{"paper":{"title":"The class of $(P_7,C_4,C_5)$-free graphs: decomposition, algorithms, and $\\chi$-boundedness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Irena Penev, Kathie Cameron, Shenwei Huang, Vaidy Sivaraman","submitted_at":"2018-03-08T21:41:47Z","abstract_excerpt":"As usual, $P_n$ ($n \\geq 1$) denotes the path on $n$ vertices, and $C_n$ ($n \\geq 3$) denotes the cycle on $n$ vertices. For a family $\\mathcal{H}$ of graphs, we say that a graph $G$ is $\\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to any graph in $\\mathcal{H}$. We present a decomposition theorem for the class of $(P_7,C_4,C_5)$-free graphs; in fact, we give a complete structural characterization of $(P_7,C_4,C_5)$-free graphs that do not admit a clique-cutset. We use this decomposition theorem to show that the class of $(P_7,C_4,C_5)$-free graphs is $\\chi$-bounded by a linear"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.03315","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":""},"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"}