{"paper":{"title":"Excluding Pairs of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alex Scott, Maria Chudnovsky, Paul Seymour","submitted_at":"2013-02-04T19:40:36Z","abstract_excerpt":"For a graph $G$ and a set of graphs $\\mathcal{H}$, we say that $G$ is {\\em $\\mathcal{H}$-free} if no induced subgraph of $G$ is isomorphic to a member of $\\mathcal{H}$. Given an integer $P>0$, a graph $G$, and a set of graphs $\\mathcal{F}$, we say that $G$ {\\em admits an $(\\mathcal{F},P)$-partition} if the vertex set of $G$ can be partitioned into $P$ subsets $X_1,..., X_P$, so that for every $i \\in \\{1,..., P\\}$, either $|X_i|=1$, or the subgraph of $G$ induced by $X_i$ is $\\{F\\}$-free for some $F \\in \\mathcal{F}$.\n  Our first result is the following. For every pair $(H,J)$ of graphs such tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.0812","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"}