{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:N4SIVA2FGPENM2UXRSAA3KLCNP","short_pith_number":"pith:N4SIVA2F","canonical_record":{"source":{"id":"1509.04944","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-09-16T15:13:08Z","cross_cats_sorted":[],"title_canon_sha256":"8dfdc12e71f7e51834c4d245f7c0300a82eab6960bda8ff66d4763b844f8b754","abstract_canon_sha256":"22d851d6bc3182a10604b4f4f28c9a1cd0a7e0ff315f7beef4a6390552d3f0ae"},"schema_version":"1.0"},"canonical_sha256":"6f248a834533c8d66a978c800da9626bf07e14ec622e95ccee6577bd2b8198bf","source":{"kind":"arxiv","id":"1509.04944","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.04944","created_at":"2026-05-18T01:32:51Z"},{"alias_kind":"arxiv_version","alias_value":"1509.04944v1","created_at":"2026-05-18T01:32:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.04944","created_at":"2026-05-18T01:32:51Z"},{"alias_kind":"pith_short_12","alias_value":"N4SIVA2FGPEN","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"N4SIVA2FGPENM2UX","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"N4SIVA2F","created_at":"2026-05-18T12:29:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:N4SIVA2FGPENM2UXRSAA3KLCNP","target":"record","payload":{"canonical_record":{"source":{"id":"1509.04944","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-09-16T15:13:08Z","cross_cats_sorted":[],"title_canon_sha256":"8dfdc12e71f7e51834c4d245f7c0300a82eab6960bda8ff66d4763b844f8b754","abstract_canon_sha256":"22d851d6bc3182a10604b4f4f28c9a1cd0a7e0ff315f7beef4a6390552d3f0ae"},"schema_version":"1.0"},"canonical_sha256":"6f248a834533c8d66a978c800da9626bf07e14ec622e95ccee6577bd2b8198bf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:51.294432Z","signature_b64":"B5HdYOer7KPwigAeN57BojRD/Co6TVBAPJbHPCICAPCLXjU54KEuhds3/Bqc+1PVj88gMCs8wfBfjHn1Zu7pCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f248a834533c8d66a978c800da9626bf07e14ec622e95ccee6577bd2b8198bf","last_reissued_at":"2026-05-18T01:32:51.293854Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:51.293854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.04944","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:32:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SnR6JldIG1DsYU23RouJ0oahhMqxJnR8KDZFYMOgNrw4dLp/k8svILigKAEsrlCfbipu2Vk7szjS4cZtHvZ6AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T00:37:26.003236Z"},"content_sha256":"eb3143673e6a053a9dc969dc5cfadcc14173bd3ec276c726d4dddd546a6c5f46","schema_version":"1.0","event_id":"sha256:eb3143673e6a053a9dc969dc5cfadcc14173bd3ec276c726d4dddd546a6c5f46"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:N4SIVA2FGPENM2UXRSAA3KLCNP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Convexities in Some Special Graph Classes ---New Results in AT-free Graphs and Beyond","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Hsiang-Hsuan Liu, Hung-Lung Wang, Ton Kloks, Wing-Kai Hon, Yue-Li Wang","submitted_at":"2015-09-16T15:13:08Z","abstract_excerpt":"We study convexity properties of graphs. In this paper we present a linear-time algorithm for the geodetic number in tree-cographs. Settling a 10-year-old conjecture, we prove that the Steiner number is at least the geodetic number in AT-free graphs. Computing a maximal and proper monophonic set in $\\AT$-free graphs is NP-complete. We present polynomial algorithms for the monophonic number in permutation graphs and the geodetic number in $P_4$- sparse graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.04944","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:32:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f527HP/sho6Q2lWVsWk155nX9vqmxfUnFiy7Zrmr/B1NH6HC5BOqTVxAiRYW4JSj/xAIufDTuID3/Fr5ogTbAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T00:37:26.003606Z"},"content_sha256":"a615057996ae4750deb2935849345f2571e663dd62c4ab0f7a42d277c38e5dca","schema_version":"1.0","event_id":"sha256:a615057996ae4750deb2935849345f2571e663dd62c4ab0f7a42d277c38e5dca"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N4SIVA2FGPENM2UXRSAA3KLCNP/bundle.json","state_url":"https://pith.science/pith/N4SIVA2FGPENM2UXRSAA3KLCNP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N4SIVA2FGPENM2UXRSAA3KLCNP/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-12T00:37:26Z","links":{"resolver":"https://pith.science/pith/N4SIVA2FGPENM2UXRSAA3KLCNP","bundle":"https://pith.science/pith/N4SIVA2FGPENM2UXRSAA3KLCNP/bundle.json","state":"https://pith.science/pith/N4SIVA2FGPENM2UXRSAA3KLCNP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N4SIVA2FGPENM2UXRSAA3KLCNP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:N4SIVA2FGPENM2UXRSAA3KLCNP","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":"22d851d6bc3182a10604b4f4f28c9a1cd0a7e0ff315f7beef4a6390552d3f0ae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-09-16T15:13:08Z","title_canon_sha256":"8dfdc12e71f7e51834c4d245f7c0300a82eab6960bda8ff66d4763b844f8b754"},"schema_version":"1.0","source":{"id":"1509.04944","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.04944","created_at":"2026-05-18T01:32:51Z"},{"alias_kind":"arxiv_version","alias_value":"1509.04944v1","created_at":"2026-05-18T01:32:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.04944","created_at":"2026-05-18T01:32:51Z"},{"alias_kind":"pith_short_12","alias_value":"N4SIVA2FGPEN","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"N4SIVA2FGPENM2UX","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"N4SIVA2F","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:a615057996ae4750deb2935849345f2571e663dd62c4ab0f7a42d277c38e5dca","target":"graph","created_at":"2026-05-18T01:32:51Z","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 study convexity properties of graphs. In this paper we present a linear-time algorithm for the geodetic number in tree-cographs. Settling a 10-year-old conjecture, we prove that the Steiner number is at least the geodetic number in AT-free graphs. Computing a maximal and proper monophonic set in $\\AT$-free graphs is NP-complete. We present polynomial algorithms for the monophonic number in permutation graphs and the geodetic number in $P_4$- sparse graphs.","authors_text":"Hsiang-Hsuan Liu, Hung-Lung Wang, Ton Kloks, Wing-Kai Hon, Yue-Li Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-09-16T15:13:08Z","title":"Convexities in Some Special Graph Classes ---New Results in AT-free Graphs and Beyond"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.04944","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:eb3143673e6a053a9dc969dc5cfadcc14173bd3ec276c726d4dddd546a6c5f46","target":"record","created_at":"2026-05-18T01:32:51Z","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":"22d851d6bc3182a10604b4f4f28c9a1cd0a7e0ff315f7beef4a6390552d3f0ae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-09-16T15:13:08Z","title_canon_sha256":"8dfdc12e71f7e51834c4d245f7c0300a82eab6960bda8ff66d4763b844f8b754"},"schema_version":"1.0","source":{"id":"1509.04944","kind":"arxiv","version":1}},"canonical_sha256":"6f248a834533c8d66a978c800da9626bf07e14ec622e95ccee6577bd2b8198bf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6f248a834533c8d66a978c800da9626bf07e14ec622e95ccee6577bd2b8198bf","first_computed_at":"2026-05-18T01:32:51.293854Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:51.293854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"B5HdYOer7KPwigAeN57BojRD/Co6TVBAPJbHPCICAPCLXjU54KEuhds3/Bqc+1PVj88gMCs8wfBfjHn1Zu7pCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:51.294432Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.04944","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eb3143673e6a053a9dc969dc5cfadcc14173bd3ec276c726d4dddd546a6c5f46","sha256:a615057996ae4750deb2935849345f2571e663dd62c4ab0f7a42d277c38e5dca"],"state_sha256":"767906f766762bfda61cd8fa84f51f082a0fc5e784b476a8ea13244b0fee922d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fj+ZWU+8WU1LGIWaOd+hZm7yqBChHywS/elfIdGFf0H3Oc4cj9ATS9Q+IMPs151vaxMRLorKZZ+NEEr6EW8BDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T00:37:26.005555Z","bundle_sha256":"6f8dd230c0f7c7bd4edb688bb40490d7c5cf5f9e51e320862238204f5cd155c3"}}