The problem of subgraph counting asks for the number of occurrences of a pattern graph $H$ as a subgraph of a host graph $G$ and is known to be computationally challenging: it is $\#W[1]$-hard even when $H$ is restricted to simple structures such as cliques or paths. Curticapean and Marx (FOCS'14) show that if the graph $H$ has vertex cover number $τ$, subgraph counting has time complexity $O(|H|^{2^{O(τ)}} |G|^{τ+ O(1)})$. This raises the question of whether this upper bound can be improved for input graphs $G$ from a restricted family of graphs. Earlier work by Eppstein~(IPL'94) shows that this is indeed possible, by proving that when $G$ is a $d$-degenerate graph and $H$ is a biclique of arbitrary size, subgraph counting has time complexity $O(d 3^{d/3} |G|)$. We show that if the input is restricted to $d$-degenerate graphs, the upper bound of Curticapean and Marx can be improved for a family of graphs $H$ that includes all bicliques and satisfies a property we call $(c,d)$-locatable. Importantly, our algorithm's running time only has a polynomial dependence on the size of~$H$. A key feature of $(c,d)$-locatable graphs $H$ is that they admit a vertex cover of size at most $cd$. We further characterize $(1,d)$-locatable graphs, for which our algorithms achieve a linear running time dependence on $|G|$, and we establish a lower bound showing that counting graphs which are barely not $(1,d)$-locatable is already $\#\text{W}[1]$-hard. We note that the restriction to $d$-degenerate graphs has been a fruitful line of research leading to two very general results (FOCS'21, SODA'25) and this creates the impression that we largely understand the complexity of counting substructures in degenerate graphs. However, all aforementioned results have an exponential dependency on the size of the pattern graph $H$.

翻译：子图计数问题要求计算模式图$H$在宿主图$G$中作为子图出现的次数，已知该问题在计算上具有挑战性：即使将$H$限制为简单结构（如团或路径），该问题也是$\#W[1]$-难的。Curticapean和Marx（FOCS'14）证明，若图$H$的顶点覆盖数为$τ$，则子图计数的时间复杂度为$O(|H|^{2^{O(τ)}} |G|^{τ+ O(1)})$。这引发了一个问题：对于来自受限图族的输入图$G$，该上界是否可改进。Eppstein（IPL'94）的早期工作表明这确实是可能的，其证明了当$G$是$d$-退化图且$H$是任意大小的二分团时，子图计数的时间复杂度为$O(d 3^{d/3} |G|)$。我们证明，若输入限制为$d$-退化图，则对于一类包含所有二分团且满足我们称为$(c,d)$-可定位性质的图$H$，Curticapean和Marx的上界可被改进。重要的是，我们算法的运行时间仅对$H$的大小具有多项式依赖。$(c,d)$-可定位图$H$的一个关键特征是它们允许大小至多为$cd$的顶点覆盖。我们进一步刻画了$(1,d)$-可定位图，对此类图我们的算法实现了对$|G|$的线性运行时间依赖，并建立了一个下界，表明对几乎不满足$(1,d)$-可定位的图进行计数已是$\#\text{W}[1]$-难的。我们注意到，对$d$-退化图的限制已成为一条富有成果的研究路线，并催生了两项非常一般性的结果（FOCS'21, SODA'25），这给人留下我们已基本理解退化图中子结构计数复杂度的印象。然而，所有上述结果均对模式图$H$的大小具有指数依赖。