Counting Small Induced Subgraphs: Scorpions Are Easy but Not Trivial

We consider the parameterized problem $\#$IndSub$(Φ)$ for fixed graph properties $Φ$: Given a graph $G$ and an integer $k$, this problem asks to count the number of induced $k$-vertex subgraphs satisfying $Φ$. Dörfler et al. [Algorithmica 2022] and Roth et al. [SICOMP 2024] conjectured that $\#$IndSub$(Φ)$ is $\#$W[1]-hard for all non-meager properties $Φ$, i.e., properties that are nontrivial for infinitely many $k$. This conjecture has been confirmed for several restricted types of properties, including all hereditary properties [STOC 2022] and all edge-monotone properties [STOC 2024]. In this work, we refute this conjecture by showing that scorpion graphs, certain $k$-vertex graphs which were introduced more than 50 years ago in the context of the evasiveness conjecture, can be counted in time $O(n^4)$ for all $k$. A simple variant of this construction results in graph properties that achieve arbitrary intermediate complexity assuming ETH. We formulate an updated conjecture on the complexity of $\#$IndSub$(Φ)$ that correctly captures the complexity status of scorpions and related constructions.