{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:K2VO4ICC65LDTYUXCLQK7AYW6S","short_pith_number":"pith:K2VO4ICC","schema_version":"1.0","canonical_sha256":"56aaee2042f75639e29712e0af8316f4a2747a715e3141e5b65123394cfa4c2f","source":{"kind":"arxiv","id":"1805.06058","version":2},"attestation_state":"computed","paper":{"title":"Asymptotically Optimal Bounds for (t,2) Broadcast Domination on Finite Grids","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Timothy W. Randolph","submitted_at":"2018-05-15T22:47:40Z","abstract_excerpt":"Let $G=(V,E)$ be a graph and $t,r$ be positive integers. The \\emph{signal} that a tower vertex $T$ of signal strength $t$ supplies to a vertex $v$ is defined as $sig(T,v)=max(t-dist(T,v),0),$ where $dist(T,v)$ denotes the distance between the vertices $v$ and $T$. In 2015 Blessing, Insko, Johnson, and Mauretour defined a \\emph{$(t,r)$ broadcast dominating set}, or simply a \\emph{$(t,r)$ broadcast}, on $G$ as a set $\\mathbb{T}\\subseteq V$ such that the sum of all signals received at each vertex $v \\in V$ from the set of towers $\\mathbb{T}$ is at least $r$. The $(t,r)$ broadcast domination numbe"},"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":"1805.06058","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2018-05-15T22:47:40Z","cross_cats_sorted":[],"title_canon_sha256":"0ae60863cf78756d23ef980468f7587cfd47e730749a5465e990c62f88f1a1f6","abstract_canon_sha256":"4e7f7704f2a6053351a1ee85946537701bac45b8e654d5e6fbab28ccff00cb0c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:48.012638Z","signature_b64":"gdDi0QulgWqpnTc/3zy/Epg6l4G0uv0+IVclfdwVvNc7/dy6LyJgvj4O5xW1t6gkdrjp1KrHyXIDGVtcKhvzAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56aaee2042f75639e29712e0af8316f4a2747a715e3141e5b65123394cfa4c2f","last_reissued_at":"2026-05-17T23:58:48.012106Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:48.012106Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotically Optimal Bounds for (t,2) Broadcast Domination on Finite Grids","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Timothy W. Randolph","submitted_at":"2018-05-15T22:47:40Z","abstract_excerpt":"Let $G=(V,E)$ be a graph and $t,r$ be positive integers. The \\emph{signal} that a tower vertex $T$ of signal strength $t$ supplies to a vertex $v$ is defined as $sig(T,v)=max(t-dist(T,v),0),$ where $dist(T,v)$ denotes the distance between the vertices $v$ and $T$. In 2015 Blessing, Insko, Johnson, and Mauretour defined a \\emph{$(t,r)$ broadcast dominating set}, or simply a \\emph{$(t,r)$ broadcast}, on $G$ as a set $\\mathbb{T}\\subseteq V$ such that the sum of all signals received at each vertex $v \\in V$ from the set of towers $\\mathbb{T}$ is at least $r$. The $(t,r)$ broadcast domination numbe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.06058","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":""},"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":"1805.06058","created_at":"2026-05-17T23:58:48.012167+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.06058v2","created_at":"2026-05-17T23:58:48.012167+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.06058","created_at":"2026-05-17T23:58:48.012167+00:00"},{"alias_kind":"pith_short_12","alias_value":"K2VO4ICC65LD","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_16","alias_value":"K2VO4ICC65LDTYUX","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_8","alias_value":"K2VO4ICC","created_at":"2026-05-18T12:32:33.847187+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/K2VO4ICC65LDTYUXCLQK7AYW6S","json":"https://pith.science/pith/K2VO4ICC65LDTYUXCLQK7AYW6S.json","graph_json":"https://pith.science/api/pith-number/K2VO4ICC65LDTYUXCLQK7AYW6S/graph.json","events_json":"https://pith.science/api/pith-number/K2VO4ICC65LDTYUXCLQK7AYW6S/events.json","paper":"https://pith.science/paper/K2VO4ICC"},"agent_actions":{"view_html":"https://pith.science/pith/K2VO4ICC65LDTYUXCLQK7AYW6S","download_json":"https://pith.science/pith/K2VO4ICC65LDTYUXCLQK7AYW6S.json","view_paper":"https://pith.science/paper/K2VO4ICC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.06058&json=true","fetch_graph":"https://pith.science/api/pith-number/K2VO4ICC65LDTYUXCLQK7AYW6S/graph.json","fetch_events":"https://pith.science/api/pith-number/K2VO4ICC65LDTYUXCLQK7AYW6S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/K2VO4ICC65LDTYUXCLQK7AYW6S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/K2VO4ICC65LDTYUXCLQK7AYW6S/action/storage_attestation","attest_author":"https://pith.science/pith/K2VO4ICC65LDTYUXCLQK7AYW6S/action/author_attestation","sign_citation":"https://pith.science/pith/K2VO4ICC65LDTYUXCLQK7AYW6S/action/citation_signature","submit_replication":"https://pith.science/pith/K2VO4ICC65LDTYUXCLQK7AYW6S/action/replication_record"}},"created_at":"2026-05-17T23:58:48.012167+00:00","updated_at":"2026-05-17T23:58:48.012167+00:00"}