{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:W6DUEXJNMJJ7CE3LNUMADU4RTO","short_pith_number":"pith:W6DUEXJN","canonical_record":{"source":{"id":"1503.06025","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-03-20T09:40:45Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"38e0d193b9af626c293768296b6f18f30c7450e2b68b64b3512f503b975b776d","abstract_canon_sha256":"a451b94d1cc330f6cef8ef4cac272adb72155cf10256bc93d585cfa834a28716"},"schema_version":"1.0"},"canonical_sha256":"b787425d2d6253f1136b6d1801d3919ba29fed7e380175a0da968e1799e37f52","source":{"kind":"arxiv","id":"1503.06025","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.06025","created_at":"2026-05-18T02:20:48Z"},{"alias_kind":"arxiv_version","alias_value":"1503.06025v1","created_at":"2026-05-18T02:20:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.06025","created_at":"2026-05-18T02:20:48Z"},{"alias_kind":"pith_short_12","alias_value":"W6DUEXJNMJJ7","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"W6DUEXJNMJJ7CE3L","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"W6DUEXJN","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:W6DUEXJNMJJ7CE3LNUMADU4RTO","target":"record","payload":{"canonical_record":{"source":{"id":"1503.06025","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-03-20T09:40:45Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"38e0d193b9af626c293768296b6f18f30c7450e2b68b64b3512f503b975b776d","abstract_canon_sha256":"a451b94d1cc330f6cef8ef4cac272adb72155cf10256bc93d585cfa834a28716"},"schema_version":"1.0"},"canonical_sha256":"b787425d2d6253f1136b6d1801d3919ba29fed7e380175a0da968e1799e37f52","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:48.887065Z","signature_b64":"Eca3kWfFAkKr/5wCoGcpK9XAUgNr1uZI9og4CPJQBDlAg6khK9s9tVOboRcJC6UIwzUcAvMk59QjCFvQ861+DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b787425d2d6253f1136b6d1801d3919ba29fed7e380175a0da968e1799e37f52","last_reissued_at":"2026-05-18T02:20:48.885913Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:48.885913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.06025","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":"2026-05-18T02:20:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OQ4lL2hsBi5m0i93WrhgyoEK400sWbJqQIUK8zLFA+A6+LRFTrWJFLV9ECBaLare4gly0VhRMz9V8dT67S1uCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T10:57:12.092853Z"},"content_sha256":"924334f6e365af584e5acf8202cf35e3ab908a3c35cfef4edee1dff4eaa073fb","schema_version":"1.0","event_id":"sha256:924334f6e365af584e5acf8202cf35e3ab908a3c35cfef4edee1dff4eaa073fb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:W6DUEXJNMJJ7CE3LNUMADU4RTO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weighted Efficient Domination in Classes of $P_6$-free Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Andreas Brandstadt, T. Karthick","submitted_at":"2015-03-20T09:40:45Z","abstract_excerpt":"In a graph $G$, an efficient dominating set is a subset $D$ of vertices such that $D$ is an independent set and each vertex outside $D$ has exactly one neighbor in $D$. The Minimum Weight Efficient Dominating Set (Min-WED) problem asks for an efficient dominating set of total minimum weight in a given vertex-weighted graph; the Maximum Weight Efficient Dominating Set (Max-WED) problem is defined similarly. The Min-WED/Max-WED is known to be $NP$-complete for $P_7$-free graphs, and is known to be polynomial time solvable for $P_5$-free graphs. However, the computational complexity of the Min-WE"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06025","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":"2026-05-18T02:20:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2G2hwm/ATS66y3KovHFphrWpgOOa8PQRH7O7TCldqsbeJdFy6shqYK1NNGyzpPkkbwK/5DFeycVuSkFTL9kRCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T10:57:12.093332Z"},"content_sha256":"fd5002000eff1bca2cc34a61a64269d74113d9abaa81bd9701e5489ef4677cc0","schema_version":"1.0","event_id":"sha256:fd5002000eff1bca2cc34a61a64269d74113d9abaa81bd9701e5489ef4677cc0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W6DUEXJNMJJ7CE3LNUMADU4RTO/bundle.json","state_url":"https://pith.science/pith/W6DUEXJNMJJ7CE3LNUMADU4RTO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W6DUEXJNMJJ7CE3LNUMADU4RTO/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":"2026-05-31T10:57:12Z","links":{"resolver":"https://pith.science/pith/W6DUEXJNMJJ7CE3LNUMADU4RTO","bundle":"https://pith.science/pith/W6DUEXJNMJJ7CE3LNUMADU4RTO/bundle.json","state":"https://pith.science/pith/W6DUEXJNMJJ7CE3LNUMADU4RTO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W6DUEXJNMJJ7CE3LNUMADU4RTO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:W6DUEXJNMJJ7CE3LNUMADU4RTO","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":"a451b94d1cc330f6cef8ef4cac272adb72155cf10256bc93d585cfa834a28716","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-03-20T09:40:45Z","title_canon_sha256":"38e0d193b9af626c293768296b6f18f30c7450e2b68b64b3512f503b975b776d"},"schema_version":"1.0","source":{"id":"1503.06025","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.06025","created_at":"2026-05-18T02:20:48Z"},{"alias_kind":"arxiv_version","alias_value":"1503.06025v1","created_at":"2026-05-18T02:20:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.06025","created_at":"2026-05-18T02:20:48Z"},{"alias_kind":"pith_short_12","alias_value":"W6DUEXJNMJJ7","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"W6DUEXJNMJJ7CE3L","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"W6DUEXJN","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:fd5002000eff1bca2cc34a61a64269d74113d9abaa81bd9701e5489ef4677cc0","target":"graph","created_at":"2026-05-18T02:20:48Z","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":"In a graph $G$, an efficient dominating set is a subset $D$ of vertices such that $D$ is an independent set and each vertex outside $D$ has exactly one neighbor in $D$. The Minimum Weight Efficient Dominating Set (Min-WED) problem asks for an efficient dominating set of total minimum weight in a given vertex-weighted graph; the Maximum Weight Efficient Dominating Set (Max-WED) problem is defined similarly. The Min-WED/Max-WED is known to be $NP$-complete for $P_7$-free graphs, and is known to be polynomial time solvable for $P_5$-free graphs. However, the computational complexity of the Min-WE","authors_text":"Andreas Brandstadt, T. Karthick","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-03-20T09:40:45Z","title":"Weighted Efficient Domination in Classes of $P_6$-free Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06025","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":"sha256:924334f6e365af584e5acf8202cf35e3ab908a3c35cfef4edee1dff4eaa073fb","target":"record","created_at":"2026-05-18T02:20:48Z","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":"a451b94d1cc330f6cef8ef4cac272adb72155cf10256bc93d585cfa834a28716","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-03-20T09:40:45Z","title_canon_sha256":"38e0d193b9af626c293768296b6f18f30c7450e2b68b64b3512f503b975b776d"},"schema_version":"1.0","source":{"id":"1503.06025","kind":"arxiv","version":1}},"canonical_sha256":"b787425d2d6253f1136b6d1801d3919ba29fed7e380175a0da968e1799e37f52","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b787425d2d6253f1136b6d1801d3919ba29fed7e380175a0da968e1799e37f52","first_computed_at":"2026-05-18T02:20:48.885913Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:20:48.885913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Eca3kWfFAkKr/5wCoGcpK9XAUgNr1uZI9og4CPJQBDlAg6khK9s9tVOboRcJC6UIwzUcAvMk59QjCFvQ861+DA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:20:48.887065Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.06025","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:924334f6e365af584e5acf8202cf35e3ab908a3c35cfef4edee1dff4eaa073fb","sha256:fd5002000eff1bca2cc34a61a64269d74113d9abaa81bd9701e5489ef4677cc0"],"state_sha256":"6f50ff2deeec88a894a211ac17a05847b0e5eb903819bafab899bfb16393b061"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fmiKzsqgATuIPF0s2UycR6vx9Hxx+uJeX6jNc3BJN/IPNyVNyWX3l/dsrhbS/PBBLFflrs9uWK5fVlLYQJQ0DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T10:57:12.097116Z","bundle_sha256":"0b8cc4a26198b23ef5c04ec8b04a0fc90189fb96c0958b24ef1781dae925453c"}}