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Counting Small Induced Subgraphs: Scorpions Are Easy but Not Trivial

Authors Radu Curticapean , Simon Döring , Daniel Neuen

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Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph algorithms
Keywords
  • induced subgraphs
  • counting complexity
  • parameterized complexity
  • scorpions

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Abstract

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.

Cite As Get BibTex

Radu Curticapean, Simon Döring, and Daniel Neuen. Counting Small Induced Subgraphs: Scorpions Are Easy but Not Trivial. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 96:1-96:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.96

Author Details

Radu Curticapean
  • University of Regensburg, Germany
  • IT University of Copenhagen, Denmark
Simon Döring
  • Max Planck Institute for Informatics, Saarbrücken, Germany
  • Saarland University (SIC), Saarbrücken, Germany
Daniel Neuen
  • Max Planck Institute for Informatics, Saarbrücken, Germany

Funding

The research is funded by the European Union (ERC, CountHom, 101077083). Views and opinions expressed are those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

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