Parameterised and Fine-Grained Subgraph Counting, Modulo 2

Abstract

AbstractGiven a class of graphs $${\mathcal {H}}$$ H , the problem $$\oplus \text {{Sub}}({\mathcal {H}})$$ ⊕ Sub ( H ) is defined as follows. The input is a graph $$H\in {\mathcal {H}}$$ 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 $${\mathcal {H}}$$ H the problem $$\oplus \text {{Sub}}({\mathcal {H}})$$ ⊕ Sub ( H ) is fixed-parameter tractable (FPT), i.e., solvable in time $$f(|H|)\cdot |G|^{O(1)}$$ f ( | H | ) · | G | O ( 1 ) . Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that $$\oplus \text {{Sub}}({\mathcal {H}})$$ ⊕ Sub ( H ) is FPT if and only if the class of allowed patterns $${\mathcal {H}}$$ H is matching splittable, which means that for some fixed B, every $$H \in {\mathcal {H}}$$ 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 $${\mathcal {H}}$$ H , and (II) all tree pattern classes, i.e., all classes $${\mathcal {H}}$$ H such that every $$H\in {\mathcal {H}}$$ H ∈ H is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I).