Counting Small Induced Subgraphs: Scorpions Are Easy but Not Trivial

TL;DR

The paper challenges the long-standing belief that every non-meager graph property $ ext{Φ}$ makes $ ext{ extnormal{#IndSub}}( ext{Φ})$ $ ext{ extnormal{#W[1]}}$-hard by introducing scorpion graphs that enable polynomial-time counting of induced subgraphs. It generalizes to $oldsymbol{ extell}$-scorpions, yielding a spectrum of complexities and ETH/SETH-based lower bounds, and connects these results to linear combinations of non-induced counts via the alternating enumerator. Central to the work is a subgraph-basis perspective that motivates an updated conjecture: hardness is tied to unbounded vertex-cover in the basis elements, with scorpions occupying a key role as fossils in this basis. The results reveal a nuanced landscape where some non-meager properties admit efficient counting, while others remain hard under standard complexity assumptions, guiding a refined understanding of the boundary between tractability and hardness in induced-subgraph counting.

Abstract

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.

Figures (1)

• Figure 1: A graph $H$ is an $\ell$-scorpion if it has the above form: Dashed edges may be present in $H$ or not, solid edges must be present, and non-drawn edges must not be present.

Theorems & Definitions (21)

• Conjecture 1.1: DorflerRSW22FockeR24RothSW24
• Definition 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4: CurticapeanN25 & CurticapeanDN025
• Corollary 3.5
• Remark 3.6
• Corollary 3.7