Parameterised and Fine-Grained Subgraph Counting, Modulo 2 - Goldberg, Leslie Ann; Roth, Marc (Algorithmica, 2024)
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).