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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1302.0812","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-02-04T19:40:36Z","cross_cats_sorted":[],"title_canon_sha256":"80358aa91d024cbbe8589367dcf5d347e2eec7eb511202a5909f33a5868d8d87","abstract_canon_sha256":"6b785dfadd6f96e9c30dcdf00a4add8c1d5a85a89f9e25b551c5ed52c801d6c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:34:36.230458Z","signature_b64":"/M1O48M1P+omWMSkwT/qY70Fah+4GmH3MCJ4gKZPPM5a7m5goIXXFOOytR0GI2tv4MWC63qhlcBFTi8uzz6wDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe409ede15dc64ad3210370f6ecdffe4515c67a37b2c3216fe7ec560af3d613f","last_reissued_at":"2026-05-18T03:34:36.229992Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:34:36.229992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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