Parameterised and Fine-Grained Subgraph Counting, Modulo 2

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).