk−1} is
+an edge of ∆.
+We first give an overall intuition of the proof; consider Figure 5 for an illustration. Since
+Tcol is isomorphic to T, there must be a (unique) path connecting T1 and T2 in ˆG (recall
+
+L. A. Goldberg and M. Roth
+23
+that, since Tcol is edge-colourful and since every edge in T1 and T2 has a different colour —
+see (C) in Definition 42 — Tcol must contain all edges in T1 and T2). We claim that this
+path must follow the outer cycle in ˆG, in which case the designated rays in R of length d
+at the junctions must follow the inwards direction and thus induce τ. To see why the path
+connecting T1 and T2 must follow the outer cycle, first recall that Vj is the subset of V (G)
+coloured by c with vj. Then recall that the path between Vj and Vj+1 along the outer cycle
+in ˆG has length ℓj ≥ 2d + 1. Hence the designated rays in R cannot be used to cover all
+edge colours in the path between Vj and Vj+1.
+We next provide a rigorous argument. Let
+S := V (T1) ∪ V (T2) ∪ {x0, . . . , xi1−1} ∪ {xik+1, . . . , xk+1}.
+Note that S is a subset of V (T) \ V (H) hence it is a subset of V (T) and of V ( ˆG).
+We first claim that every fork and every ray of length > d of T must be fully contained
+in the subgraph of T induced by S. This claim follows from the definition of closed strong
+C-gadgets. In particular, the condition of being closed implies that neither xi1 nor xik is a
+fork.
+As a consequence, every fork and every ray of length greater than d of Tcol must be
+contained in the subgraph of ˆG induced by S as well. Additionally, this implies that none of
+the vertices in Tcol[G] can be a fork or the source of a ray of length > d in Tcol — otherwise,
+Tcol would have either more forks or more rays of length > d than T, contradicting the fact
+that Tcol and T are isomorphic.
+Recall that V1, . . . , Vk denote the subsets of vertices of G that are coloured by c with
+v1, . . . , vk. Now let P be the (unique) path P in Tcol that connects T1 with T2. Then, starting
+with V1 and ending with Vk, the path P must pass through a sequence of colour classes
+V1 = Vj1, Vj2, . . . , Vjt = Vk of G. The following claim formalises the idea that this sequence
+must correspond to the Hamilton cycle v1, . . . , vk in ∆.
+Claim:
+We have t = k and Vji = Vi for each i ∈ [k].
+Before proving the claim, we show that it implies the lemma. Since, from the claim, P
+must follow the outer cycle, the fracture ρ = ρ(Tcol) induced by Tcol must split the inner
+paths of length 2d (otherwise Tcol would contain a cycle). However, since there are no sources
+or rays of length greater than d outside of S in Tcol, ρ must split all of the inner length-2d
+paths at the central vertex m(e). Furthermore, it cannot split additional vertices since this
+would disconnect Tcol. Thus, ρ is the fracture τ, concluding the proof. ■
+To conclude the proof, we now prove the claim. Note first that P cannot pass through
+any of the colour classes Vi more than once as this would cause Tcol to use an edge-colour
+multiple times. Next assume for contradiction that P misses some colour class Va for some
+a ∈ [2, k − 1] (i.e., we assume that t < k). Since Tcol is a connected tree containing all of the
+edge colours in Q there must be an index ji ̸= a and a vertex u ∈ Vji ∩ P such that Tcol
+contains a (unique) path Pu from u to a vertex w ∈ Va. In order to get the contradiction,
+root Tcol at u. Construct a subtree Tcol(u) of Tcol as follows: For each neighbour x of u
+except the ancestor of w on the path from u, we delete x and all of its descendants. Observe
+that the edge colours of Tcol(u) are disjoint from the edge-colours of P and that V (Tcol(u))
+is disjoint from S. Now, if Tcol(u) is a path, then (using that ℓi > 2d), we obtain that u is
+the source of a ray in Tcol of length greater than d, contradicting the fact that every ray of
+length > d of Tcol is in the subgraph of ˆG induced by S. Otherwise, Tcol(u) contains a fork,
+contradicting the fact that all forks of Tcol are in the subgraph of ˆG induced by S.
+Having established that t = k and that no Vi is visited more than once, it remains to
+show that P visits the colour classes in the correct order, that is Vji = Vi for each i ∈ [k].
+
+24
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+Figure 5 Illustration of Lemma 43: The only possibility for an edge-colourful copy of T to be
+embedded in ˆG is depicted in red.
+Assume for contradiction that this is not the case, which allows us to set
+m := min{i ∈ [k] | Vji ̸= Vi} − 1 .
+Note that m ≥ 1 since j1 = 1. Let zm ∈ Vm ∩ P and zm+1 ∈ Vm+1 ∩ P and recall that G
+contains colour classes U 1
+m, . . . , U ℓm−1
+m
+corresponding to the path
+Pm = vm, u1
+m, . . . , uℓm−1
+m
+, vm+1
+
+L. A. Goldberg and M. Roth
+25
+in Q (see Definition 37). Let us now define the subtrees Tcol(m) and Tcol(m + 1):
+For Tcol(m) we root Tcol at zm and for each neighbour x of zm in Tcol, we delete x and all
+of its descendants unless x ∈ U 1
+m.
+For Tcol(m + 1) we root Tcol at zm+1 and for each neighbour x of zm+1 in Tcol, we delete
+x and all of its descendants unless x ∈ U ℓm−1
+m
+.
+Note that at least one of Tcol(m) and Tcol(m + 1) must have depth greater than d (if rooted
+at zm and zm+1, respectively), since ℓm > 2d and Tcol is edge-colourful with respect to ˆγ,
+that is, we have to make sure that we cover all of the edge colours
+{vm, u1
+m}, {u1
+m, u2
+m}, . . . , {uℓm−1
+m
+, vm+1}
+Finally, regardless of which one of the two subtrees has depth greater than d, we will find
+either a fork, or the source of a ray of length greater than d outside of the set S, yielding
+the desired contradiction and concluding the proof of the claim, and hence the proof of the
+lemma.
+◀
+We are now able to prove the main lemma of this subsection.
+▶ Lemma 44. ⊕Emb((Q
+♯
+τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)).
+Proof. We start with the following claim from [30].
+Claim: A colour-preserving embedding ϕ ∈ Emb((Q
+♯
+τ, cτ) → (G, c)) is uniquely defined
+by its image (which is a subgraph of (G, c)).
+For convenience, we provide a proof of the claim: Consider in image (G′, c′) of ϕ where G′
+is a subgraph of G and c′ = c |V (G′). Let e = {u, v} be an edge of G′ Then c′(e) = {c(u), c(v)}
+is an edge of Q since c is a Q-colouring. Recall that Q
+♯
+τ is Q-coloured by the function
+cτ that maps wB to w for each w ∈ V (Q) and block B ∈ τw. Now recall the definition
+of fractured graphs (Definition 8) and let B1 and B2 be the blocks of τc(u) and τc(v) that
+contain c(e). Then, since ϕ is an embedding, it maps c(u)B1 to u and c(v)B2 to v. Since Q
+does not have isolated vertices, continuing this process over all edges of G′ defines ϕ. This
+concludes the proof of the claim. ■
+By the claim, it is sufficient to construct a bijection b from elements in ColSub(T → ( ˆG, ˆγ))
+to subgraphs (G′, c′) that are images of embeddings in Emb((Q
+♯
+τ, cτ) → (G, c)). Given
+Tcol ∈ ColSub(T → ( ˆG, ˆγ)) we set b(Tcol) := (Tcol[G], c(Tcol)) where c(Tcol) is the colouring of
+vertices of Tcol[G] which agrees with ˆγ on the edges of Tcol[G]. In the rest of the proof, we
+show that b is the desired bijection.
+First, we have to show that for all Tcol, (Tcol[G], c(Tcol)) is the image of an embedding in
+Emb((Q
+♯
+τ, cτ) → (G, c)). To this end, recall that Tcol[G] induces a fracture ρ = ρ(Tcol) of Q.
+By the definition of ρ, Tcol[G] and Q
+♯
+ρ are isomorphic and this isomorphism preserves the
+colours so cρ agrees with ˆγ on the edges of Q
+♯
+ρ. This implies that cρ and c(Tcol) are the
+same. So (Tcol[G], c(Tcol)) is the image of an embedding in Emb((Q
+♯
+ρ, cρ) → (G, c)). Finally,
+Lemma 43 guarantees that ρ = τ.
+Second, we will show that b is injective. To this end, let Tcol1 ̸= Tcol2 ∈ ColSub(T → ( ˆG, ˆγ)).
+Since Tcol1 and Tcol2 must both fully contain V (T) \ C(H), and since both are edge-colourful
+(see Definition 42), the only possibility for Tcol1 and Tcol2 not being equal is that they disagree
+on G, that is, Tcol1[G] ̸= Tcol2[G]. This proves b to be injective.
+Finally, we will show that b is surjective: Given any (G′, c′) that is the image of an
+embedding ϕ ∈ Emb((Q
+♯
+τ, cτ) → (G, c)), we construct Tcol(G′, c′) ∈ ColSub(T → ( ˆG, ˆγ))
+with b(Tcol(G′, c′)) = (G′, c′) as follows. Observe first that G′ is isomorphic to T[C(H)] since
+
+26
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+Q
+♯
+τ is, by definition of τ, isomorphic to T[C(H)]: Splitting the inner paths of length 2d in
+Q at their central vertices yields precisely T[C(H)]. Then Tcol(G′, c′) is obtained by adding
+the remainder of T to (G′, c′):
+1. We add to (G′, c′) all vertices in V (T) \ C(H) (see (B) in Definition 42).
+2. We add all edges between vertices in V (T) \ C(H) that are present in ˆG (see (C) in
+Definition 42).
+3. Finally, we connect a vertex in z in V (T) \ C(H) with a vertex w in G′ if and only if z
+and w are connected in ˆG (see (D) in Definition 42).
+The resulting subgraph Tcol(G′, c′) of ˆG is clearly edge-colourful and isomorphic to T,
+concluding the proof.
+◀
+We are now able to establish hardness of ⊕Sub(T ) in case of unbounded C-number.
+▶ Lemma 45. Let T be a recursively enumerable class of trees of unbounded C-number.
+Then ⊕Sub(T ) is ⊕W[1]-hard.
+Proof. Assume first that T contains trees with 2-paths of unbounded length. In this case
+we reduce from the problem of counting k-cycles, modulo 2, which was shown ⊕W[1]-hard
+in [11]. In the first step, this problem reduces to the problem of counting s-t-paths of length
+k, modulo 2 as shown in Lemma 5.2 in the full version [28] of [27]. In the second and final
+step, we can easily reduce from the problem of counting s-t-paths of length k, modulo 2, to
+⊕Sub(T ), as shown in Figure 6: Concretely, let (G, s, t, k) be a problem instance. Since T
+contains trees with 2-paths of unbounded length, we can find, in time only depending on k,
+a tree T in T containing a 2-path x0, x1, . . . , xk+1, xk+2 of length k + 2. Let furthermore T1
+and T2 be the subtrees of T as depicted in Figure 6. We construct a graph G′ from G in two
+steps as follows: First, we add fresh vertices x0 and xk+2 and edges {x0, s} and {t, xk+2}.
+Second, we add T1 and T2 and identify their roots with x0 and xk+2, respectively. The
+construction is depicted in Figure 6 as well. Now let A be the set of subgraphs of G′ that are
+isomorphic to T and that contain all edges of T1 and T2. It is easy to see that the cardinality
+of A is equal to the number of s-t-paths of length k in G. Thus it suffices to compute |A|
+mod 2, using an oracle for ⊕Sub(T ). This can be achieved by a simple application of the
+inclusion-exclusion principle: Setting S = E(T1) ∪ E(T2), we have
+|A| =
+�
+J⊆S
+(−1)|J| · #Sub(T → G′ \ J) ,
+(7)
+where G′ \ J is the graph obtained from G′ by deleting all edges in J. We can conclude the
+reduction by observing that the number of terms in (7) only depends on T and thus on k,
+and that our oracle to ⊕Sub(T ) allows us to evaluate (7) modulo 2.
+For the remainder of the proof we can thus assume that the length of any 2-path in any
+tree in T is bounded by a constant d. Since T has unbounded C-number, we obtain that the
+trees in T contain C-gadgets of order d of unbounded length. By Corollary 35 we obtain
+that for any positive integer k, there is a value d′ in the range 1 ≤ d′ ≤ d such that there is
+a tree Tk in T which contains a strong C-gadget of order d′ with k junctions.
+Let C be a class of cubic Hamiltonian graphs of unbounded treewidth. Assume w.l.g.
+that, for each k, the class C contains at most one graph with k vertices; otherwise we just
+keep one k-vertex graph with the largest treewidth among all k-vertex graphs in C. For each
+∆ ∈ C set T∆ := T|V (∆)|, that is T∆ is contained in T and contains a strong C-gadget H∆
+with at least |V (∆)| junctions. Recall Definition 37 and set
+Q := {Q(∆, T∆, H∆) | ∆ ∈ C} .
+
+L. A. Goldberg and M. Roth
+27
+Observe that Q(∆, T∆, H∆) contains as minor the graph obtained from ∆ by removing one
+edge. Since the removal of a single edge can decrease the treewidth only by a constant, and
+since treewidth is minor-monotone, we have that Q has unbounded treewidth.
+By Theorem 12 the problem ⊕cp-Hom(Q) is therefore ⊕W[1]-hard. Thus it suffices to
+show that
+⊕cp-Hom(Q) ≤fpt
+T ⊕Sub(T ) .
+In the first step, we reduce the computation of ⊕Hom((Q, idQ) → ⋆) to the computation
+of ⊕Emb((Q
+♯
+τ, cτ) → ⋆); here, τ is the fracture defined in Definition 37. To this end, it was
+shown in [29] that
+⊕Emb((Q
+♯
+τ, cτ) → ⋆) =
+�
+ρ≥τ
+µ(τ, ρ) · ⊕Hom((Q
+♯
+ρ, cρ) → ⋆) ,
+(8)
+where the relation “≥” and the Möbius function µ are over the lattice of fractures. We
+omit introducing these objects in detail, since we only require that the coefficient of the
+term ⊕Hom((Q
+♯
+⊤, c⊤) → ⋆) (which is equal to ⊕Hom((Q, idQ) → ⋆)) in the above linear
+combination was shown in [29] to be equal to
+�
+v∈V (Q)
+(−1)|τv|−1 · (|τv| − 1)! .
+Since each partition τv has at most two blocks, the above term is odd. Thus, by Lemma 14, we
+can evaluate the term ⊕Hom((Q
+♯
+⊤, c⊤) → ⋆) if we can evaluate the entire linear combination,
+that is, if we can evaluate ⊕Emb((Q
+♯
+τ, cτ) → ⋆). It thus remains to show how we can evaluate
+⊕Emb((Q
+♯
+τ, cτ) → ⋆) using our oracle for ⊕Sub(T ).
+To this end, we use Lemma 44: Given any Q = Q(∆, T∆, H∆)-coloured graph (G, c)
+for which we want to compute ⊕Emb((Q
+♯
+τ, cτ) → (G, c)), we first construct ( ˆG, ˆγ) as in
+Definition 42. Then Lemma 44 yields that
+⊕Emb((Q
+♯
+τ, cτ) → (G, c)) = ⊕ColSub(T∆ → ( ˆG, ˆγ)).
+Finally, by Lemma 15 we can compute ⊕ColSub(T∆ → ( ˆG, ˆγ)) in FPT time using an
+oracle for ⊕Sub(T∆ → ⋆). Since the size of T∆ only depends on Q, and since, with input Q
+we can find T∆ (recall that T is recursively enumerable) this yields indeed a parameterised
+Turing-reduction and the proof is concluded.
+◀
+4.2
+Unbounded Star Number
+We will use the same strategy as in Subsection 4.1: Given a tree T with large star number,
+we start with a properly chosen cubic graph ∆, and we construct a graph Q depending on ∆
+and T which contains ∆ as a minor. Then we show that for any Q-coloured graph (G, c),
+we can construct an edge-coloured graph ( ˆG, ˆγ) such that ⊕ColSub(T → ( ˆG, ˆγ)) is equal to
+⊕Emb((Q
+♯
+τ, cτ) → (G, c)) for a particular fracture τ.
+To this end, let T be a tree with star number (at least) 6k for some positive integer k. By
+definition of the star number, there is a d ≥ 3 such that T contains a vertex s which is the
+source of 6k rays R1, . . . , R6k of length precisely d. For each i ∈ [6k], let Ri = s, r1
+i , . . . , rd
+i .
+Furthermore, let Ts be the subtree of T obtained by deleting the vertices r1
+i , . . . , rd
+i for each
+i ∈ [6k]; consider Figure 7 for an illustration.
+
+28
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+Figure 6 Reduction from counting s-t-paths of length k, modulo 2, in a graph G to counting
+copies of a tree T with a 2-path of length at least k + 2.
+Figure 7 A tree with Sd(T) ≥ 6k.
+▶ Definition 46 (Q). Let ∆ be cubic graph on k vertices. We obtain Q from ∆ by substituting
+each vertex v by a gadget depicted in Figure 8. Afterwards, we connect the gadgets as follows:
+If {v, x} is an edge of ∆, then we identify the vertex vx in the gadget of v and the vertex xv
+in the gadget of x.
+▶ Observation 47. ∆ is a minor of Q.
+The fracture τ of Q that we will be interested in is defined as follows; Figure 9 depicts
+the fractured graph Q
+♯
+τ.
+▶ Definition 48 (τ). Let Q be the graph defined in Definition 46.
+For each edge {v, x} of ∆, the graph Q contains a vertex vx(= xv), which has degree 2.
+We let τvx be the partition consisting of 2 singleton blocks.
+For each vertex v of ∆, the vertices v3 and v5 have degree 2 in Q. We let τv3 and τv5 be
+the partitions consisting of 2 singleton blocks.
+For each vertex v of ∆, the vertices v2, v4 and v6 have degree 3 in Q. For each i ∈ {2, 4, 5}
+we let τvi be the partition consisting of one block of size 2 corresponding to the edges
+incident to vi from the left and the right, and one block of size 1 corresponding to the
+edge incident to vi from below.
+
+L. A. Goldberg and M. Roth
+29
+Figure 8 The construction of Q; the vertices v1, . . . , v6 on the gadget of v are emphasized.
+Figure 9 Illustration of the fractured graph Q
+♯
+τ via fracturing the vertex gadgets.
+For all other vertices u of Q, we let τu be the partition consisting only of one block.
+Analogously to the notion of a core in the case of unbounded C-number, we will identify
+a specific subgraph of the tree T and we will use it to define the graph ˆG later.
+▶ Definition 49 (V ′). Let V ′ be the vertex subset of T defined as follows:
+V ′ :=
+�
+� �
+i∈[6k]
+V (Ri)
+�
+� \ {s} .
+Furthermore, we set E′ := E(T[V ′]).
+Observe that T[V ′] is a (disjoint) union of 6k paths of length d − 1, where the vertices of
+the i-th path are r1
+i , . . . , rd
+i . Observe further that V (T) = V (Ts) ˙∪V ′ and that
+E(T) = E′ ˙∪ E(Ts) ˙∪ {{s, r1
+i } | i ∈ [6k]} .
+(9)
+Next, note that the edges of Q can be decomposed into 6k paths, each of length d − 1:
+There are k vertices of ∆. For each vertex v ∈ V (∆) the graph Q contains, by definition, a
+gadget corresponding to v, the edges of which can be decomposed into 6 paths P 1
+v , . . . , P 6
+v
+of length d − 1 (formally, the fractured graph Q
+♯
+τ yields precisely this decomposition; see
+Figure 9). Additionally, for each v ∈ V (∆) and i ∈ [6], the first vertex of P i
+v is chosen to be
+vi as depicted in Figure 8.
+
+30
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+▶ Definition 50 (γ, γE). We define a function γ : T[V ′] → V (Q) as follows. Recall that T[V ′]
+is the union 6k paths P ′
+j := r1
+j, . . . , rd
+j for j ∈ [6k]. Fix any bijection b : [6k] → V (∆) × [6].
+Then γ maps P ′
+j to P i
+v, where b(j) = (v, i). In particular, we enforce that the first vertices
+of the paths are mapped onto each other, that is, γ(r1
+j) := vi. Additionally, we define
+γE : E′ → E(Q) by mapping e to γ(e).
+▶ Observation 51. The function γ is an edge-bijective homomorphism from T[V ′] to Q.
+Specifically, γE is a bijection.
+Now let (G, c) be a Q-coloured graph. We state the following explicitly, since it will be
+crucial in our reduction.
+▶ Observation 52. Let (G, c) be a Q-coloured graph. The mapping cE ◦ γ−1
+E
+is a map from
+E(G) to E′.
+Let us now construct a graph ˆG from a Q-coloured graph G; an illustration is provided
+in Figure 10.
+▶ Definition 53 (( ˆG, ˆγ)). Let (G, c) be a Q-coloured graph. The graph ˆG is an edge-coloured
+graph, with colouring ˆγ : E( ˆG) → E(T), constructed as follows:
+(A)
+The graph ˆG contains G as a subgraph. For each e ∈ E(G) we set ˆγ(e) = γ−1
+E (cE(e)).
+(B)
+The vertex set of ˆG is the union of V (G) and V (Ts), and pairs of vertices in V (Ts) are
+connected by an edge in ˆG if and only they are adjacent in T. For each such edge e,
+ˆγ(e) = e.
+(C)
+The remaining edges of ˆG are defined as follows. For each edge e = {s, r1
+j} ∈ E(T), we
+connect s to all vertices in G that are coloured (by c) with γ(r1
+j) (see Definition 50), and
+for each of those newly added edges e′ we set ˆγ(e′) := e
+Observe that ˆγ colours the edges of ˆG with E(T); the cases (A), (B), and (C) correspond,
+respectively, to the sets E′, E(Ts) and {{s, r1
+i } | i ∈ [6k]} (see Equation (9)). Similarly to
+the case of unbounded C-gadgets, for each element Tcol ∈ ColSub(T → ( ˆG, ˆγ)) the induced
+subgraph
+Tcol[G] := Tcol[V (Tcol) ∩ V (G)]
+of Tcol is an edge-colourful subgraph in G, that is, Tcol[G] contains precisely one edge per
+edge-colour of G under the edge colouring ˆγ hence it contains precisely one edge per edge-
+colour of G under cE. As shown in Section 3 in the full version [30] of [31], Tcol[G] thus
+induces a fracture ρ = ρ(Tcol) of Q: Two edges {v, w} and {v, y} of Q are in the same block
+in the partition ρv corresponding to vertex v of Q if and only if the edges of Tcol[G] that are
+coloured γ−1
+E ({v, w}) and γ−1
+E ({v, y}) are adjacent. In what follows, we show that ρ must
+always be equal to τ(∆, T, H) (see Definition 48).
+▶ Lemma 54. For every Tcol ∈ ColSub(T → ( ˆG, ˆγ)) we have that ρ(Tcol) = τ.
+Proof. Let Tcol ∈ ColSub(T → ˆG, ˆγ). Since Tcol must include each of the edge colours given
+by ˆγ (precisely) once, we have that Tcol must fully contain Ts. Note that Ts fully contains T
+except for 6k rays of length d, and the only way to attach those rays in ˆG is via the vertex s.
+Now consider the subgraph Tcol[G + s] of Tcol defined as follows:
+Tcol[G + s] := Tcol[(V (Tcol) ∩ V (G)) ∪ {s}] .
+Since Tcol includes all edge colours given by ˆγ, we have that s must have degree 6k in
+Tcol[G + s]: By (C) in Definition 53, the vertex s must be connected (within Tcol[G + s]) to
+one vertex in each of the colour classes Vi = c−1(vi) for v ∈ V (∆) and i ∈ [6]. Additionally,
+this implies the following:
+
+L. A. Goldberg and M. Roth
+31
+Figure 10 The construction of ˆG. The graph G within ˆG is depicted in black.
+
+32
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+▶ Observation 55. Tcol[G + s] is isomorphic to the d-stretch of K1,6k with s at the centre.
+In the remainder of the proof, we will show that the only way for Tcol to (colourfully)
+embed the 6k rays of length d is as depicted in Figure 11. Note that this will conclude the
+proof since the induced fracture of the depicted embedding is τ.
+Hence we proceed with proving the claim. We first consider, for each edge {v, x} ∈ E(∆),
+the vertex vx = (xv) of Q (see Definition 46 and Figure 8). The vertex vx has two neighbours
+nv and nx in Q, where nv denotes the neighbour in the gadget of v and nx denotes the
+neighbour in the gadget of x. Recall that we write Vx = c−1(vx), Nv = c−1(nv), Nx =
+c−1(nx) ⊆ V (G) for their colour class within G (and thus within ˆG). Since Tcol is edge-
+colourful, it must contain precisely one edge ev between Vx and Nv and one edge ex between
+Vx and Nx (see (A) in Definition 53). Now observe that every vertex in Vx has distance (at
+least) d to s within ˆG. This has two crucial consequences:
+First, the endpoints of ev and ex inside Vx cannot be equal: Otherwise, they could not be
+part of a ray of length precisely d with source s, and this would contradict the previous
+observation that Tcol[G + s] is isomorphic to the d-stretch of K1,6k with s at the centre
+(Observation 55).
+Hence, second, the endpoints of ev and ex inside Vx both have degree 1. Consequently,
+they must be the endpoints of two of the rays of length d. However, the only way for this
+to be true is them each being connected to s as depicted in Figure 11; in all other cases,
+Tcol[G + s] cannot be isomorphic to the d-stretch of K1,6k with s at the centre.
+The second consequence implies that the edge colours corresponding to the edges in the paths
+P 2
+v , P 4
+v , and P 6
+v are covered for each v (recall that Tcol must include each edge colour precisely
+once). Thus, the only possibility to include the remaining edge colours corresponding to the
+paths P 1
+v , P 3
+v , and P 5
+v while keeping Tcol[G + s] being isomorphic to the d-stretch of K1,6k, is
+to embed, for each gadget, the remaining 3 rays of length d as depicted in Figure 11. This
+concludes the proof.
+◀
+We are now able to prove the main lemma of this section.
+▶ Lemma 56. ⊕Emb((Q
+♯
+τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)).
+Proof. Thanks to Lemma 54, the proof is similar to the proof of Lemma 44: Colour-
+preserving embeddings in Emb((Q
+♯
+τ, cτ) → (G, c)) are uniquely identified by their image,
+and a bijection b from ColSub(T → ( ˆG, ˆγ)) to images of colour-preserving embeddings in
+Emb((Q
+♯
+τ, cτ) → (G, c)) is given by b : Tcol �→ Tcol[G].
+◀
+Similarly to the proof in Section 4.1, Lemma 56 is sufficient for hardness.
+▶ Lemma 57. Let T be a recursively class of trees of unbounded star number. Then ⊕Sub(T )
+is ⊕W[1]-hard.
+Proof. The proof is almost identical to the proof of Lemma 45, with the exception that we
+use Q, τ, ˆG, and ˆγ as defined in the current section, and that we rely on Lemma 56 for the
+identity
+⊕Emb((Q
+♯
+τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)).
+The remainder of the proof transfers verbatim.
+◀
+
+L. A. Goldberg and M. Roth
+33
+Figure 11 Illustration of the unique way to colourfully embed T into ˆG. The induced fracture
+is τ.
+
+34
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+4.3
+Unbounded Fork number
+We will rely on the same high-level strategy as the one that we used when the C-number
+or star number was unbounded: Given a tree T with large a-b-fork number, we start
+with a properly chosen cubic graph ∆, and we construct a graph Q which depends on T
+and ∆, and which contains ∆ as a minor. Afterwards, we show that for any Q-coloured
+graph (G, c) we can construct an edge-coloured graph ( ˆG, ˆγ) where the co-domain of ˆγ is
+E(T) such that #ColSub(T → ( ˆG, ˆγ)) is equal (modulo 2) to #Emb((Q
+♯
+τ, cτ) → (G, c)) for
+a particular fracture τ of Q. However, proving this equality will be more involved than
+it was in the previous cases: In Sections 4.1 and 4.2, we were able to prove, implicitly,
+that #ColSub(T → ( ˆG, ˆγ)) = #Emb((Q
+♯
+τ, cτ) → (G, c)), that is, we were able to establish
+equality, rather than equality modulo 2. In the current case, we are not able to prove equality
+and must therefore rely on parity arguments, which makes the case slightly more involved.
+We start by fixing the following:
+Positive integers k, a and b with a ≤ b and k ≥ 2.
+A tree T with Fa,b(T) ≥ 2k. By definition of forks (Definition 29), T contains designated
+sources s1
+1, s2
+1, . . . , s1
+k, s2
+k such that for each (i, j) ∈ [k] × [2], the source sj
+i is the source of
+two (distinct) rays Fa(i, j) of length a and Fb(i, j) of length b. Additionally degNL(sj
+i) = 1.
+We assume w.l.o.g. that the designated sources are ordered by their leaf-degrees, that is
+degL(s1
+1) ≥ degL(s2
+1) ≥ · · · ≥ degL(s1
+k) ≥ degL(s2
+k) .
+(10)
+Consider Figure 12 for an illustration of T, its designated sources, and the rays Fa(i, j)
+and Fb(i, j).
+A k-vertex bipartite cubic graph ∆ with vertices V (∆) = {v1, . . . , vk}.
+A proper 3-edge-colouring C : E(∆) → {s, m, ℓ} of ∆.6
+We first note that, since there are at least 2k ≥ 4 sources in T, any pair of distinct sources
+must not be adjacent: Otherwise, the tree T would either be disconnected, or one of the
+sources would have degNL at least 2, both of which is a contradiction.
+▶ Observation 58. For any distinct pair (i, j) ̸= (i′, j′) we have that sj
+i and sj′
+i′ are not
+adjacent in T.
+Next, we define the graph Q.
+▶ Definition 59 (Q). The graph Q is obtained from ∆ and C via substituting vi by the
+gadget depicted in Figure 13 for each i ∈ [k]. Afterwards, for every edge e = {vi, vj} of ∆ we
+identify the vertex coloured with C(e) in the gadget of vi with the vertex coloured with C(e)
+in the gadget of vj.
+While Definition 59 will be useful in our proofs, we note the following easier equivalent
+way to define Q.
+▶ Observation 60. The graph Q is obtained from ∆ and C by substituting each edge of
+colour s (of ∆) with a path of length 2a, each edge of colour m with a path of length 2b, and
+each edge of colour ℓ with a path of length 2(a + b). Consequently, ∆ is a minor of Q.
+The fracture τ of Q that we will be interested in is defined as follows; Figure 14 depicts
+the fractured graph Q
+♯
+τ.
+6 That is, C(e1) ̸= C(e2) whenever e1 ̸= e2 share a vertex. Note that every cubic bipartite graph has a
+3-edge-colouring by Hall’s Theorem.
+
+L. A. Goldberg and M. Roth
+35
+Figure 12 A tree T with Fa,b(T) ≥ 2k. Note that the parents of the sj
+i are not necessarily
+distinct. The rays Fa(i, j) and Fb(i, j) are depicted in red.
+v1
+i
+v2
+i
+a
+ℓ
+b
+s
+a
+m
+b
+Figure 13 A vertex gadget in the construction of Q in Definition 59. A dashed line labelled with
+a (resp. b) depicts a path of length a (resp. b).
+
+36
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+Figure 14 The fractured graph Q
+♯
+τ. Note that the illustration only depicts the fracturing of a
+single vertex gadget.
+▶ Definition 61 (τ). Let Q be the graph defined in Definition 59.
+For each edge e = {vi, vj} of ∆, there is a vertex C(e) ∈ {s, m, ℓ} of degree 2 that connects
+the gadgets of vi and vj. We let τC(e) be the partition consisting of two singleton blocks.
+For each vertex vi of ∆, the gadget of vi in Q contains the vertex v1
+i of degree 3 which is
+connected to s via a path of length a, to m via a path of length b, and to ℓ via a path of
+length a + b. Let es, em, and eℓ be the first edges on those paths. We set
+τvi = {{es, em}, {eℓ}} .
+For all other vertices u of Q, we let τu be the partition consisting only of one block.
+Next we identify specific substructures of T that will be necessary in the construction of ˆG.
+▶ Definition 62. Recall that sj
+i with (i, j) ∈ [k] × [2] are the designated sources of T.
+T ′ is the graph obtained from T by deleting, for each (i, j) ∈ [k] × [2], the designated
+source sj
+i as well as all rays with source sj
+i.
+For each (i, j) ∈ [k] × [2], pj
+i is the neighbour of sj
+i which is not contained in a ray. Note
+that pj
+i is unique by definition of forks. Note that pj
+i ∈ V (T ′) and that the pj
+i are not
+necessarily pairwise distinct.
+For each (i, j) ∈ [k] × [2], dj
+i = degL(sj
+i) − 2, that is, dj
+i is the number of rays with source
+sj
+i minus 2. Note that dj
+i ≥ 0 since each sj
+i is the source of Fa(i, j) and Fb(i, j).
+F :=
+�
+(i,j)∈[k]×[2]
+(Fa(i, j) ∪ Fb(i, j)) , that is, F is the subset of V (T) that contains the
+vertices of the rays Fa(i, j) and Fb(i, j) (which includes sj
+i) for each (i, j) ∈ [k] × [2].
+E′ := E(T[F]).
+An illustration of these notions is given in Figure 12.
+
+L. A. Goldberg and M. Roth
+37
+Observe that T[F] is a disjoint union of 2k paths of length a + b. Specifically, for each
+(i, j) ∈ [k] × [2] it contains the path
+F j
+i := T[Fa(i, j) ∪ Fb(i, j)] .
+It turns out that Q is isomorphic to a quotient graph of T[F], since for each vertex vi of ∆,
+the vertex gadget of vi decomposes into two paths of length a + b. In fact, this decomposition
+is given by the fractured graph Q
+♯
+τ (see Figure 14). Formally, we have the following:
+▶ Observation 63. T[F] ∼= Q
+♯
+τ ∼= 2kPa+b.
+Similarly to the previous two cases, we introduce functions γ and γE which we will need
+for defining the edge-colours of ˆG.
+▶ Definition 64 (γ, γE). We define a function γ : F → V (Q) as follows:
+1. For each i ∈ [k], γ maps F 1
+i to the (a + b)-path in the gadget of vi from s to m, such that
+γ(s1
+i ) = v1
+i .
+2. For each i ∈ [k], γ maps F 2
+i to the (a + b)-path in the gadget of vi from v1
+i to ℓ, such that
+γ(s2
+i ) = v2
+i .
+Furthermore, we write γE : E′ → E(Q) by setting γE({x, y}) := {γ(x), γ(y)}.
+Note that the definition of γE is well-defined since γ is a homomorphism by Observation 63.
+Concretely, γ can be viewed as the composition of an isomorphism from T[F] to Q
+♯
+τ and
+the Q-colouring cτ of Q
+♯
+τ (see Definition 9). Furthermore, γE is clearly a bijection. Hence,
+similarly to the previous sections, we point out the following:
+▶ Observation 65. Let (G, c) be a Q-coloured graph. The mapping cE ◦ γ−1
+E
+is a map from
+E(G) to E′.
+We are now able construct a graph ˆG from a Q-coloured graph G; an illustration is
+provided in Figure 15.
+▶ Definition 66 (( ˆG, ˆγ)). Let (G, c) be a Q-coloured graph. The pair ( ˆG, ˆγ) is an edge-coloured
+graph constructed as follows, where the co-domain of ˆγ is E(T).
+(A)
+The graph ˆG contains G as a subgraph. For each e ∈ E(G), define ˆγ(e) = γ−1
+E (cE(e)).
+(B)
+The vertex set of ˆG is the union of V (G) and V (T) \ F.
+(C)
+Pairs of vertices in V (T)\F are connected by an edge in ˆG if and only if they are adjacent
+in T. For each such edge e, we set ˆγ(e) = e.
+(D)
+The remaining edges of ˆG are defined as follows. For each edge e ∈ E(T) that connects a
+vertex z ∈ V (T) \ F to a vertex y ∈ F there are corresponding edges in ˆG. These edges
+connect z to all vertices g ∈ V (G) such that c(g) = γ(y) For each such edge e′ in ˆG,
+ˆγ(e′) = e.
+In (D), the only edges in T connecting z ∈ V (T) \ F to a vertex y ∈ F satisfy that y is one
+of the designated sources sj
+i, and z is either pj
+i ∈ V (T ′) or z is contained in one of the dj
+i
+rays with source sj
+i that are not Fa(i, j) or Fb(i, j) (see Definition 62).
+Similarly to the other cases, for each element Tcol ∈ ColSub(T → ( ˆG, ˆγ)) the induced
+subgraph Tcol[G] := Tcol[V (Tcol) ∩ V (G)] of Tcol is an edge-colourful subgraph in G. Also,
+Tcol[G] induces a fracture ρ = ρ(Tcol) of Q as follows. First, recall that G is Q-coloured by c,
+and that G is contained in ˆG (see (A) in Definition 66). Next note that Tcol[G] is a subgraph
+of G that contains each edge colour in the image of cE ◦ γ−1
+E
+precisely once. Since γE is a
+bijection from E′ to E(Q), we can thus equivalently view Tcol[G] as a subgraph of G that
+contains each edge colour in the image of cE precisely once. This fact allows us to define
+ρ(T ) in terms of the function cE as follows.
+
+38
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+Figure 15 The graph ˆG. Depicted in the centre is the part of G (within ˆG) that is coloured with
+the vertices of the i-th vertex gadget of Q. Depicted in black are the subtree T ′ of T (left), and,
+as dashed lines, the inner edges of the d1
+i + d2
+i rays incident to s1
+i and s2
+i (right) — here, the inner
+edges are those that are not incident to the sources s1
+i and s2
+i . Each edge of ˆG fully contained in the
+black part has a unique colour w.r.t. ˆγ (see Definition 66 (C)). Pairs consisting of remaining edges
+have the same colour (w.r.t. ˆγ) if and only if they are depicted with the same colour.
+
+L. A. Goldberg and M. Roth
+39
+Figure 16 Illustration of the condition that yields invalid trees at (i, 1) (below) and (i, 2) (above).
+Edges contained in E′ are coloured red.
+▶ Definition 67 (ρ(Tcol)). Let Tcol be an element of ColSub(T → ( ˆG, ˆγ)).
+The fracture
+ρ = ρ(Tcol) of Q is defined as follows. Two edges {v, w} and {v, y} of Q are in the same
+block in the partition ρv corresponding to vertex v of Q if and only if the edges of Tcol[G] that
+are coloured by cE with {v, w} and {v, y} are incident.
+With ( ˆG, ˆγ) defined, we can finally state formally the goal of this section. Recall that
+(G, c) is a Q-coloured graph.
+▶ Lemma 68. Suppose that |c−1(v)| is odd for each v ∈ V (Q). Then ⊕ColSub(T → ( ˆG, ˆγ)) =
+⊕Emb((Q
+♯
+τ, cτ) → (G, c)).
+The proof requires some additional set-up. In particular, we need the condition that
+|c−1(v)| is odd to deal with the case in which what we call “invalid trees” arise. To this
+end, recall that V j
+i = c−1(vj
+i ) denotes the set of vertices in G that are coloured by c with vj
+i .
+Since G is a subgraph of ˆG (see Definition 66), we slightly abuse notation and write V j
+i also
+for the subset of vertices in ˆG corresponding to V j
+i in G.
+▶ Definition 69. Let Tcol ∈ ColSub(T → ( ˆG, ˆγ)) and let (i, j) ∈ [k] × [2]. We call Tcol invalid
+at (i, j) if the following two conditions are met:
+(I)
+Tcol contains precisely two vertices x and y in V j
+i .
+(II)
+x is adjacent to pj
+i and not incident in Tcol to any edge coloured with a colour in E′ (see
+Definition 66 (A)).
+Otherwise Tcol is called valid at (i, j). We call Tcol an invalid tree if there exists a pair
+(i, j) ∈ [k] × [2] such that Tcol is invalid at (i, j). Otherwise, we call Tcol a valid tree. We
+write ColSubval(T → ( �G, ˆγ)) for the set of all valid Tcol in ColSub(T → ( �G, �γ)).
+Consider Figure 16 for an illustration of Definition 69.
+▶ Lemma 70. Suppose that |c−1(v)| is odd for each v ∈ V (Q). Then the number of invalid
+trees Tcol ∈ ColSub(T → ( ˆG, ˆγ)) is even.
+
+40
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+Proof. For the proof, given two tuples (i, j) and (i′, j′) in [k] × [2] we write (i′, j′) < (i, j)
+if (i′, j′) is lexicographically smaller than (i, j).
+Write T (i, j) for the set of all Tcol ∈
+ColSub(T → ( ˆG, ˆγ)) that are invalid at (i, j) but valid on all pairs (i′, j′) < (i, j). We will
+prove that T (i, j) is even for all (i, j) ∈ [k] × [2]; this is sufficient for the lemma to hold.
+Hence fix (i, j), let Tcol ∈ T (i, j), and let x and y be as in Definition 69. Since V j
+i = c−1(vj
+i )
+and for j ∈ [2], vj
+i is a vertex of Q, the assumption in the statement of the lemma implies that
+|V j
+i | is odd. Since x and y are distinct vertices in V j
+i , V j
+i contains additional vertices other
+than x and y. Fix a vertex x′ ∈ V j
+i \ {x, y}. Obtain T ′
+col from Tcol by deleting x (including
+edges incident to x) and by adding x′ and the edge {x′, u} for every u that was adjacent to
+x — this is well-defined since x is not incident to any edge coloured with a colour in E′, and
+by construction of ˆG (see Definition 66 (C) and (D)) whenever {x, u} ∈ E( ˆG) is an edge not
+coloured with a colour in E′, then {x′, u} ∈ E( ˆG) for every x′ ∈ V j
+i . Additionally, {x, u}
+and {x′, u} have the same edge-colour. Hence, clearly, T ′
+col an edge-colourful subgraph of ˆG
+that is isomorphic to Tcol (and thus to T). For this reason, we obtain that T ′
+col ∈ T (i, j).
+More generally, the observation that T ′
+col ∈ T (i, j) allows us to define an equivalence
+relation on T (i, j): Let Tcol and T ′
+col be elements of T (i, j), and let x and x′ be the vertices
+in Tcol and T ′
+col that satisfy (II) in Definition 69. We set Tcol and T ′
+col to be equivalent if and
+only if one can obtained from the other by switching x with x′ as defined above. The size of
+one equivalence class is precisely |V j
+i | − 1 = |c−1(vj
+i )| − 1, which is even by the premise of
+the lemma.
+◀
+For the proof of Lemma 68, we need to establish some facts about rays and 2-paths of
+elements Tcol ∈ ColSubval(T → ( ˆG, ˆγ)), which are those Tcol ∈ ColSub(T → ( ˆG, ˆγ)) that are
+valid. We encapsulate these facts in the next section.
+4.3.1
+The Proof of Lemma 68
+We first note that, thanks to Lemma 70, it suffices to prove that
+#ColSubval(T → ( ˆG, ˆγ)) = #Emb((Q
+♯
+τ, cτ) → (G, c)) .
+This requires some preparation. We first fix the following objects (recall the definitions of
+2-path, Definition 27 and ray, Definition 28).
+Tcol is an element of ColSubval(T → ( ˆG, ˆγ))
+Tcol[G] is the graph obtained from Tcol[V (Tcol)∩V (G)] with isolated vertices removed. (In
+fact, our proof will show that, for valid trees Tcol ∈ ColSubval(T → ( ˆG, ˆγ)), the induced
+subgraph Tcol[V (Tcol) ∩ V (G)] cannot have isolated vertices. However, at the current
+point of the proof, it is easiest to just remove them.)
+For any positive integer t, Rt is the set of length-t rays in T. Pt is the set of length-t
+2-paths in T that are not rays.
+For any positive integer t, Rt
+col is the set of length-t rays in Tcol and Pt
+col is the set of
+2-paths in Tcol that are not rays. Note that |Rt| = |Rt
+col| and |Pt| = |Pt
+col| for all t since
+T and Tcol are isomorphic.
+We will also rely on the following notion of external rays and 2-paths.
+▶ Definition 71. A 2-path P of Tcol is called external if the following two conditions are
+satisfied.
+Except for the endpoints, none of the vertices of P is contained in V (G).
+P does not contain an edge of G.
+
+L. A. Goldberg and M. Roth
+41
+Definition 71 applies whether or not P is a ray. The following lemmas establish that all
+2-paths of Tcol of length greater than b must be external.
+▶ Lemma 72. Suppose that t is an integer that is greater than b. Suppose that, for all t′ > t,
+every 2-path in Rt′
+col ∪ Pt′
+col is external. Then every 2-path in Rt
+col ∪ Pt
+col is external.
+Proof. We first construct a bijection f from Rt to Rt
+col.
+We will use this bijection to
+argue that every ray in Rt
+col is external. In order to define the bijection, consider a ray
+R = r0, r1, . . . , rt in Rt. Since t > b ≥ a, R is not one of the designated rays Fa(i, j) and
+Fb(i, j). If r0 is not among the designated sources sj
+i, then, by the construction of ˆG, R
+is contained in T ′ and thus R ∈ Rt
+col. In this case R is external and we set f(R) := R.
+Alternatively, suppose that r0 = sj
+i for some i and j. Then R must be one of the dj
+i black
+rays in Figure 12 (see Definition 62). By the construction of ˆG and the fact that Tcol is
+edge-colourful, there is a vertex x ∈ V j
+i such that Tcol contains the path x, r1, . . . , rt. In
+Tcol, as in T, the vertices r1, . . . , rt−1 have degree 2 and the vertex rt has degree 1. Vertex
+x cannot have degree 1 in Tcol since this would disconnect Tcol. Also, vertex x x cannot
+have degree 2: To see this, assume for contradiction that x has degree 2. Then there is an
+integer t′ > t and a ray R′ ∈ Rt′
+col the last vertices of which are x, r1, . . . , rt. Since x is not an
+endpoint of the ray and since x ∈ V (G), the ray R′ is not external, contradicting the premise
+of the lemma. Hence x has degree at least 3 and therefore f(R) := x, r1, . . . , rt is an external
+ray of Tcol. The function f is injective by construction. Since Tcol and T are isomorphic,
+|Rt| = |Rt
+col| and thus f is a bijection. Since the image of f only contains external rays, we
+have shown that every element of Rt
+col is external.
+Every ray in the image of f has the property that its degree-1 endpoint is not contained
+in V (G). Since the image of f is Rt
+col, we obtain
+(∗) Every ray in Rt
+col has the property that its degree-1 endpoint is not contained in V (G).
+To complete the proof, we show that every 2-path in Pt
+col is external. Following the same
+strategy that we used before, we construct a bijection g from Pt to Pt
+col. Every 2-path in
+the range of g is external, so we will conclude that every element of Pt is external. In order
+to define the bijection, consider a 2-path P = p0, . . . , pt in Pt. If neither of the endpoints
+of P is among the designated sources sj
+i, then P is contained in T ′ and thus P ∈ Pt. In
+this case, P is external and we set g(P) := P. If exactly one endpoint of P is among the
+designated sources, say p0 = sj
+i, then there is a vertex x ∈ V j
+i such that x, p1, . . . , pt is a
+path in Tcol. The vertices p1, . . . , pt−1 have degree 2 in Tcol (as in T) and the vertex pt has
+degree at least 3.
+If x has degree 1 in Tcol, the ray R = pt, . . . , p1, x is in Tcol, and its degree-1 endpoint x
+is in V (G), contradicting (∗). Hence x cannot have degree 1 in Tcol. Similarly, x cannot
+have degree 2, since this would create a 2-path longer than t in Tcol that is not external,
+which contradicts the premise of the lemma.
+Hence x has degree at least 3, and thus
+g(P) := x, p1, . . . , pt is an external 2-path in Pt
+col.
+For the last case, suppose that both endpoints of P are among the designated sources,
+say p0 = sj
+i and pt = sj′
+i′ . Then there are x and y in, respectively, V j
+i and V j′
+i′ such that
+x, p1, . . . , pt−1, y is a path in Tcol. Again, p1, . . . , pt−1 must all have degree 2 in Tcol as well.
+We show that both x and y have degree at least 3 in Tcol: If both have degree 1, then
+Tcol is disconnected. If one of them has degree 1 and the other one has degree at least 3,
+then we created a ray of length t whose degree-1 endpoint in in V (G), contradicting (∗).
+If one has degree 1 and the other one has degree 2, then we found a ray longer than t
+which is not external, contradicting the premise of the lemma. If one has degree 2 and
+the other has degree at least 2, then there is a non-external 2-path longer than t, again
+
+42
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+contradicting the premise of the lemma. Thus, as desired, both must have degree at least 3.
+Therefore, g(P) := x, p1, . . . , pt−1, y is an external 2-path in Pt
+col. The function g is injective
+by construction. Since Tcol and T are isomorphic, |Pt| = |Pt
+col| and thus g is a bijection.
+Since the image of g only contains external 2-paths, we have shown that every element of
+Pt
+col is external, concluding the proof.
+◀
+▶ Lemma 73. Suppose that t is an integer that is greater than b. Then every 2-path in
+Rt
+col ∪ Pt
+col is external.
+Proof. Let tmax be the maximum integer for which Rtmax ∪ Ptmax is nonempty. Let Φt be the
+proposition “t ≤ b or every 2-path in Rt
+col ∪ Pt
+col is external”.
+We will show by induction on tmax−t that Φt holds. The base case arises when tmax−t = 0,
+so t = tmax. If tmax ≤ b then Φt is satisfied. Otherwise, for each t′ > t, the set Rt′
+col ∪ Pt′
+col is
+empty and we can invoke Lemma 72 to conclude that Φt holds.
+For the induction step, consider t such that tmax − t ≥ 1. By the induction hypothesis,
+Φt′ holds for all t′ ∈ {t + 1, . . . , tmax}. If t ≤ b then Φt holds. Otherwise, for all t′ > t > b,
+we know from Φt′ that every 2-path in Rt′
+col ∪ Pt′
+col is external. We can then apply Lemma 72
+to conclude that every 2-path in Rt
+col ∪ Pt
+col is external.
+◀
+Before proceeding with the proof of Lemma 68, we provide an overview of the central
+steps of the proof. Recall that it suffices to prove that
+#ColSubval(T → ( ˆG, ˆγ)) = #Emb((Q
+♯
+τ, cτ) → (G, c))
+and that we have a fixed an element Tcol of ColSubval(T → ( ˆG, ˆγ)) and proved various
+properties about it.
+(1) Our goal is to show that Tcol is embedded in ˆG in the following manner (see Figure 17).
+For each (i, j) ∈ [k] × [2], Tcol contains a ray Ra(i, j) of length a and a ray Rb(i, j) of
+length b; those rays correspond to the designated rays Fa(i, j) and Fb(i, j) in T (recall
+that T and Tcol are isomorphic.)
+a. T ′ is part of Tcol.
+b. For every i ∈ [k] and j ∈ [2], the vertices pj
+i in T ′ is connected to a vertex wj
+i of G
+with c(wj
+i ) = vj
+i = γ(sj
+i). In Tcol, the vertex wj
+i is the source of dj
+i rays other than
+Ra(i, j) and Rb(i, j). The vertices of these dj
+i rays are not in T ′ and are not in G.
+The edge colours of the edges in these rays in ˆγ are the same as the edge-names in T
+(see Definition 66 (C)).
+c. The length-a ray Ra(i, 1) is a path in Tcol from w1
+i to the vertex ua(i, 1) of G with
+some colour c(ua(i, 1)) (a vertex of Q). This colour c(ua(i, 1)) corresponds to the
+vertex “s” in the gadget of the vertex vi of ∆ (see Definition 59 and Figure 13).
+d. The length-b ray Rb(i, 1) is a path in Tcol from w1
+i to the vertex ub(i, 1) of G with
+some colour c(ub(i, 1)) (a vertex of Q). This colour c(ub(i, 1)) corresponds to the
+vertex “m” in the gadget of the vertex vi of ∆ (see Definition 59 and Figure 13).
+e. The length-b ray Rb(i, 2) is a path in Tcol from w2
+i to the vertex ub(i, 2) of G with
+some colour c(ub(i, 2)) (a vertex of Q). This colour c(ub(i, 2)) corresponds to the
+vertex “ℓ” in the gadget of the vertex vi of ∆ (see Definition 59 and Figure 13).
+f. The length-a ray Ra(i, 2) is a path in Tcol from w2
+i to the vertex ua(i, 2) ̸= w1
+i of G
+with some colour c(ua(i, 2)) = γ(s1
+i ) = v1
+i (recall that the colour is a vertex of Q).
+g. For every edge e = {vi, vi′} in ∆, ua(i, 1) ̸= ua(i′, 1), ub(i, 1) ̸= ub(i′, 1) and ub(i, 2) ̸=
+ub(i′, 2).
+
+L. A. Goldberg and M. Roth
+43
+(2) We now make some observations about the fracture ρ = ρ(Tcol) from Definition 67, given
+that Tcol is embedded in ˆG as described in Item (1).
+The definition of Q (Definition 59) tells us that, for every edge e = {vi, vi′} in ∆,
+there is a degree-2 vertex y of Q that connects the gadgets of vi and vi′. Vertex y
+corresponds to the vertex C(e) ∈ {s, m, ℓ} in the two gadgets. Suppose without loss of
+generality that C(e) = s. The other cases are similar. From (1c) the colour C(e) = s
+is the same as c(ua(i, 1)) and c(ua(i′, 1)). From (1b) c(w1
+i ) = v1
+i and c(w1
+i′) = v1
+i′.
+Since Tcol is colourful and the embedding is as in (1), the edges of the ray from w1
+i to
+ua(i, 1) have different edge colours to the ray from w1
+i′ to ua(i′, 1). Thus, the edge in
+G in the first ray that is adjacent to ua(i, 1) (call it ei) has a different colour from the
+edge n G in the second ray that is adjacent to ua(i′, 1) (call it ei′). Concretely, we
+have cE(ei) = {s, x} and cE(ei′) = {s, x′} where x and x′ are the neighbours of s (in
+Q) in the gadgets of vi and vi′, respectively. By (1g) we have ua(i, 1) ̸= ua(i′, 1) and
+thus, by definition of ρ (Definition 67), ρy consists of two singleton blocks. Similar
+arguments show that ρ coincides with τ (see Definition 61) at every vertex of Q that
+corresponds to vertex “s”, “ℓ” or “m” in any gadget corresponding to any vertex vi
+of ∆.
+We now continue with the vertices v1
+i for i ∈ [k] of Q. See Figure 13 for the gadget
+containing v1
+i in Q and Figure 17 for the graph ˆG. We will use “s”, “ℓ” and “m” as
+the names of these vertices in the gadget containing v1
+i . The vertex v1
+i has degree
+3 and is connected to s via a path of length a, to m via a path of length b and to
+ℓ via a path of length a + b. Let ys, ym, and yℓ be the successors of v1
+i on those
+paths, that is, the edges incident to v1
+i in Q are es := {v1
+i , ys}, em := {v1
+i , ym}, an
+eℓ := {v1
+i , yℓ}. Now, by (1c) and (1d), the edges of Tcol that are coloured (by cE) with
+es and em are {w1
+i , ra} and {w1
+i , rb}, where ra and rb are the successors of w1
+i on the
+rays Ra(i, 1) and Rb(i, 1), respectively. Furthermore, by (1f), the edge of Tcol that
+is coloured (by cE) with eℓ is {ua(i, 2), ˆr} where ˆr is the vertex in the ray Ra(i, 2)
+that is adjacent to ua(i, 2). Since ua(i, 2) ̸= w1
+i (by (1f)), the edge {ua(i, 2), ˆr} is not
+incident to either {w1
+i , ra} or {w1
+i , rb}. Thus ρv1
+i = {{es, em}, {eℓ}} which coincides
+with τv1
+i by Definition 61. So τ and ρ coincide at vertex v1
+i .
+Next are the vertices v2
+i for i ∈ [k] (see Figure 13). This case is easy. If Tcol is
+embedded as described in (1) (see Figure 17), then, for each i ∈ [k], there is only one
+vertex of Tcol which is coloured by c with colour v2
+i . This vertex is w2
+i . Thus every
+edge of Tcol whose edge colour includes v2
+i is incident to w2
+i . Hence ρv2
+i only consists
+of one block, which coincides with τv2
+i by Definition 61.
+Finally, every remaining vertex of Q (see Figure 13) has degree 2. Let y be such a
+vertex and let y1 and y2 be the neighbours of y. Then the edges of Tcol coloured by cE
+with {y, y1} and {y, y2} must be successive edges on one of the rays Ra(i, 1), Rb(i, 1),
+Ra(i, 2), or Rb(i, 2). So these successive edges are both incident to the vertex of the
+ray that is coloured y by c. Thus ρy only consists of one block, which coincides with
+τy.
+Since we have shown that the fractures ρ and τ coincide at every vertex of Q, we conclude
+that ρ = τ.
+(3) We next explain why it is useful to have ρ = τ.
+Recall that our goal is to prove
+that #ColSubval(T → ( ˆG, ˆγ)) = #Emb((Q
+♯
+τ, cτ) → (G, c)) and that Tcol is an element
+of ColSubval(T → ( ˆG, ˆγ)). Our method will be to show that the function β defined by
+
+44
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+β(Tcol) = Tcol[G] is a bijection from ColSubval(T → ( ˆG, ˆγ)) to Emb((Q
+♯
+τ, cτ) → (G, c)).
+It will turn out that this implies that the embedding ρ coincides with τ.
+(4) In order to prove Item (1) we will proceed as follows.
+(i) We show that all 2-paths (including rays) of Tcol are external, except for 2k rays of
+length b and 2k rays of length a. Note that we already established this claim for
+2-paths of lengths greater than b in Lemma 73.
+(ii) Then we show that Tcol contains two degree-1 vertices in each of the vertex sets L
+and M of G (within ˆG) — see Figure 17, recalling that, for each vertex gadget, the
+sets L and M denote the vertex subsets of G that are coloured by c with ℓ and m.
+The point of this is that we will also prove that Tcol has two degree-1 vertices in S
+(Item 4iv) — this will split off the part of Tcol corresponding to a single gadget, so
+we will only have to study the embedding of Tcol within each gadget. We prove
+the claim about L and M by using the fact that Tcol is isomorphic to T and that
+all 2-paths longer than b are external. This implies that if vi and vi′ are the two
+vertices of ∆ sharing this gadget then the 2-paths between V 2
+i and V 2
+i′ are covered
+by two rays in Tcol, both of which end in L.
+(iii) We next show that the degree-1 vertices in (4ii) are the endpoints of 2k rays of
+length b. We have already seen that for each of the k gadgets the endpoints of
+these rays are in L and M. For the i’th gadget, the sources are in V 1
+i and V 2
+i If
+b > a then we show that all remaining 2-paths of length b and also all 2-paths with
+lengths in a + 1, . . . , b − 1 are external. The proof of this claim relies on the same
+arguments as the proof of Lemma 73.
+(iv) Next, we show that for each gadget, Tcol contains two degree-1 vertices in S — see
+Figure 17. The proof uses the fact that all 2-paths longer than a that are not
+covered by (4iii) are external.
+(v) We next show that the degree-1 vertices in (4iv) are the endpoints of 2k rays of
+length a. We have already seen that for each of the k gadgets the endpoints of
+these rays are in S. For the i’th gadget, the source is in V 1
+i .
+(vi) The remaining details of the proof rely on the fact that the tree Tcol is valid.
+We now provide the proof in detail; for convenience, we also restate the lemma.
+▶ Lemma 68. Suppose that |c−1(v)| is odd for each v ∈ V (Q). Then ⊕ColSub(T → ( ˆG, ˆγ)) =
+⊕Emb((Q
+♯
+τ, cτ) → (G, c)).
+Proof. We will prove that for any Tcol ∈ ColSubval(T → ( ˆG, ˆγ)), Item (1) of the proof overview
+holds.
+Using this fact and the argument from Item (2) of the proof overview, we conclude that
+for any Tcol ∈ ColSubval(T → ( ˆG, ˆγ)), ρ(Tcol) = τ.
+Recall that every edge-colourful subgraph of G induces a fracture of Q.
+Let G′ be an element of Emb((Q
+♯
+τ, cτ) → (G, c)).
+This means that G′ is an edge-
+colourful subgraph of G that induces τ. We wish to see how G′ can be extended to some
+Tcol
+′ ∈ ColSubval(T → ( ˆG, ˆγ)). We know from Item (1) that any Tcol
+′′ ∈ ColSubval(T → ( ˆG, ˆγ))
+can only be embedded in ˆG in one way, so G′ can only be extended in one way. The details
+are as follows. We claim that there is only one possible extension because T ′ has to be
+included and item (b) of (1) ensures that, for each j ∈ [2], the vertex pj
+i is connected to wj
+i .
+The rest of (1) shows the unique way to include the rays, so the extension is unique.
+Let β be the function from ColSubval(T → ( ˆG, ˆγ)) that maps any element Tcol to Tcol[G].
+Note that Tcol[G] ∈ Emb((Q
+♯
+τ, cτ) → (G, c)) since ρ(Tcol) = τ and ρ(Tcol) is a function of
+
+L. A. Goldberg and M. Roth
+45
+Figure 17 An embedding Tcol of T in ˆG that yields the fracture τ. We will show that this is the
+only way to embed T in ˆG in such a way that each edge-colour is used precisely once. Note that
+dashed lines depict paths in Tcol, and solid lines depict edges in Tcol.
+Tcol[G]. Let β′ be the function that maps an element of Emb((Q
+��
+τ, cτ) → (G, c)) to its unique
+extension in ColSubval(T → ( ˆG, ˆγ)). Note that β◦β′ and β′◦β are both the identity. Therefore
+β is a bijection and |ColSubval(T → ( ˆG, ˆγ))| = |Emb((Q
+♯
+τ, cτ) → (G, c))|.The lemma follows
+from Lemma 70.
+To finish the proof, we will fix Tcol ∈ ColSubval(T → ( ˆG, ˆγ)) and we will show that Item (1)
+of the proof overview holds. Part (a) of (1) is trivial since Tcol is edge-colourful so it contains
+T ′. The first sentence of (b) is also trivial. We will next focus on (c)–(g), noting along the
+way when the rest of (b) is proved.
+Recall from Definition 59 that, for each i ∈ [k], the graph Q contains
+for each vertex vj such that ∆ has an edge e = {vi, vj} with C(e) = m, a path Pi,j of
+length 2b from v1
+i to v1
+j , and
+for each vertex vj such that ∆ has an edge e = {vi, vj} with C(e) = ℓ, a path Pi,j of
+length 2b from v2
+i to v2
+j .
+
+46
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+Recall from Definition 6 that cE maps edges of G to edges of Q. Furthermore, G is a
+subgraph of ˆG, see Definition 66 (A). Let Tcol(i, j) be the subgraph of Tcol[G] induced by
+edges e of G such that cE(e) is in the path Pi,j
+By construction, Tcol(i, j) is the union of some number of paths. We will next argue that
+it is the union of exactly two disjoint length-b paths:
+If Tcol(i, j) has more than two components then at least one component is disconnected
+from T ′ in Tcol, contradicting the fact that Tcol is a tree.
+If Tcol(i, j) is a single path then it is contained in a 2-path of length at least 2b. Since
+this 2-path contains an edge in G, it is not external (Definition 71). This contradicts
+Lemma 73.
+If Tcol(i, j) is the union of exactly two disjoint paths, one of which has length larger
+than b then this larger 2-path is contained in a 2-path that is not external contradicting
+Lemma 73
+What we have shown is that T(i, j) consists of two length-b paths. For some t ∈ {1, 2},
+one of these paths is from V t
+i and the other is from V t
+j . To be more precise and to fix the
+notation for t = 1, we have now shown that, for each i ∈ [k], Tcol[G] contains a path Rb(i, 1)
+of length b that starts at a vertex w1
+i ∈ V 1
+i . We refer to the other end of this path as ub(i, 1).
+The vertex ub(i, 1) has degree 1 and is contained in M (i.e., in c−1(m)). We next argue that
+w1
+i has degree at least 3 in Tcol. (See Figure 18.)
+If w1
+i has degree 1 in Tcol then Tcol is disconnected, contradicting the fact that it is a tree.
+If w1
+i has degree 2 in Tcol, then Tcol has a ray of length at least b + 1 that is not external,
+which is again a contradiction.
+By the same reasoning, Tcol contains a ray Rb(i, 2) of length b that starts at a vertex w2
+i ∈ V 2
+i
+and ends at a vertex ub(i, 2). The ray Rb(i, 2) is contained in Tcol[G].
+We have just finished parts (d) and (e) of (1) and the part of (g) that concerns length b.
+So what we have shown corresponds to Figure 18. We would now like to prove parts (c) and
+(f) but unfortunately these are more difficult because we have to show where the rays with
+lengths between a and b are embedded so that we can argue about where the length-a rays
+are embedded.
+Define ˆR := �k
+i=1{Rb(i, 1), Rb(i, 2)}. Recall that k, a, and b are positive integers with
+a ≤ b and k ≥ 2 and that T has Fa,b(T) ≥ 2k and Tcol ∼= T. Also, Rb
+col is the set of length-b
+rays in Tcol and Rb is the set of length-b rays in T. (See Figure 12.) Using the notation that
+we have established, we will prove the following claims.
+Claim 1: Let P ∈ (Rb
+col \ ˆR) ∪ Pb
+col. If a < b then P is external.
+We prove Claim 1 for the case where P ∈ Rb
+col \ ˆR. The other case is similar but easier.
+Observe that |Rb| ≥ 2k since Fa,b(T) ≥ 2k. So Rb can be partitioned as follows
+Rb[S] is the set of the 2k length-b rays Fb(i, j) whose sources are s1
+1, . . . , s2
+k and which
+are depicted as red dashed lines in Figure 12.
+Rb[T] = Rb \ Rb[S] contains the remaining rays of length b.
+Our goal is to show that all rays in Rb
+col \ ˆR are external. To do this, we first show that
+|Rb[T]| = |Rb
+col \ ˆR| and we then provide an injection from Rb[T] to Rb
+col \ ˆR in which all
+elements of the range are external rays.
+To show that |Rb[T]| = |Rb
+col \ ˆR|, first note that |Rb| = |Rb
+col| because T and Tcol are
+isomorphic. We further have |Rb[S]| = | ˆR| = 2k.
+We next define the (injective) map from Rb[T] to Rb
+col\ ˆR . For any ray R = r0, r1, . . . , rb ∈
+Rb[T] we proceed as follows.
+
+L. A. Goldberg and M. Roth
+47
+Figure 18 Illustration of the embedding of Tcol after the rays of length b are analysed. Solid
+lines depict edges, dashed lines depict paths, and dash-dotted lines depict sequences of edges (the
+identification of the endpoints of which we have not yet been determined). Note that both Rb(i, 1)
+and Rb(i, 2) must be of length b. Except for those two rays, the identification of endpoints of
+the remaining edges that are incident to G (within ˆG) has not been determined yet either; this is
+depicted by the dotted circles inside the colour classes. The fracture ρ induced by Tcol will depend
+on the identification of the edges of Tcol, both endpoints of which lie in G. The goal is to show that
+the endpoints have to be identified precisely as depicted in Figure 17.
+
+48
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+If r0 is not among the designated sources sj
+i, then R is fully contained in T ′ (see Figure 12)
+and thus R is a ray in Tcol. We map R to itself. Note that R is external since it is fully
+contained in T ′.
+Otherwise, r0 = sj
+i and R is one of the rays depicted as black dashed lines in Figure 12.
+Since Tcol is edge-colourful, and by construction of ˆG, Tcol contains a path R′ = x, r1, . . . , rb
+where x ∈ V j
+i . (See Figure 15.) If x has degree 1 in Tcol then Tcol is disconnected, which
+is not true. If x has degree 2 in Tcol then Tcol has a non-external ray which is longer
+than b, which is also a contradiction by Lemma 73. Thus, x has degree at least 3 in Tcol,
+and R′ is an external ray. We map R to R′.
+This concludes the proof of Claim 1 for the case where P ∈ Rb
+col \ ˆR. ■
+Claim 2: Suppose that there is an integer t′ such that a < t′ < b. Suppose that P ∈ Rt′
+col∪Pt′
+col.
+Then P is external.
+In order to explain the proof of Claim 2, recall that we have established the following
+facts about 2-paths in Tcol in Lemma 73 and Claim 1.
+Every 2-path of length greater than b is external.
+Every 2-path of length b is either a ray in ˆR or is external.
+With those 2-paths covered, the proof of Claim 2 is analogous to the proof of Lemma 73. ■
+Using Claims 1 and 2 we will now prove parts (c) and (f) of (1). For each 2-path whose
+length is larger than a, we have already shown that it is in ˆR or we have shown that it is
+external. In order to prove (c) we will show that, for each edge {vi, vi′} of ∆ with colour s,
+the sequence of edges in Tcol between V 1
+i and V 1
+i′ is the union of two disjoint length-a rays.
+This is formalised as follows.
+Note that for each edge {vi, vj} of ∆ coloured by the 3-edge-colouring C with s, there is
+a path Pi,j of length 2a from v1
+i to v1
+j . Recall that cE maps edges of G to edges of Q. We
+write Tcol(i, j) for the subgraph of Tcol[G] induced by edges e of G such that cE(e) is in the
+path Pi,j. By construction, Tcol(i, j) is the union of some number of paths. We will next
+argue that it is the union of exactly two disjoint length-a paths:
+If Tcol(i, j) has more than two components then at least one component is disconnected
+from T ′ in Tcol, contradicting the fact that Tcol is a tree.
+If Tcol(i, j) is a single path then it is contained in a 2-path of length at least 2a. Since
+this 2-path contains an edge in G, it is not external (Definition 71). Additionally, it is
+not included in ˆR. This contradicts the aforementioned fact that each 2-paths of length
+at least a + 1 is external or included in the set ˆR.
+If Tcol(i, j) is the union of exactly two disjoint paths, one of which has length larger
+than a, then this larger path yields a contradiction similarly to the previous case.
+What we have shown is that T(i, j) consists of two length-a paths. One of these paths
+is from V 1
+i and the other is from V 1
+j . To be more precise and to fix the notation, we have
+now shown that, for each i ∈ [k], Tcol[G] contains a path Ra(i, 1) of length a that starts at a
+vertex ˆw1
+i ∈ V 1
+i . We refer to the other end of this path as ua(i, 1). The vertex ua(i, 1) has
+degree 1 and is contained in S (i.e., in c−1(s)). So we have established Part (c) of item (1).
+Consider Figure 19 for an illustration of all the information we gathered so far. (The vertices
+labelled zj
+i and the edge set Ea
+i in the figure will be discussed below).
+To finish the proof of item (1) we will show part (f) and the rest of part (b). We take these
+together. Recall that for every i ∈ [k] there is a path P a
+i = v1
+i , y1, . . . , ya−1, v2
+i of length a in
+Q from v1
+i to v2
+i . Since Tcol is edge-colourful, it includes each of the colours of the edges on
+this path exactly once — these colours are γ−1
+E ({v1
+i , y1}),γ−1
+E ({y1, y2}), . . . ,γ−1
+E ({ya−1, v2
+i }).
+
+L. A. Goldberg and M. Roth
+49
+Figure 19 Depiction of the embedding of Tcol as established after Claim 2 (in the proof of
+Lemma 68).
+Solid lines depict edges, dashed lines depict paths, and dash-dotted lines depict
+sequences of edges (the identification of the endpoints of which has not yet been determined). Note
+that we have not yet determined how the endpoints inside of the colour classes V 1
+i and V 2
+i are
+identified either; this is depicted by the dotted circles inside these colour classes. Proving that
+the embedding of Tcol is as depicted in Figure 17 requires us to show that all endpoints in V 2
+i are
+identified, and that all endpoints in V 1
+i , except for x1
+i , are identified.
+
+50
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+Under the edge colouring cE, the same edges of Tcol are coloured with the colours {v1
+i , y1},
+{y1, y2}, . . . , {ya−1, v2
+i }.
+Let e1, . . . , ea be the edges of Tcol with those colours; we write Ea
+i for this set of edges
+(as is depicted in Figure 19). We let x1
+i be the vertex of Tcol which is contained in V 1
+i and
+incident to e1, and we let x2
+i be the vertex of Tcol which is contained in V 2
+i and incident to ea.
+Let z1
+i and z2
+i be the vertices of Tcol in V 1
+i and V 2
+i that are adjacent to p1
+i and p2
+i — those
+vertices are depicted in Figure 19 and we point out that, a priori, x1
+i might be equal to to z1
+i
+and x2
+i might be equal to z2
+i .
+Claim 3: There are no vertices in V (Tcol) ∩ V 1
+i other than z1
+i , x1
+i , w1
+i , ˆw1
+i and vertices in
+the d1
+i rays.
+To prove Claim 3, assume for contradiction that z is such a vertex. Recall that V 1
+i is an
+independent set (because vertices in V 1
+i all receive the same colour under c.) Since Tcol is
+connected, z has a neighbour outside of V 1
+i but all of the edge colours incident to V 1
+i are
+already used. ■
+The proof of the following claim is similar.
+Claim 4: There are no vertices in V (Tcol) ∩ V 2
+i other than z2
+i , x2
+i , w2
+i , and vertices in the d2
+i
+rays. ■
+Claim 5: Both z1
+i and z2
+i have degree at least 3 in Tcol. We prove the claim for z1
+i ; an
+analogous argument applies for z2
+i . Assume first for contradiction that z1
+i has degree 1. Since
+Tcol is connected, Claim 2.5 implies that |V (Tcol)∩V 1
+i | = 2 so x1
+i = w1
+i = ˆw1
+i and the depicted
+vertices in the d1
+i rays are also identified with this vertex. By Definition 69, Tcol is invalid,
+giving a contradiction.
+Now assume for contradiction that z1
+i has degree 2. We consider two subcases:
+z1
+i is identical to x1
+i . Then Tcol is disconnected, which yields a contradiction.
+z1
+i is identical to w1
+i or ˆw1
+i . This is an immediate contradiction since sources cannot have
+degree 2 (recall that we already established Ra(i, 1) and Rb(i, 2) to be rays).
+zi is incident to the first edges of one of the additional d1
+i outgoing paths. However, in
+this case, Tcol can only be connected if there is precisely one further vertex of Tcol in V 1
+i
+that is incident to all outgoing edges not covered by z1
+i . However, in this case, Tcol is an
+invalid tree, yielding the desired contradiction.
+Since the three cases above are exhaustive, the proof of Claim 5 is concluded. ■
+Next we need the following property:
+Claim 6: Let t be a positive integer. If t < a then each ray in Rt
+col is external.
+For the proof, recall that |Rt| = |Rt
+col| since T and Tcol are isomorphic. Note that each
+ray R of length t of T is either fully contained in T ′, or it is one of the dj
+i black rays for some
+(i, j) ∈ [k] × [2]. (See Figure 12) If R is fully contained in T ′, then R is also contained in
+Rt
+col and it is external.
+If R = r0, r1, . . . , rt is one of the dj
+i black rays, then Tcol contains a path R′ = y0, r1, . . . , rt
+for some y0 ∈ V j
+i . Suppose that y0 has degree at least 3 in Tcol. Then, as in Claim 1, R′ is
+then an external ray, and we are finished. We next consider the case where y0 has degree 1
+or 2 in Tcol.
+If the degree is 1, then Tcol is disconnected, leading to a contradiction. If the degree is 2,
+then y0 ̸= zj
+i by Claim 5. Thus, the only way for Tcol not being disconnected is y0 = xj
+i and
+Tcol[Ea
+i ] is a path. However, then we obtained a ray of length at least a + t which is neither
+external, nor in the set ˆR. Thus, we obtain a contradiction by either Claim 2 (a + t < b), or
+by Claim 1 (a + t = b), or by Lemma 73 (a + t > b). This concludes the proof of Claim 6. ■
+
+L. A. Goldberg and M. Roth
+51
+Next, observe that Tcol cannot connect z1
+i and z2
+i via a path within G, that is, via a path
+containing the edges Ea
+i : Otherwise Tcol would contain a cycle since p1
+i and p2
+i are connected
+by a path within T ′. We will see that z1
+i and z2
+i are sources of Tcol.
+Let S be the set of all sources of T. Consider the multi-set of leaf-degrees of T
+degL(S) := {{degL(s) | s ∈ S}} .
+Let Scol be the set of all sources of Tcol and let degL(Scol) be the muti-set of leaf-degrees Tcol.
+Since Tcol and T are isomorphic, the multi-sets degL(S) and degL(Scol) are equal.
+Suppose that s ∈ S is a source of T not among the designated sources sj
+i. Then s is
+contained in T ′, and it is also a source of Tcol. Since all of the zj
+i have degree at least 3 in
+Tcol (by Claim 5), they cannot be part of further rays with source s in Tcol so s has the same
+leaf-degree in T and in Tcol.
+We next show that for each i ∈ [k], the set V 1
+i ∪ V 2
+i contains at least 2 sources of Tcol:
+Either z1
+i is a source or it is connected by a 2-path within Tcol[G] to another source. However,
+the only vertices reachable in Tcol[G] from z1
+i that can have degree at least 3 are contained in
+V 2
+i . Similarly, either x2
+i is a source or it is connected by a 2-path within Tcol[G] to a source
+in V 1
+i . We have already seen that z1
+i cannot be connected to z2
+i within Tcol[G]. Thus the
+sources reachable from z1
+i and z2
+i within Tcol[G] must be distinct, and we have shown that
+for each i ∈ [k], the set V 1
+i ∪ V 2
+i contains at least 2 sources of Tcol.
+Since Tcol and T have the same number of sources, and since 2k sources of T are not
+contained in T ′, we have thus shown that for each i ∈ [k], the set V 1
+i ∪ V 2
+i contains precisely
+2 sources of Tcol; let us denote those 2 sources by ˆz1
+i and ˆz2
+i .
+Now, consider the following subsets of S and Scol:
+S′ := {s1
+1, s2
+1, . . . , s1
+k, s2
+k} is the set of designated sources.
+S′
+col := {ˆz1
+1, ˆz2
+1, . . . , ˆz1
+k, ˆz2
+k} is the set of sources of Tcol in G (within ˆG).
+Since we already know that degL(S \ S′) = degL(Scol \ S′
+col) (those are the sources in T ′), we
+require degL(S′) = degL(S′
+col) for T and Tcol to be isomorphic.
+What follows is the final claim within the proof of this lemma.
+Claim 7: For all i ∈ [k], the following five conditions are satisfied:
+{z1
+i , z2
+i } = {ˆz1
+i , ˆz2
+i }, that is, z1
+i and z2
+i are the two sources in V 1
+i ∪ V 2
+i .
+Tcol contains precisely 2 vertices in V 1
+i : One is z1
+i and one is x1
+i .
+x1
+i has degree 1. Further, z1
+i , w1
+i , ˆw1
+i and all the endpoints of the d1
+i rays are the same.
+Tcol contains precisely 1 vertex in V 2
+i . Further, z2
+i , x2
+i , w2
+i and all endpoints of the d2
+i
+rays are the same.
+Tcol[Ea
+i ] is a ray with source z2
+i (= x2
+i = w2
+i ).
+Before proving Claim 7, we point out that (1b) and (1f) follow immediately from Claim
+7; see Figure 17 and observe that Tcol[Ea
+i ] becomes the ray Ra(i, 2), and x1
+i becomes the
+endpoint ua(i, 2) of Ra(i, 2) for each i ∈ [k]. Thus the proof of this lemma is concluded if
+Claim 7 is proved, which is done below:
+We first show that {z1
+i , z2
+i } = {ˆz1
+i , ˆz2
+i } for each i ∈ [k]. Let Φ = �
+s∈S′ degL(s) and
+Φcol = �
+s∈S′
+col degL(s). Observe that degL(S′) = degL(S′
+col) implies Φ = Φcol.
+We start by observing that
+degL(ˆz1
+i ) + degL(ˆz2
+i ) ≤ (d1
+i + 2) + (d2
+i + 1) + 2.
+There are d1
+i rays from V 1
+i and also Ra(i, 1) and Rb(i, 1). There are d2
+i rays from V 2
+i and
+also Rb(i, 2). There is also Ea
+i which could form two rays.
+
+52
+Parameterised and Fine-grained Subgraph Counting, modulo 2
+We next show that Ea
+i cannot form two rays. Assume for contradiction that is does.
+Since Tcol is connected, z1
+i , w1
+i , ˆw1
+i , x1
+i and all the endpoints of the d1
+i rays are identical,
+and z2
+i , w2
+i , x2
+i and all the endpoints of the d2
+i rays are identical.
+Now, if Tcol[Ea
+i ] would be the disjoint union of two rays of length less than a with sources
+z1
+i and z2
+i then those rays are non-external rays of length less than a, contradicting Claim
+6. We have now shown
+degL(ˆz1
+i ) + degL(ˆz2
+i ) ≤ (d1
+i + 2) + (d2
+i + 2) .
+(11)
+Next, note that by definition of the dj
+i (see Figure 12), the following holds:
+(d1
+i + 2) + (d2
+i + 2) = degL(s1
+i ) + degL(s2
+i )
+(12)
+We have now shown that
+degL(ˆz1
+i ) + degL(ˆz2
+i ) ≤ degL(s1
+i ) + degL(s2
+i ).
+Finally, we will show that z1
+i and z2
+i are sources to finish the first bullet point.
+Consider z1
+i , and recall that is has degree at least 3 by Claim 5, and assume for contradic-
+tion that it is not a source of Tcol. Then z1
+i = x1
+i , and Tcol[Ea
+i ] is a path, and x2
+i is source
+(since it is the only vertex in V (Tcol) ∩ V 2
+i that might have degree at least 3, except for
+z2
+i ). Note that this also implies that z2
+i is a source. Thus {ˆz1
+i , ˆz2
+i } = {x2
+i , z2
+i }. In this
+case, we have
+degL(ˆz1
+i ) + degL(ˆz2
+i ) ≤ d2
+i + 1 < degL(s1
+i ) + degL(s2
+i ) .
+Consequently, using (11) and (12), we have Φcol < Φ, which is a contradiction. Thus z1
+i
+is a source of Tcol, and a similar argument shows that z2
+i is a source of Tcol as well.
+We now prove the remaining items. In what follows, using the previous bulleted item, we
+can assume that w.l.o.g. ˆz1
+i = z1
+i and ˆz2
+i = z2
+i for all i ∈ [k]. First, recall that we ordered
+the sj
+i by their leaf-degrees, that is
+degL(s1
+1) ≥ degL(s2
+1) ≥ · · · ≥ degL(s2
+k) ≥ 2 .
+If x1
+1 were equal to z1
+1, then Tcol can only be connected if there is only one vertex in V 1
+1 ,
+that is, all edges incident to V 1
+1 are in fact incident to z1
+1. However, in that case, we have
+degL(z1
+1) = degL(s1
+1) + 1 (by construction of ˆG), and thus the multi-sets cannot be equal
+anymore. Hence x1
+1 ̸= z1
+1.
+If x1
+1 had degree 2, then there would have been a ray of length at least a+1 that originates
+in V 2
+1 (otherwise Tcol would have been disconnected). However, this ray would neither
+be external, nor among the rays in ˆR, contradicting either Lemma 73 or the previous
+sequence of claims. Finally, if x1
+1 had degree at least 3, then Tcol would have contained
+more sources than T, which also yields a contradiction.
+This shows that x1
+1 has degree 1. However, this implies that Tcol can only contain one vertex
+in V 2
+i ; otherwise Tcol would be disconnected. Note that we have just proved the remaining
+items of Claim 7 for i = 1. Additionally, we have shown that degL(z1
+1) = degL(s1
+1) and
+degL(z2
+1) = degL(s2
+1) Hence we can remove those two numbers from the multi-sets and
+continue recursively with i = 2. This concludes the proof of Claim 7, and thus the proof
+of the overall lemma.
+◀
+We are now ready to conclude the case for trees of unbounded fork number.
+
+L. A. Goldberg and M. Roth
+53
+▶ Lemma 74. Let T be a recursively enumerable class of trees of unbounded fork number.
+Then ⊕Sub(T ) is ⊕W[1]-hard.
+Proof. We proceed similarly to Lemma 44. However, we have to take care of some subtleties.
+First, we start with a class C of cubic bipartite graphs of unbounded treewidth. Next, we
+wish to rely on Lemma 68 to obtain the identity
+⊕Emb((Q
+♯
+τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)),
+where τ is the fracture defined in Definition 61. Unfortunately, Lemma 68 only yields the
+above identity if, for each v ∈ V (Q), |c−1(v)| is odd, that is, each colour class of vertices of
+G has odd cardinality. However, this property can easily be achieved. Let (G′, c′) be the
+Q-coloured graph obtained from (G, c) by adding to each even colour class one fresh isolated
+vertex. Since Q
+♯
+τ does not have isolated vertices, this operation does not change the number
+of colour-preserving embeddings. In combination with Lemma 68 we thus obtain
+⊕Emb((Q
+♯
+τ, cτ) → (G, c)) = ⊕Emb((Q
+♯
+τ, cτ) → (G′, c′)) = ⊕ColSub(T → ( ˆG′, ˆγ)).
+From here on, we can proceed analogously to the proof of Lemma 44.
+◀
+4.4
+The Dichotomy Theorem for Trees
+We are now able to prove Theorem 5, i.e., an exhaustive and explicit parameterised complexity
+classification for counting trees modulo 2:
+▶ Theorem 5. Let T be a recursively enumerable class of trees. If T is matching splittable,
+then ⊕Sub(T ) is fixed-parameter tractable. Otherwise ⊕Sub(T ) is ⊕W[1]-complete.
+Proof. The fixed-parameter tractability result, as well as the fact that ⊕Sub(T ) is always
+contained in ⊕W[1] were both shown in [11]. Hence, it remains to prove ⊕W[1]-hardness if
+T is not matching splittable.
+By Lemma 32 each class T of trees that is not matching splittable has unbounded
+C-number, unbounded star number, or unbounded fork number. Finally, each of these three
+cases yields ⊕W[1]-hardness as established by Lemmas 44, 56, and 74.
+◀
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