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We study the cop number of geometric graphs. For points $x_1, ..., x_n \\in \\R^2$, and $r \\in \\R^+$, the vertex set of the geometric graph $G(x_1, ..., x_n; r)$ is the graph on these $n$ points, with $x_i, x_j$ adjacent when $ \\norm{x_i -x_j} \\leq r$. We prove that $c(G) \\leq 9$ for any connected geometric gr"},"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":"1108.2549","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-12T02:05:56Z","cross_cats_sorted":[],"title_canon_sha256":"4a7bc0260ee0166b715beadc508ce7095377a84b70d6f67175b038b15eaf1ee3","abstract_canon_sha256":"b797bb105a68a537de2b3537c51f54034d3656ba4273ddac06a8dca1157bfaea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:13:54.952354Z","signature_b64":"Gytz+GEHDKleHmLFk5h8OPgYsaPkKmse165Gw7jBdsuRarGlONm+7c0OR8EauGkEYxCAJdNssHGrD8hfMBcmCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ca99cf16f50d2713a4e8be90b4e75409967af6bcb474bcbabfd3c27de99f6c1","last_reissued_at":"2026-05-18T04:13:54.951802Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:13:54.951802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cops and Robbers on Geometric Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan Frieze, Andrew Beveridge, Andrzej Dudek, Tobias M\\\"uller","submitted_at":"2011-08-12T02:05:56Z","abstract_excerpt":"Cops and robbers is a turn-based pursuit game played on a graph $G$. One robber is pursued by a set of cops. In each round, these agents move between vertices along the edges of the graph. The cop number $c(G)$ denotes the minimum number of cops required to catch the robber in finite time. We study the cop number of geometric graphs. For points $x_1, ..., x_n \\in \\R^2$, and $r \\in \\R^+$, the vertex set of the geometric graph $G(x_1, ..., x_n; r)$ is the graph on these $n$ points, with $x_i, x_j$ adjacent when $ \\norm{x_i -x_j} \\leq r$. We prove that $c(G) \\leq 9$ for any connected geometric gr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.2549","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":"1108.2549","created_at":"2026-05-18T04:13:54.951897+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.2549v4","created_at":"2026-05-18T04:13:54.951897+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.2549","created_at":"2026-05-18T04:13:54.951897+00:00"},{"alias_kind":"pith_short_12","alias_value":"BSUZZ4LPKDJH","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_16","alias_value":"BSUZZ4LPKDJHCOSO","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_8","alias_value":"BSUZZ4LP","created_at":"2026-05-18T12:26:24.575870+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/BSUZZ4LPKDJHCOSORPUQWTTVIC","json":"https://pith.science/pith/BSUZZ4LPKDJHCOSORPUQWTTVIC.json","graph_json":"https://pith.science/api/pith-number/BSUZZ4LPKDJHCOSORPUQWTTVIC/graph.json","events_json":"https://pith.science/api/pith-number/BSUZZ4LPKDJHCOSORPUQWTTVIC/events.json","paper":"https://pith.science/paper/BSUZZ4LP"},"agent_actions":{"view_html":"https://pith.science/pith/BSUZZ4LPKDJHCOSORPUQWTTVIC","download_json":"https://pith.science/pith/BSUZZ4LPKDJHCOSORPUQWTTVIC.json","view_paper":"https://pith.science/paper/BSUZZ4LP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.2549&json=true","fetch_graph":"https://pith.science/api/pith-number/BSUZZ4LPKDJHCOSORPUQWTTVIC/graph.json","fetch_events":"https://pith.science/api/pith-number/BSUZZ4LPKDJHCOSORPUQWTTVIC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BSUZZ4LPKDJHCOSORPUQWTTVIC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BSUZZ4LPKDJHCOSORPUQWTTVIC/action/storage_attestation","attest_author":"https://pith.science/pith/BSUZZ4LPKDJHCOSORPUQWTTVIC/action/author_attestation","sign_citation":"https://pith.science/pith/BSUZZ4LPKDJHCOSORPUQWTTVIC/action/citation_signature","submit_replication":"https://pith.science/pith/BSUZZ4LPKDJHCOSORPUQWTTVIC/action/replication_record"}},"created_at":"2026-05-18T04:13:54.951897+00:00","updated_at":"2026-05-18T04:13:54.951897+00:00"}