{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:NHSBHGH6JBF4WCSCGEUYBJBSD3","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b9123b1e095ccc91c875bde5445aa69b9ab49c20154b1ace1e5f1bd71029ab39","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2017-06-27T20:58:26Z","title_canon_sha256":"b436579e99118ca50a578e4268b3b8511eb7b475a91fb126c3433359148b63a7"},"schema_version":"1.0","source":{"id":"1706.09043","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.09043","created_at":"2026-05-18T00:41:20Z"},{"alias_kind":"arxiv_version","alias_value":"1706.09043v1","created_at":"2026-05-18T00:41:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.09043","created_at":"2026-05-18T00:41:20Z"},{"alias_kind":"pith_short_12","alias_value":"NHSBHGH6JBF4","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"NHSBHGH6JBF4WCSC","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"NHSBHGH6","created_at":"2026-05-18T12:31:31Z"}],"graph_snapshots":[{"event_id":"sha256:b957c28213d2201d3638488856cabee3df1fac2cb36e8a0117d2cf75aad0d1e3","target":"graph","created_at":"2026-05-18T00:41:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"A vertex or edge in a graph is critical if its deletion reduces the chromatic number of the graph by 1. We consider the problems of deciding whether a graph has a critical vertex or edge, respectively. We give a complexity dichotomy for both problems restricted to $H$-free graphs, that is, graphs with no induced subgraph isomorphic to $H$. Moreover, we show that an edge is critical if and only if its contraction reduces the chromatic number by 1. Hence, we also obtain a complexity dichotomy for the problem of deciding if a graph has an edge whose contraction reduces the chromatic number by 1.","authors_text":"Bernard Ries, Christophe Picouleau, Dani\\\"el Paulusma","cross_cats":["cs.DM","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2017-06-27T20:58:26Z","title":"Critical Vertices and Edges in $H$-free Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.09043","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0615bda8ac665cae250a525a288e0fb3685d39db016194ba82525722752408fb","target":"record","created_at":"2026-05-18T00:41:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b9123b1e095ccc91c875bde5445aa69b9ab49c20154b1ace1e5f1bd71029ab39","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2017-06-27T20:58:26Z","title_canon_sha256":"b436579e99118ca50a578e4268b3b8511eb7b475a91fb126c3433359148b63a7"},"schema_version":"1.0","source":{"id":"1706.09043","kind":"arxiv","version":1}},"canonical_sha256":"69e41398fe484bcb0a42312980a4321ec47f463844ae191ff1a64fca3e988491","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"69e41398fe484bcb0a42312980a4321ec47f463844ae191ff1a64fca3e988491","first_computed_at":"2026-05-18T00:41:20.058743Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:20.058743Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HsiMHgfDv1xsEtiiJ6QrECDBImi257UbUQRujYfodA+XkeWWEulHI2rwKZSHKxDM2YCFGeFXCB6idtQ4BDXoCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:20.059408Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.09043","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0615bda8ac665cae250a525a288e0fb3685d39db016194ba82525722752408fb","sha256:b957c28213d2201d3638488856cabee3df1fac2cb36e8a0117d2cf75aad0d1e3"],"state_sha256":"afdf806f4d2ecb786927d79db7d60bfcbe1d4b8965dcd83296717f5094955ee4"}