{"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/inversion-diameter-and-treewidth","title":"Inversion Diameter and Treewidth","arxiv_id":"2407.15384","date":"2024-07-22","proceeding":null,"authors":["Yichen Wang","Haozhe Wang","Yuxuan Yang","Mei Lu"],"abstract":"In an oriented graph $\\overrightarrow{G}$, the inversion of a subset $X$ of vertices is the operation that reverses the orientation of all arcs with both end-vertices in $X$. The inversion graph of a graph $G$, denoted by $\\mathcal{I}(G)$, is the graph whose vertices are orientations of $G$ in which two orientations $\\overrightarrow{G_1}$ and $\\overrightarrow{G_2}$ are adjacent if and only if there is an inversion transforming $\\overrightarrow{G_1}$ into $\\overrightarrow{G_2}$.The inversion diameter of a graph $G$ is the diameter of its inversion graph $\\mathcal{I}(G)$, denoted by $\\mathrm{diam}(\\mathcal{I}(G))$.Havet, H\\\"orsch, and Rambaud~(2024) first proved that for $G$ of treewidth $k$, $\\mathrm{diam}(\\mathcal{I}(G)) \\le 2k$, and that there are graphs of treewidth $k$ with inversion diameter $k+2$.In this paper, we construct graphs of treewidth $k$ with inversion diameter $2k$, which implies that the previous upper bound $\\mathrm{diam}(\\mathcal{I}(G)) \\le 2k$ is tight.Moreover, for graphs with maximum degree $\\Delta$, Havet, H\\\"orsch, and Rambaud~(2024) proved $\\mathrm{diam}(\\mathcal{I}(G)) \\le 2\\Delta-1$ and conjectured that $\\mathrm{diam}(\\mathcal{I}(G)) \\le \\Delta$. We prove the conjecture when $\\Delta=3$ with the help of computer calculations.","url_abs":"https://arxiv.org/abs/2407.15384v1","url_pdf":"https://arxiv.org/pdf/2407.15384v1.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":"inversion-diameter-and-treewidth","repo_url":"https://github.com/handsome12138/InversionDiameter","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}