We study the parameterized complexity of #IndSub($\Phi$), where given a graph $G$ and an integer $k$, the task is to count the number of induced subgraphs on $k$ vertices that satisfy the graph property $\Phi$. Focke and Roth [STOC 2022] completely characterized the complexity for each $\Phi$ that is a hereditary property (that is, closed under vertex deletions): #IndSub($\Phi$) is #W[1]-hard except in the degenerate cases when every graph satisfies $\Phi$ or only finitely many graphs satisfy $\Phi$. We complement this result with a classification for each $\Phi$ that is edge monotone (that is, closed under edge deletions): #IndSub($\Phi$) is #W[1]-hard except in the degenerate case when there are only finitely many integers $k$ such that $\Phi$ is nontrivial on $k$-vertex graphs. Our result generalizes earlier results for specific properties $\Phi$ that are related to the connectivity or density of the graph. Further, we extend the #W[1]-hardness result by a lower bound which shows that #IndSub($\Phi$) cannot be solved in time $f(k) \cdot |V(G)|^{o(\sqrt{\log k/\log\log k})}$ for any function $f$, unless the Exponential-Time Hypothesis (ETH) fails. For many natural properties, we obtain even a tight bound $f(k) \cdot |V(G)|^{o(k)}$; for example, this is the case for every property $\Phi$ that is nontrivial on $k$-vertex graphs for each $k$ greater than some $k_0$.

翻译：我们研究#IndSub($\Phi$)的参数复杂性，其中给定图$G$和整数$k$，任务是计数满足图性质$\Phi$的$k$个顶点上的诱导子图数量。Focke和Roth [STOC 2022]完全刻画了每个遗传性质（即在顶点删除下封闭）$\Phi$的复杂性：除退化情形（即每个图都满足$\Phi$，或仅有有限多个图满足$\Phi$）外，#IndSub($\Phi$)是#W[1]-难的。我们对此结果进行补充，对每个边单调性质（即在边删除下封闭）$\Phi$进行分类：#IndSub($\Phi$)是#W[1]-难的，除非在退化情形，即仅有有限多个整数$k$使得$\Phi$在$k$顶点图上非平凡。我们的结果推广了先前关于与图的连通性或密度相关的特定性质$\Phi$的结论。此外，我们通过一个下界扩展了#W[1]-难的结果，该下界表明除非指数时间假设（ETH）失败，否则任何函数$f$都无法在时间$f(k) \cdot |V(G)|^{o(\sqrt{\log k/\log\log k})}$内求解#IndSub($\Phi$)。对于许多自然性质，我们甚至获得了紧界$f(k) \cdot |V(G)|^{o(k)}$；例如，对于每个大于某个$k_0$的$k$值，在$k$顶点图上非平凡的每个性质$\Phi$均满足此条件。