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Let c_{infty}(G) denote the number of cops needed to capture the robber in a graph G in this variant. We characterize graphs G with c_{infty}(G)=1, and give an O(|V(G)|^2) algorithm for their detection. We prove a lower bound for c_{infty} of expander graphs, and use it to prove three things. The first is that if np > 4.2 log n then the random graph G = G(n,p) asymptotically almost sure"},"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":"1105.2851","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-13T23:29:29Z","cross_cats_sorted":[],"title_canon_sha256":"73e47ffb4a2f6254d27617b25b953aa0f06278a5458d25a0589cad6bad605dfe","abstract_canon_sha256":"0e3d4d90e827e7e9d153d6f6870aa0659fa5aabde2541baae9936f6fdf02b173"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:19.553094Z","signature_b64":"f1zdHMXFGztFUDGYUDhyTAEspYq7LOm2vH0raFM4NxTBt1/ZGVUApLwEAJuK5eZ5/6RJvs2NLZLw+PHIaIShCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"248a5bce5147fe4b2ee11067f1c6df0d0babe3d2f573cb0c5906f2dead015e1e","last_reissued_at":"2026-05-18T02:56:19.552476Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:19.552476Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cops and Robber Game with a Fast Robber on Expander Graphs and Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abbas Mehrabian","submitted_at":"2011-05-13T23:29:29Z","abstract_excerpt":"We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e. can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c_{infty}(G) denote the number of cops needed to capture the robber in a graph G in this variant. We characterize graphs G with c_{infty}(G)=1, and give an O(|V(G)|^2) algorithm for their detection. We prove a lower bound for c_{infty} of expander graphs, and use it to prove three things. 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