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This shows that every planar graph of order $n$ has a 4-degenerate induced subgraph of order more than $8/9 \\cdot n$. We also consider a local variation of this problem and show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more ve"},"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":"1305.6195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-27T12:32:29Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"fb42cd8062caf719ebf68de293c8028500783e5662262308954222d2a8ad2910","abstract_canon_sha256":"350aca2152cceefaf94f7e65b6a0cee814cb6ddf5cc6a674591070ef2273eb3f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:23.841133Z","signature_b64":"Nvoyab+X61RC/mShpjm5JI4sz2IkrTXid+Pt+l9SnA7LKkKKWCUdH7tIIGa2UGgXRIQY+iDfI5koqHF3K1XmAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"149137f78b9b3240399ff37306bd074ad3ff5bd90d70d8873cbd03c7a646b7a5","last_reissued_at":"2026-05-18T03:11:23.840495Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:23.840495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Maximum 4-degenerate subgraph of a planar graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"J\\'an Maz\\'ak, Robert Luko\\v{t}ka, Xuding Zhu","submitted_at":"2013-05-27T12:32:29Z","abstract_excerpt":"A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. 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