Counting Small Induced Subgraphs: Hardness of Symmetry-Based Properties

Abstract

Jerrum and Meeks (TOCT, JCSS 2015) introduced the counting problems $\text{IndSub}(\Phi)$ for fixed graph properties $\Phi$: Given an input graph $G$ and $k\in\mathbb N$, count the $k$-vertex subsets $S \subseteq V(G)$ such that the induced subgraph $G[S]$ satisfies $\Phi$. For recursively enumerable $\Phi$, it is known that $\text{IndSub}(\Phi)$ is either #W[1]-hard or fixed-parameter tractable. A direct classification depending on $\Phi$ however still remains open. In particular, the status was open for the property of graphs without nontrivial automorphisms, also mentioned in a very recent survey on parameterized counting by Roth (Comput.~Sci.~Rev.~2026). This is a natural property that evades all currently known techniques for proving #W[1]-hardness, including a general toolkit based on Fourier analysis that was very recently introduced by Curticapean and Neuen (SODA~2025). In this paper, we show that counting induced $k$-vertex graphs without nontrivial automorphisms is #W[1]-hard by constructing ``clique scaffolds'', i.e., problem-specific restrictions of the property that enable a reduction from the $k$-clique problem. More generally, we show that for every finite group $Q$, counting $k$-vertex induced subgraphs with automorphism group $Q$ is #W[1]-hard.