2026-08-17T12:28:44Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:24565 2025-12-16T14:00:54Z ddc:004 open_access

Counting Small Induced Subgraphs: Scorpions Are Easy but Not Trivial Curticapean, Radu Döring, Simon Neuen, Daniel induced subgraphs counting complexity parameterized complexity scorpions In the parameterized problem #IndSub(Φ) for fixed graph properties Φ, given as input a graph G and an integer k, the task is to compute 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]. We refute this conjecture by showing that induced k-vertex graphs that are scorpions can be counted in time O(n⁴) for all k. Scorpions were introduced more than 50 years ago in the context of the evasiveness conjecture. A simple variant of this construction results in graph properties that achieve arbitrary intermediate complexity assuming ETH. Moreover, we formulate an updated conjecture on the complexity of #IndSub(Φ) that correctly captures the complexity status of scorpions and related constructions. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Radu Curticapean and Simon Döring and Daniel Neuen 2025 Is Part Of LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2025.96 urn:nbn:de:0030-drops-245651 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.96 eng https://creativecommons.org/licenses/by/4.0/legalcode