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).

翻译：给定图类$\mathcal{H}$，问题$\oplus\mathsf{Sub}(\mathcal{H})$定义如下。输入为一个图$H\in \mathcal{H}$以及任意图$G$，需计算$G$中与$H$同构的子图个数模$2$。本研究旨在确定哪些$\mathcal{H}$类能使问题$\oplus\mathsf{Sub}(\mathcal{H})$具有固定参数可解性（FPT），即存在时间$f(|H|)\cdot |G|^{O(1)}$的求解算法。Curticapean、Dell和Husfeldt（ESA 2021）猜想：$\oplus\mathsf{Sub}(\mathcal{H})$为FPT当且仅当允许的模式类$\mathcal{H}$是“匹配可分裂的”，即存在固定常数$B$，使得每个$H\in \mathcal{H}$可通过删除至多$B$个顶点转化为匹配图（所有顶点度数不超过1的图）。假设随机指数时间假设成立，我们证明了该猜想对（I）所有遗传模式类$\mathcal{H}$，以及（II）所有树模式类（即每个$H\in \mathcal{H}$均为树的类）成立。此外，针对遗传模式类（I），我们还给出了几乎紧的细粒度上下界。