{"paper":{"title":"Colouring $(sP_1+P_5)$-Free Graphs: a Mim-Width Perspective","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DM","math.CO"],"primary_cat":"cs.DS","authors_text":"Daniel Paulusma, Jake Horsfield, Nick Brettell","submitted_at":"2020-04-10T13:12:32Z","abstract_excerpt":"We prove that the class of $(K_t,sP_1+P_5)$-free graphs has bounded mim-width for every $s\\geq 0$ and $t\\geq 1$, and that there is a polynomial-time algorithm that, given a graph in the class, computes a branch decomposition of constant mim-width. A large number of \\NP-complete graph problems become polynomial-time solvable on graph classes with bounded mim-width and for which a branch decomposition is quickly computable. The $k$-Colouring problem is an example of such a problem. For this problem, we may assume that the input graph is $K_{k+1}$-free. Then, as a consequence of our result, we ob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2004.05022","kind":"arxiv","version":2},"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/2004.05022/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"}