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The cardinality redundance of $G$ is the minimum of $CR(S)$ taken over all dominating sets $S$. A set that achieves $CR(G)$ is a $\\gamma_{cr}$-set, and the size of the minimum $\\gamma_{cr}$-set is $\\gamma_{cr}(G)$. We give the maximum number of edges in a graph with a given number of vertices and given cardinality redundance."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1810.08657","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-19T19:13:33Z","cross_cats_sorted":[],"title_canon_sha256":"104deecdefeca06bebafa1cedcbc715e873d02533314e64dc530949e14f26e68","abstract_canon_sha256":"ad61303df0abca9ead6d8b36bc49cfa5607fb2d1d5d10f33e066f4dff1e20844"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:56.459823Z","signature_b64":"LDU6n72PWXq/RzKxQpxPadWQxGoDH2K6/8j9j4MX+jcK7190uXFZB2+Ud/I/5sLqOU+/7wBhyZQ79BTUSNQoCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9fc9fc161302cc48b314c3ca8ccf68bdca53c8027f00c4f8e8a4f7d50704f361","last_reissued_at":"2026-05-17T23:43:56.459117Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:56.459117Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extremal Problems Related to the Cardinality Redundance of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel McGinnis, Nathan Shank","submitted_at":"2018-10-19T19:13:33Z","abstract_excerpt":"A dominating set of a graph $G$ is a set of vertices $D$ such that for all $v \\in V(G)$, either $v \\in D$ or $(v,d) \\in E(G)$ for some $d \\in D$. The cardinality redundance of a vertex set $S$, $CR(S)$, is the number of vertices in $V(G)$ such that $|N[x] \\cap S| \\geq 2$. The cardinality redundance of $G$ is the minimum of $CR(S)$ taken over all dominating sets $S$. A set that achieves $CR(G)$ is a $\\gamma_{cr}$-set, and the size of the minimum $\\gamma_{cr}$-set is $\\gamma_{cr}(G)$. We give the maximum number of edges in a graph with a given number of vertices and given cardinality redundance."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.08657","kind":"arxiv","version":4},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1810.08657","created_at":"2026-05-17T23:43:56.459234+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.08657v4","created_at":"2026-05-17T23:43:56.459234+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.08657","created_at":"2026-05-17T23:43:56.459234+00:00"},{"alias_kind":"pith_short_12","alias_value":"T7E7YFQTALGE","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_16","alias_value":"T7E7YFQTALGERMYU","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_8","alias_value":"T7E7YFQT","created_at":"2026-05-18T12:32:53.628368+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/T7E7YFQTALGERMYUYPFIZT3IXX","json":"https://pith.science/pith/T7E7YFQTALGERMYUYPFIZT3IXX.json","graph_json":"https://pith.science/api/pith-number/T7E7YFQTALGERMYUYPFIZT3IXX/graph.json","events_json":"https://pith.science/api/pith-number/T7E7YFQTALGERMYUYPFIZT3IXX/events.json","paper":"https://pith.science/paper/T7E7YFQT"},"agent_actions":{"view_html":"https://pith.science/pith/T7E7YFQTALGERMYUYPFIZT3IXX","download_json":"https://pith.science/pith/T7E7YFQTALGERMYUYPFIZT3IXX.json","view_paper":"https://pith.science/paper/T7E7YFQT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.08657&json=true","fetch_graph":"https://pith.science/api/pith-number/T7E7YFQTALGERMYUYPFIZT3IXX/graph.json","fetch_events":"https://pith.science/api/pith-number/T7E7YFQTALGERMYUYPFIZT3IXX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T7E7YFQTALGERMYUYPFIZT3IXX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T7E7YFQTALGERMYUYPFIZT3IXX/action/storage_attestation","attest_author":"https://pith.science/pith/T7E7YFQTALGERMYUYPFIZT3IXX/action/author_attestation","sign_citation":"https://pith.science/pith/T7E7YFQTALGERMYUYPFIZT3IXX/action/citation_signature","submit_replication":"https://pith.science/pith/T7E7YFQTALGERMYUYPFIZT3IXX/action/replication_record"}},"created_at":"2026-05-17T23:43:56.459234+00:00","updated_at":"2026-05-17T23:43:56.459234+00:00"}