{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:V2RDAL7URV4GZAGCOLFXRBVHJL","short_pith_number":"pith:V2RDAL7U","canonical_record":{"source":{"id":"1708.00188","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-08-01T07:26:18Z","cross_cats_sorted":[],"title_canon_sha256":"39f961d3ad66be0b7fa8ef515330f4dc533da7a55a41752311ef3f494d51a09c","abstract_canon_sha256":"370345a9c52ac98c1b182a2bbc0dc2ff73c84208b7fa5ab0eb8afc697a5643c4"},"schema_version":"1.0"},"canonical_sha256":"aea2302ff48d786c80c272cb7886a74ac6bf2c677a5bc8471241456c16e53c1b","source":{"kind":"arxiv","id":"1708.00188","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.00188","created_at":"2026-05-18T00:39:05Z"},{"alias_kind":"arxiv_version","alias_value":"1708.00188v1","created_at":"2026-05-18T00:39:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.00188","created_at":"2026-05-18T00:39:05Z"},{"alias_kind":"pith_short_12","alias_value":"V2RDAL7URV4G","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"V2RDAL7URV4GZAGC","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"V2RDAL7U","created_at":"2026-05-18T12:31:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:V2RDAL7URV4GZAGCOLFXRBVHJL","target":"record","payload":{"canonical_record":{"source":{"id":"1708.00188","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-08-01T07:26:18Z","cross_cats_sorted":[],"title_canon_sha256":"39f961d3ad66be0b7fa8ef515330f4dc533da7a55a41752311ef3f494d51a09c","abstract_canon_sha256":"370345a9c52ac98c1b182a2bbc0dc2ff73c84208b7fa5ab0eb8afc697a5643c4"},"schema_version":"1.0"},"canonical_sha256":"aea2302ff48d786c80c272cb7886a74ac6bf2c677a5bc8471241456c16e53c1b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:05.850895Z","signature_b64":"MJFBpiEgc3CjaacZiM8pUbkL7z2Y4z1Gc2zcvDsfgTBS2oy2wjUOM7YhviP9q2QlrSjLCNASi0ADpm4ZGv0PCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aea2302ff48d786c80c272cb7886a74ac6bf2c677a5bc8471241456c16e53c1b","last_reissued_at":"2026-05-18T00:39:05.850387Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:05.850387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1708.00188","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":"2026-05-18T00:39:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZeTivAj6vvMvpVl6mdi3t9ATtuNttbXV4rhPBsZ9wdYPyFpRARRqR45UCkRFXNBAUZOkMhKOPUFZ0j4DYgo6AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T19:22:26.882122Z"},"content_sha256":"639046f8e1ed153a132e09347750d371abf51ebb5ec22b1e63ff5b300d153998","schema_version":"1.0","event_id":"sha256:639046f8e1ed153a132e09347750d371abf51ebb5ec22b1e63ff5b300d153998"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:V2RDAL7URV4GZAGCOLFXRBVHJL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On outer-connected domination for graph products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"A. Shakiba, M. Hashemipour, M. R. Hooshmandasl","submitted_at":"2017-08-01T07:26:18Z","abstract_excerpt":"An outer-connected dominating set for an arbitrary graph $G$ is a set $\\tilde{D} \\subseteq V$ such that $\\tilde{D}$ is a dominating set and the induced subgraph $G [V \\setminus \\tilde{D}]$ be connected. In this paper, we focus on the outer-connected domination number of the product of graphs. We investigate the existence of outer-connected dominating set in lexicographic product and Corona of two arbitrary graphs, and we present upper bounds for outer-connected domination number in lexicographic and Cartesian product of graphs. Also, we establish an equivalent form of the Vizing's conjecture f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.00188","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":"2026-05-18T00:39:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BOEvN0MzSaGOr3w5sItiPFpdvA7AU1V1Z0wkblLptEFgzYpQ0pA1QbbzFM1+jmVfMOd8Z7jieaZXUmuiyl6JDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T19:22:26.882473Z"},"content_sha256":"6c8cdce269d741f83fc1e7ce5de80b5f12664a8e3353b65925e0250dc9543ac5","schema_version":"1.0","event_id":"sha256:6c8cdce269d741f83fc1e7ce5de80b5f12664a8e3353b65925e0250dc9543ac5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V2RDAL7URV4GZAGCOLFXRBVHJL/bundle.json","state_url":"https://pith.science/pith/V2RDAL7URV4GZAGCOLFXRBVHJL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V2RDAL7URV4GZAGCOLFXRBVHJL/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":"2026-06-27T19:22:26Z","links":{"resolver":"https://pith.science/pith/V2RDAL7URV4GZAGCOLFXRBVHJL","bundle":"https://pith.science/pith/V2RDAL7URV4GZAGCOLFXRBVHJL/bundle.json","state":"https://pith.science/pith/V2RDAL7URV4GZAGCOLFXRBVHJL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V2RDAL7URV4GZAGCOLFXRBVHJL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:V2RDAL7URV4GZAGCOLFXRBVHJL","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":"370345a9c52ac98c1b182a2bbc0dc2ff73c84208b7fa5ab0eb8afc697a5643c4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-08-01T07:26:18Z","title_canon_sha256":"39f961d3ad66be0b7fa8ef515330f4dc533da7a55a41752311ef3f494d51a09c"},"schema_version":"1.0","source":{"id":"1708.00188","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.00188","created_at":"2026-05-18T00:39:05Z"},{"alias_kind":"arxiv_version","alias_value":"1708.00188v1","created_at":"2026-05-18T00:39:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.00188","created_at":"2026-05-18T00:39:05Z"},{"alias_kind":"pith_short_12","alias_value":"V2RDAL7URV4G","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"V2RDAL7URV4GZAGC","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"V2RDAL7U","created_at":"2026-05-18T12:31:49Z"}],"graph_snapshots":[{"event_id":"sha256:6c8cdce269d741f83fc1e7ce5de80b5f12664a8e3353b65925e0250dc9543ac5","target":"graph","created_at":"2026-05-18T00:39:05Z","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":"An outer-connected dominating set for an arbitrary graph $G$ is a set $\\tilde{D} \\subseteq V$ such that $\\tilde{D}$ is a dominating set and the induced subgraph $G [V \\setminus \\tilde{D}]$ be connected. In this paper, we focus on the outer-connected domination number of the product of graphs. We investigate the existence of outer-connected dominating set in lexicographic product and Corona of two arbitrary graphs, and we present upper bounds for outer-connected domination number in lexicographic and Cartesian product of graphs. Also, we establish an equivalent form of the Vizing's conjecture f","authors_text":"A. Shakiba, M. Hashemipour, M. R. Hooshmandasl","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-08-01T07:26:18Z","title":"On outer-connected domination for graph products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.00188","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":"sha256:639046f8e1ed153a132e09347750d371abf51ebb5ec22b1e63ff5b300d153998","target":"record","created_at":"2026-05-18T00:39:05Z","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":"370345a9c52ac98c1b182a2bbc0dc2ff73c84208b7fa5ab0eb8afc697a5643c4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-08-01T07:26:18Z","title_canon_sha256":"39f961d3ad66be0b7fa8ef515330f4dc533da7a55a41752311ef3f494d51a09c"},"schema_version":"1.0","source":{"id":"1708.00188","kind":"arxiv","version":1}},"canonical_sha256":"aea2302ff48d786c80c272cb7886a74ac6bf2c677a5bc8471241456c16e53c1b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aea2302ff48d786c80c272cb7886a74ac6bf2c677a5bc8471241456c16e53c1b","first_computed_at":"2026-05-18T00:39:05.850387Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:39:05.850387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MJFBpiEgc3CjaacZiM8pUbkL7z2Y4z1Gc2zcvDsfgTBS2oy2wjUOM7YhviP9q2QlrSjLCNASi0ADpm4ZGv0PCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:39:05.850895Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.00188","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:639046f8e1ed153a132e09347750d371abf51ebb5ec22b1e63ff5b300d153998","sha256:6c8cdce269d741f83fc1e7ce5de80b5f12664a8e3353b65925e0250dc9543ac5"],"state_sha256":"eb1214cd48cd67dc7c45445be17cb228321d3c6f6dfaeb29ddd12ac16553d221"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GZpUSi1lOo6nF1TfXSbECz/OYoLRi2aIooG3BUu7WG3CNyevwht+lj0Q+G31EaJ98U0q5d+P1WE2cCQ2eEwxAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T19:22:26.884292Z","bundle_sha256":"9f0e38bffaeb1898ac891501497d1525666f82605b8159b792ea10940435c75c"}}