Parameterised and Fine-Grained Subgraph Counting, Modulo 2

Abstract

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