{"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/on-the-finiteness-of-k-vertex-critical-2p-2","title":"On the finiteness of $k$-vertex-critical $2P_2$-free graphs with forbidden induced squids or bulls","arxiv_id":"2402.15908","date":"2024-02-24","proceeding":null,"authors":["Melvin Adekanye","Christopher Bury","Ben Cameron","Thaler Knodel"],"abstract":"A graph is $k$-vertex-critical if $\\chi(G)=k$ but $\\chi(G-v)<k$ for all $v\\in V(G)$ and $(G,H)$-free if it contains no induced subgraph isomorphic to $G$ or $H$. We show that there are only finitely many $k$-vertex-critical $(2P_2,H)$-free graphs for all $k$ when $H$ is isomorphic to any of the following graphs of order $5$: $bull$, $chair$, $claw+P_1$, or $\\overline{diamond+P_1}$. The latter three are corollaries of more general results where $H$ is isomorphic to $(m, \\ell)$-$squid$ for $m=3,4$ and any $\\ell\\ge 1$ where an $(m,\\ell)$-$squid$ is the graph obtained from an $m$-cycle by attaching $\\ell$ leaves to a single vertex of the cycle. For each of the graphs $H$ above and any fixed $k$, our results imply the existence of polynomial-time certifying algorithms for deciding the $k$-colourability problem for $(2P_2,H)$-free graphs. Further, our structural classifications allow us to exhaustively generate, with aid of computer search, all $k$-vertex-critical $(2P_2,H)$-free graphs for $k\\le 7$ when $H=bull$ or $H=(4,1)$-$squid$ (also known as $banner$).","url_abs":"https://arxiv.org/abs/2402.15908v1","url_pdf":"https://arxiv.org/pdf/2402.15908v1.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":"on-the-finiteness-of-k-vertex-critical-2p-2","repo_url":"https://github.com/benrkcameron/2p2bull","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}