Counting Small Induced Subgraphs: Hardness of Symmetry-Based Properties

2026-06-30

AI Summary

This work investigates the parameterized counting complexity of $k$-vertex induced subgraphs satisfying a fixed graph property $\Phi$, with a focus on symmetry conditions dictated by the structure of their automorphism groups. By initiating from the $k$-clique problem and employing a refined parameterized reduction based on a “clique gadget” construction, the study establishes—for the first time—that counting $k$-vertex induced subgraphs whose automorphism group is exactly a given finite group $Q$ is $\#\mathbf{W}[1]$-hard for any finite group $Q$. This result not only confirms the $\#\mathbf{W}[1]$-hardness in the case of trivial automorphism groups but also generalizes it to arbitrary finite groups, thereby overcoming limitations inherent in existing Fourier-analytic approaches and resolving a long-standing open problem in this direction.

Abstract

Jerrum and Meeks (TOCT, JCSS 2015) introduced the counting problems $\text{IndSub}(Φ)$ for fixed graph properties $Φ$: 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 $Φ$. For recursively enumerable $Φ$, it is known that $\text{IndSub}(Φ)$ is either #W[1]-hard or fixed-parameter tractable. A direct classification depending on $Φ$ 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.

Problem

• induced subgraphs
• automorphism group
• parameterized counting
• #W[1]-hardness
• symmetry-based properties

Innovation

• #W[1]-hardness
• induced subgraphs
• automorphism groups
• clique scaffolds
• parameterized counting