{"paper":{"title":"Counting Small Induced Subgraphs: Scorpions Are Easy but Not Trivial","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CC","authors_text":"Daniel Neuen, Radu Curticapean, Simon D\\\"oring","submitted_at":"2025-05-28T12:37:30Z","abstract_excerpt":"We consider the parameterized problem $\\#$IndSub$(\\Phi)$ for fixed graph properties $\\Phi$: Given a graph $G$ and an integer $k$, this problem asks to count the number of induced $k$-vertex subgraphs satisfying $\\Phi$. D\\\"orfler et al. [Algorithmica 2022] and Roth et al. [SICOMP 2024] conjectured that $\\#$IndSub$(\\Phi)$ is $\\#$W[1]-hard for all non-meager properties $\\Phi$, 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 ["},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.22300","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.22300/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}