{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:VEILBZ3YLYD6KKDQRTCL2NF5AH","short_pith_number":"pith:VEILBZ3Y","canonical_record":{"source":{"id":"1903.09933","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-24T06:42:08Z","cross_cats_sorted":[],"title_canon_sha256":"56b6b632ab0866c5274c9b097a3a295b9409dd84b45965a20a24c8acfa4cf668","abstract_canon_sha256":"4ecb03974950c2f4e5658f8c2f0c94e9b5cbb1373f71391ec74bcffd0f1b9340"},"schema_version":"1.0"},"canonical_sha256":"a910b0e7785e07e528708cc4bd34bd01c9e6881465a9cff814dbfac3a2744329","source":{"kind":"arxiv","id":"1903.09933","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.09933","created_at":"2026-05-17T23:50:35Z"},{"alias_kind":"arxiv_version","alias_value":"1903.09933v1","created_at":"2026-05-17T23:50:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.09933","created_at":"2026-05-17T23:50:35Z"},{"alias_kind":"pith_short_12","alias_value":"VEILBZ3YLYD6","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VEILBZ3YLYD6KKDQ","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VEILBZ3Y","created_at":"2026-05-18T12:33:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:VEILBZ3YLYD6KKDQRTCL2NF5AH","target":"record","payload":{"canonical_record":{"source":{"id":"1903.09933","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-24T06:42:08Z","cross_cats_sorted":[],"title_canon_sha256":"56b6b632ab0866c5274c9b097a3a295b9409dd84b45965a20a24c8acfa4cf668","abstract_canon_sha256":"4ecb03974950c2f4e5658f8c2f0c94e9b5cbb1373f71391ec74bcffd0f1b9340"},"schema_version":"1.0"},"canonical_sha256":"a910b0e7785e07e528708cc4bd34bd01c9e6881465a9cff814dbfac3a2744329","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:35.033347Z","signature_b64":"w/m798k6ogtWfJn4Hrz8HkkuwZJMrbDolxoGGClDRbiSSfU+Q5NSITTiVUt0Pxg0gURd7bBUGqJ4XZ1Wwrq/Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a910b0e7785e07e528708cc4bd34bd01c9e6881465a9cff814dbfac3a2744329","last_reissued_at":"2026-05-17T23:50:35.032787Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:35.032787Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1903.09933","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":"2026-05-17T23:50:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kno0wC4fJBeKU70TAFb67sOTXx5PxavKuLe8zzJol6XOZyQElY2CMqZxFTXZAn2VFjvLYIzkg4G0U5YobZH5AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T17:05:42.753344Z"},"content_sha256":"abe81acae54bad9e31ab0a999f275be9875e7994202b4916b38e29455831903c","schema_version":"1.0","event_id":"sha256:abe81acae54bad9e31ab0a999f275be9875e7994202b4916b38e29455831903c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:VEILBZ3YLYD6KKDQRTCL2NF5AH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On weak majority dimensions of digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Soogang Eoh, Suh-Ryung Kim","submitted_at":"2019-03-24T06:42:08Z","abstract_excerpt":"In this paper, we introduce the notion of the weak majority dimension of a digraph which is well-defined for any digraph. We first study properties shared by the weak dimension of a digraph and show that a weak majority dimension of a digraph can be arbitrarily large. Then we present a complete characterization of digraphs of weak majority dimension $0$ and $1$, respectively, and show that every digraph with weak majority dimension at most two is transitive. Finally, we compute the weak majority dimensions of directed paths and directed cycles and pose open problems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.09933","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":"2026-05-17T23:50:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AEI70rnSQecpWJJhH54rkEw9IHOukKOmUK8WS0u3BvO48C1jlNfY0w6cLbWdbEDAhXItnpNXZgXjEdzQlkZCAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T17:05:42.754021Z"},"content_sha256":"605d48d05979b03502574c11569850268600b4512a5bda18a9a73d9c11d82aad","schema_version":"1.0","event_id":"sha256:605d48d05979b03502574c11569850268600b4512a5bda18a9a73d9c11d82aad"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VEILBZ3YLYD6KKDQRTCL2NF5AH/bundle.json","state_url":"https://pith.science/pith/VEILBZ3YLYD6KKDQRTCL2NF5AH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VEILBZ3YLYD6KKDQRTCL2NF5AH/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":"2026-06-08T17:05:42Z","links":{"resolver":"https://pith.science/pith/VEILBZ3YLYD6KKDQRTCL2NF5AH","bundle":"https://pith.science/pith/VEILBZ3YLYD6KKDQRTCL2NF5AH/bundle.json","state":"https://pith.science/pith/VEILBZ3YLYD6KKDQRTCL2NF5AH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VEILBZ3YLYD6KKDQRTCL2NF5AH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:VEILBZ3YLYD6KKDQRTCL2NF5AH","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":"4ecb03974950c2f4e5658f8c2f0c94e9b5cbb1373f71391ec74bcffd0f1b9340","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-24T06:42:08Z","title_canon_sha256":"56b6b632ab0866c5274c9b097a3a295b9409dd84b45965a20a24c8acfa4cf668"},"schema_version":"1.0","source":{"id":"1903.09933","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.09933","created_at":"2026-05-17T23:50:35Z"},{"alias_kind":"arxiv_version","alias_value":"1903.09933v1","created_at":"2026-05-17T23:50:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.09933","created_at":"2026-05-17T23:50:35Z"},{"alias_kind":"pith_short_12","alias_value":"VEILBZ3YLYD6","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VEILBZ3YLYD6KKDQ","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VEILBZ3Y","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:605d48d05979b03502574c11569850268600b4512a5bda18a9a73d9c11d82aad","target":"graph","created_at":"2026-05-17T23:50:35Z","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 this paper, we introduce the notion of the weak majority dimension of a digraph which is well-defined for any digraph. We first study properties shared by the weak dimension of a digraph and show that a weak majority dimension of a digraph can be arbitrarily large. Then we present a complete characterization of digraphs of weak majority dimension $0$ and $1$, respectively, and show that every digraph with weak majority dimension at most two is transitive. Finally, we compute the weak majority dimensions of directed paths and directed cycles and pose open problems.","authors_text":"Soogang Eoh, Suh-Ryung Kim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-24T06:42:08Z","title":"On weak majority dimensions of digraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.09933","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:abe81acae54bad9e31ab0a999f275be9875e7994202b4916b38e29455831903c","target":"record","created_at":"2026-05-17T23:50:35Z","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":"4ecb03974950c2f4e5658f8c2f0c94e9b5cbb1373f71391ec74bcffd0f1b9340","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-24T06:42:08Z","title_canon_sha256":"56b6b632ab0866c5274c9b097a3a295b9409dd84b45965a20a24c8acfa4cf668"},"schema_version":"1.0","source":{"id":"1903.09933","kind":"arxiv","version":1}},"canonical_sha256":"a910b0e7785e07e528708cc4bd34bd01c9e6881465a9cff814dbfac3a2744329","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a910b0e7785e07e528708cc4bd34bd01c9e6881465a9cff814dbfac3a2744329","first_computed_at":"2026-05-17T23:50:35.032787Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:50:35.032787Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"w/m798k6ogtWfJn4Hrz8HkkuwZJMrbDolxoGGClDRbiSSfU+Q5NSITTiVUt0Pxg0gURd7bBUGqJ4XZ1Wwrq/Aw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:50:35.033347Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.09933","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:abe81acae54bad9e31ab0a999f275be9875e7994202b4916b38e29455831903c","sha256:605d48d05979b03502574c11569850268600b4512a5bda18a9a73d9c11d82aad"],"state_sha256":"1496e2980eecd51d870115ba01a51c279ba4efd32c13330f17967491fc0a8f8f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wkU4wE8/ek07nizhVycEblp/mO6pwXi3VnqP4FV7WbA800rG9KWehiaW3NVf8wUcYsA9caHnq8euUL9EV6KsBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T17:05:42.758287Z","bundle_sha256":"95ffe57ba5ddb49ead1a9d8a502b035dacf01b79f28c0267388af89913f2b53e"}}