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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 tocompute, modulo $2$, the number of subgraphs of $G$ that are isomorphic to $H$.The goal of this research is to determine for which classes $\mathcal{H}$ theproblem $\oplus\mathsf{Sub}(\mathcal{H})$ 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 allowedpatterns $\mathcal{H}$ is "matching splittable", which means that for somefixed $B$, every $H \in \mathcal{H}$ can be turned into a matching (a graph inwhich every vertex has degree at most $1$) by removing at most $B$ vertices.  Assuming the randomised Exponential Time Hypothesis, we prove theirconjecture for (I) all hereditary pattern classes $\mathcal{H}$, and (II) alltree 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 thecase of hereditary patterns (I).

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