We study the problems of counting copies and induced copies of a small pattern graph $H$ in a large host graph $G$. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns $H$. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time $f(H)\cdot |G|^{O(1)}$ for some computable function $f$. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes $\mathcal{G}$ as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis: (1) Counting $k$-matchings in a graph $G\in\mathcal{G}$ is fixed-parameter tractable if and only if $\mathcal{G}$ is nowhere dense. (2) Counting $k$-independent sets in a graph $G\in\mathcal{G}$ is fixed-parameter tractable if and only if $\mathcal{G}$ is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if $\mathcal{G}$ is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting $k$-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in $F$-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting $k$-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19).

翻译：我们研究在大型宿主图 $G$ 中计数小型模式图 $H$ 的副本和诱导副本的问题。近期工作根据模式 $H$ 的结构限制完全分类了这些问题的复杂性。在本工作中，我们致力于分析受限模式与受限宿主情形下复杂性的更具挑战性任务。具体而言，我们探究何种允许的模式与宿主族蕴含固定参数可解性，即存在运行时间 $f(H)\cdot |G|^{O(1)}$ 的算法（其中 $f$ 为可计算函数）。主要结果为满足自然闭包性质的族提供了详尽且显式的复杂性分类。其中，我们将子图封闭图类 $\mathcal{G}$ 中计数小型匹配与独立集的问题确定为核心研究对象，并基于指数时间假设建立以下清晰二分性：（1）在 $G\in\mathcal{G}$ 中计数 $k$-匹配是固定参数可解的当且仅当 $\mathcal{G}$ 无处稠密；（2）在 $G\in\mathcal{G}$ 中计数 $k$-独立集是固定参数可解的当且仅当 $\mathcal{G}$ 无处稠密。此外，若 $\mathcal{G}$ 为某处稠密（即非无处稠密），我们获得了近乎紧的条件性下界。这些分类的基础情形涵盖了匹配与独立集问题上先前大量成果，包括二分图中的 $k$-匹配计数（Curticapean, Marx; FOCS 14）、$F$-可着色图中的计数（Roth, Wellnitz; SODA 20）、退化图中的计数（Bressan, Roth; FOCS 21），以及二分图中 $k$-独立集计数（Curticapean 等; Algorithmica 19）。