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Two distinct vertices $x,y\\in U$ are $U$-mutually visible if $G$ contains a shortest $x$-$y$ path that is internally disjoint from $U$. $U$ is called a mutual-visibility set of $G$ if any two vertices of $U$ are $U$-mutually visible. The mutual-visibility number $\\mu(G)$ of $G$ is the size of a largest mutual-visibility set of $G$. Let $P$ be a set of $n\\geq 3$ points in ${\\mathbb R}^2$ in general position. The disjointness graph of segments $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with"},"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":"2506.00689","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-31T19:46:38Z","cross_cats_sorted":[],"title_canon_sha256":"fd801487e37cf8448afc95aaaff7094ead1dbc486bdb7c66e54c996bfb8b2df2","abstract_canon_sha256":"4984f2ef52e9c324490a89b1221db2c3ba9b0bf63a3c66a41b1d1fc152f6b789"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:15:01.235703Z","signature_b64":"QYuLDx+REixCXAghGAPpyy3ransxzh084PN/JWk9JqfAP8CSbqaNMhQzLasxIiUCzR0N4zOe9bfLd7w2mKQ8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b60d57d5f0a6724b0782265795637551c188a7e14e777d500a5ceb89b51c47d","last_reissued_at":"2026-07-05T11:15:01.235225Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:15:01.235225Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mutual-visibility of the disjointness graph of segments in ${\\mathbb R}^2$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christophe Ndjatchi, J. Lea\\~nos, L. M. R\\'ios-Castro, M. Lomel\\'i-Haro","submitted_at":"2025-05-31T19:46:38Z","abstract_excerpt":"Let $G=(V(G),E(G))$ be a simple graph, and let $U\\subseteq V(G)$. Two distinct vertices $x,y\\in U$ are $U$-mutually visible if $G$ contains a shortest $x$-$y$ path that is internally disjoint from $U$. $U$ is called a mutual-visibility set of $G$ if any two vertices of $U$ are $U$-mutually visible. The mutual-visibility number $\\mu(G)$ of $G$ is the size of a largest mutual-visibility set of $G$. Let $P$ be a set of $n\\geq 3$ points in ${\\mathbb R}^2$ in general position. The disjointness graph of segments $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.00689","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.00689/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2506.00689","created_at":"2026-07-05T11:15:01.235298+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.00689v2","created_at":"2026-07-05T11:15:01.235298+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.00689","created_at":"2026-07-05T11:15:01.235298+00:00"},{"alias_kind":"pith_short_12","alias_value":"FNQNK7K7BJTS","created_at":"2026-07-05T11:15:01.235298+00:00"},{"alias_kind":"pith_short_16","alias_value":"FNQNK7K7BJTSJMDY","created_at":"2026-07-05T11:15:01.235298+00:00"},{"alias_kind":"pith_short_8","alias_value":"FNQNK7K7","created_at":"2026-07-05T11:15:01.235298+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/FNQNK7K7BJTSJMDYEJSXSVRXKU","json":"https://pith.science/pith/FNQNK7K7BJTSJMDYEJSXSVRXKU.json","graph_json":"https://pith.science/api/pith-number/FNQNK7K7BJTSJMDYEJSXSVRXKU/graph.json","events_json":"https://pith.science/api/pith-number/FNQNK7K7BJTSJMDYEJSXSVRXKU/events.json","paper":"https://pith.science/paper/FNQNK7K7"},"agent_actions":{"view_html":"https://pith.science/pith/FNQNK7K7BJTSJMDYEJSXSVRXKU","download_json":"https://pith.science/pith/FNQNK7K7BJTSJMDYEJSXSVRXKU.json","view_paper":"https://pith.science/paper/FNQNK7K7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.00689&json=true","fetch_graph":"https://pith.science/api/pith-number/FNQNK7K7BJTSJMDYEJSXSVRXKU/graph.json","fetch_events":"https://pith.science/api/pith-number/FNQNK7K7BJTSJMDYEJSXSVRXKU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FNQNK7K7BJTSJMDYEJSXSVRXKU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FNQNK7K7BJTSJMDYEJSXSVRXKU/action/storage_attestation","attest_author":"https://pith.science/pith/FNQNK7K7BJTSJMDYEJSXSVRXKU/action/author_attestation","sign_citation":"https://pith.science/pith/FNQNK7K7BJTSJMDYEJSXSVRXKU/action/citation_signature","submit_replication":"https://pith.science/pith/FNQNK7K7BJTSJMDYEJSXSVRXKU/action/replication_record"}},"created_at":"2026-07-05T11:15:01.235298+00:00","updated_at":"2026-07-05T11:15:01.235298+00:00"}