Parameterised and Fine-grained Subgraph Counting, modulo 2
Given a class of graphs โ, the problem โ๐ฒ๐๐ป(โ) is defined as follows. The input is a graph Hโโ 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 โ the problem โ๐ฒ๐๐ป(โ) is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|)ยท |G|^O(1). Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that โ๐ฒ๐๐ป(โ) is FPT if and only if the class of allowed patterns โ is "matching splittable", which means that for some fixed B, every H โโ 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 โ, and (II) all tree pattern classes, i.e., all classes โ such that every Hโโ is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I).
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