{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:RK2GTROVJSE35NBSOXMLO2ESSS","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":"22498bd2c73668b2b4593edc028c2aa409d3514ed9c0ba20bed2a785e352a9f9","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-22T08:30:24Z","title_canon_sha256":"709c717c54e9252b36d77255ec9086778a077b571360e17bed84fca900efbbf1"},"schema_version":"1.0","source":{"id":"1206.5082","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5082","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5082v2","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5082","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"pith_short_12","alias_value":"RK2GTROVJSE3","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RK2GTROVJSE35NBS","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RK2GTROV","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:944202149bda1c0b1e9865b73752c5c4d544628ab0b1d2428666892fa190ce32","target":"graph","created_at":"2026-05-18T03:26:32Z","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":"We show that edge-clique graphs of cocktail party graphs have unbounded rankwidth. This, and other observations lead us to conjecture that the edge-clique cover problem is NP-complete for cographs. We show that the independent set problem on edge-clique graphs of cographs. We show that the independent set problem on edge-clique graphs of graphs without odd wheels remains NP-complete. We present a PTAS for planar graphs and show that the problem is polynomial for planar graphs without triangle separators.","authors_text":"Ching-Hao Liu, Sheung-Hung Poon, Ton Kloks","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-22T08:30:24Z","title":"Independent sets in edge-clique graphs II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5082","kind":"arxiv","version":2},"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:c8d6d6cfcfb090d46852504bbaea2210602e379de2a99964c232f9525fa0fdba","target":"record","created_at":"2026-05-18T03:26:32Z","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":"22498bd2c73668b2b4593edc028c2aa409d3514ed9c0ba20bed2a785e352a9f9","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-22T08:30:24Z","title_canon_sha256":"709c717c54e9252b36d77255ec9086778a077b571360e17bed84fca900efbbf1"},"schema_version":"1.0","source":{"id":"1206.5082","kind":"arxiv","version":2}},"canonical_sha256":"8ab469c5d54c89beb43275d8b7689294bab164df29e38c4084ebd091bcc9c375","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ab469c5d54c89beb43275d8b7689294bab164df29e38c4084ebd091bcc9c375","first_computed_at":"2026-05-18T03:26:32.463288Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:26:32.463288Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GGC7fDYzSP8fo6pgEy8kuhjDinpz91HR1jiSqQB8z7jHhhtmoWy+1YlkMEnLxtcK9GVqmnBm68yj3woPYqfNAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:26:32.463929Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.5082","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c8d6d6cfcfb090d46852504bbaea2210602e379de2a99964c232f9525fa0fdba","sha256:944202149bda1c0b1e9865b73752c5c4d544628ab0b1d2428666892fa190ce32"],"state_sha256":"0166d50e0d4a6fc69d4e32acb3889d333d64ec8e0630337a0821b4ba2845ad1d"}