{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:A3LH4LVR5X2YQKBVNNUNDAJ2PR","short_pith_number":"pith:A3LH4LVR","schema_version":"1.0","canonical_sha256":"06d67e2eb1edf58828356b68d1813a7c4a30ad92d273be8be224c00b86468ddb","source":{"kind":"arxiv","id":"1508.00357","version":1},"attestation_state":"computed","paper":{"title":"Generalized power domination in WK-Pyramid Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. Vijayakumar, Seethu Varghese","submitted_at":"2015-08-03T09:55:09Z","abstract_excerpt":"The notion of power domination arises in the context of monitoring an electric power system with as few phase measurement units as possible. The $k-$power domination number of a graph $G$ is the minimum cardinality of a $k-$power dominating set ($k-$PDS) of $G$. In this paper, we determine the $k-$power domination number of WK-Pyramid networks, $WKP_{(C,L)}$, for all positive values of $k$ except for $k=C-1, C \\geq 2$, for which we give an upper bound. The $k-$propagation radius of a graph $G$ is the minimum number of propagation steps needed to monitor the graph $G$ over all minimum $k-$PDS. "},"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":"1508.00357","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-03T09:55:09Z","cross_cats_sorted":[],"title_canon_sha256":"fed141ef7a4a5bfa86bae5c1db6ce697c16b64582a4ffc147a5dab47c7946094","abstract_canon_sha256":"74b3e23d0c7b085bcb7b50828242bdc7a3fada73e59f190db2c5af382ff66609"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:58.323955Z","signature_b64":"aNBF3mhv+HL9Mxbm54FKXco5JAfyGqQGUKuzPGGno6ZhmOywUbJwNrI0Y0OnaarsLbWEll1HQzgrTxu5SdvMAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06d67e2eb1edf58828356b68d1813a7c4a30ad92d273be8be224c00b86468ddb","last_reissued_at":"2026-05-18T01:35:58.323358Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:58.323358Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generalized power domination in WK-Pyramid Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. Vijayakumar, Seethu Varghese","submitted_at":"2015-08-03T09:55:09Z","abstract_excerpt":"The notion of power domination arises in the context of monitoring an electric power system with as few phase measurement units as possible. The $k-$power domination number of a graph $G$ is the minimum cardinality of a $k-$power dominating set ($k-$PDS) of $G$. In this paper, we determine the $k-$power domination number of WK-Pyramid networks, $WKP_{(C,L)}$, for all positive values of $k$ except for $k=C-1, C \\geq 2$, for which we give an upper bound. The $k-$propagation radius of a graph $G$ is the minimum number of propagation steps needed to monitor the graph $G$ over all minimum $k-$PDS. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.00357","kind":"arxiv","version":1},"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":"1508.00357","created_at":"2026-05-18T01:35:58.323435+00:00"},{"alias_kind":"arxiv_version","alias_value":"1508.00357v1","created_at":"2026-05-18T01:35:58.323435+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.00357","created_at":"2026-05-18T01:35:58.323435+00:00"},{"alias_kind":"pith_short_12","alias_value":"A3LH4LVR5X2Y","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"A3LH4LVR5X2YQKBV","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"A3LH4LVR","created_at":"2026-05-18T12:29:10.953037+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/A3LH4LVR5X2YQKBVNNUNDAJ2PR","json":"https://pith.science/pith/A3LH4LVR5X2YQKBVNNUNDAJ2PR.json","graph_json":"https://pith.science/api/pith-number/A3LH4LVR5X2YQKBVNNUNDAJ2PR/graph.json","events_json":"https://pith.science/api/pith-number/A3LH4LVR5X2YQKBVNNUNDAJ2PR/events.json","paper":"https://pith.science/paper/A3LH4LVR"},"agent_actions":{"view_html":"https://pith.science/pith/A3LH4LVR5X2YQKBVNNUNDAJ2PR","download_json":"https://pith.science/pith/A3LH4LVR5X2YQKBVNNUNDAJ2PR.json","view_paper":"https://pith.science/paper/A3LH4LVR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1508.00357&json=true","fetch_graph":"https://pith.science/api/pith-number/A3LH4LVR5X2YQKBVNNUNDAJ2PR/graph.json","fetch_events":"https://pith.science/api/pith-number/A3LH4LVR5X2YQKBVNNUNDAJ2PR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A3LH4LVR5X2YQKBVNNUNDAJ2PR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A3LH4LVR5X2YQKBVNNUNDAJ2PR/action/storage_attestation","attest_author":"https://pith.science/pith/A3LH4LVR5X2YQKBVNNUNDAJ2PR/action/author_attestation","sign_citation":"https://pith.science/pith/A3LH4LVR5X2YQKBVNNUNDAJ2PR/action/citation_signature","submit_replication":"https://pith.science/pith/A3LH4LVR5X2YQKBVNNUNDAJ2PR/action/replication_record"}},"created_at":"2026-05-18T01:35:58.323435+00:00","updated_at":"2026-05-18T01:35:58.323435+00:00"}