We study the problems of counting the homomorphisms, counting the copies, and counting the induced copies of a $k$-vertex graph $H$ in a $d$-degenerate $n$-vertex graph $G$. Our main result establishes exhaustive and explicit complexity classifications for counting subgraphs and induced subgraphs. We show that the (not necessarily induced) copies of $H$ in $G$ can be counted in time $f(k,d)\cdot n^{\max(\mathsf{imn}(H),1)}\cdot \log n$, where $f$ is some computable function and $\mathsf{imn}(H)$ is the size of the largest induced matching of $H$. Whenever the class of allowed patterns has unbounded induced matching number, this algorithm is essentially optimal: Unless the Exponential Time Hypothesis (ETH) fails, there is no algorithm running in time $f(k,d)\cdot n^{o(\mathsf{imn}(H)/\log \mathsf{imn}(H))}$ for any function $f$. In case of counting induced subgraphs, we obtain a similar classification along the independence number $\alpha$: we can count the induced copies of $H$ in $G$ in time $f(k,d)\cdot n^{\alpha(H)}\cdot \log n$, and if the class of allowed patterns has unbounded independence number, an algorithm running in time $f(k,d)\cdot n^{o(\alpha(H)/\log \alpha(H))}$ is impossible, unless ETH fails. In the language of parameterized complexity, our results yield dichotomies in fixed-parameter tractable and $\#\mathsf{W}[1]$-hard cases if we parameterize by the size of the pattern and the degeneracy of the host graph. Our results imply that several patterns cannot be counted in time $f(k,d)\cdot n^{o(k/\log k)}$, including $k$-matchings, $k$-independent sets, (induced) $k$-paths, (induced) $k$-cycles, and induced $(k,k)$-bicliques, unless ETH fails. Those lower bounds for exact counting are complemented with new algorithms for approximate counting of subgraphs and induced subgraphs in degenerate graphs.

翻译：我们研究的是计算同质性的问题, 计算副本, 并计算一个 $k$- verdex 平面图的诱导副本, 以美元计, 以美元计, 以美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计。 我们的主要结果为计算子图和子图, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 美元计, 时间计, 美元计, 美元计, 美元计, 美元计, 美元计, 时间计, 时间计, 时间计, 美元计, (k, 美元计, 美元计, 美元计, 美元计, 时间计, 美元) 。