Parameterised and Fine-grained Subgraph Counting, modulo $2$

Given a class of graphs $\mathcal{H}$, the problem $\oplus\mathsf{Sub}(\mathcal{H})$ 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 $\oplus\mathsf{Sub}(\mathcal{H})$ is fixed-parameter tractable (FPT), i.e., solvable in time $f(|H|)\cdot |G|^{O(1)}$. Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that $\oplus\mathsf{Sub}(\mathcal{H})$ 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).