{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/incompressibility-of-h-free-edge-modification","title":"Incompressibility of H-free edge modification problems: Towards a dichotomy","arxiv_id":"2004.11761","date":"2020-04-24","proceeding":null,"authors":["Dániel Marx","R. B. Sandeep"],"abstract":"Given a graph $G$ and an integer $k$, the $H$-free Edge Editing problem is to find whether there exists at most $k$ pairs of vertices in $G$ such that changing the adjacency of the pairs in $G$ results in a graph without any induced copy of $H$. The existence of polynomial kernels for $H$-free Edge Editing received significant attention in the parameterized complexity literature. Nontrivial polynomial kernels are known to exist for some graphs $H$ with at most 4 vertices, but starting from 5 vertices, polynomial kernels are known only if $H$ is either complete or empty. This suggests the conjecture that there is no other $H$ with at least 5 vertices were $H$-free Edge Editing admits a polynomial kernel. Towards this goal, we obtain a set $\\mathcal{H}$ of nine 5-vertex graphs such that if for every $H\\in\\mathcal{H}$, $H$-free Edge Editing is incompressible and the complexity assumption $NP \\not\\subseteq coNP/poly$ holds, then $H$-free Edge Editing is incompressible for every graph $H$ with at least five vertices that is neither complete nor empty. That is, proving incompressibility for these nine graphs would give a complete classification of the kernelization complexity of $H$-free Edge Editing for every $H$ with at least 5 vertices. We obtain similar result also for $H$-free Edge Deletion. Here the picture is more complicated due to the existence of another infinite family of graphs $H$ where the problem is trivial (graphs with exactly one edge). We obtain a larger set $\\mathcal{H}$ of nineteen graphs whose incompressibility would give a complete classification of the kernelization complexity of $H$-free Edge Deletion for every graph $H$ with at least 5 vertices. Analogous results follow also for the $H$-free Edge Completion problem by simple complementation.","url_abs":"https://arxiv.org/abs/2004.11761v3","url_pdf":"https://arxiv.org/pdf/2004.11761v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"incompressibility-of-h-free-edge-modification","repo_url":"https://github.com/rbsandeep/unchurnable","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}