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We say that a function $f: V(G)\\longrightarrow \\{0,1,\\dots ,l\\}$ is a $w$-dominating function if $f(N(v))=\\sum_{u\\in N(v)}f(u)\\ge w_i$ for every vertex $v$ with $f(v)=i$. The weight of $f$ is defined to be $\\omega(f)=\\sum_{v\\in V(G)} f(v)$. Given a $w$-dominating function $f$ and any pair of adjacent vertices $v, u\\in V(G)$ with $f(v)=0$ and $f(u)>0$, the function $f_{u\\rightarrow v}$ is defined by $f_{u\\rightarrow v}(v)=1$, $f_{u\\rightarr"},"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":"2105.05199","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-05-11T17:02:41Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"230e78079e243a82afd3793df2597794c7606f7ae29b0feef7816a1ae893ec51","abstract_canon_sha256":"b86a8d10fe84b6a6bc0bb5d212ea1dc7bea68348de5d7d135adb21b599e87b80"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:39:29.401454Z","signature_b64":"PfcIYxePrIBNVIdgnxKE3ALOTq+MILstvYJ85pGjn8+LlanzhRQLrx7btN4m9kGPi7dOsUMsH0FcAaXsWtgCBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e78566ad92890825d29b7a43e4d78032dee14ff04f28ce52fd028027cf32e38d","last_reissued_at":"2026-07-05T02:39:29.401024Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:39:29.401024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"From (secure) w-domination in graphs to protection of lexicographic product graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Abel Cabrera Martinez, Alejandro Estrada Moreno, Juan Alberto Rodriguez-Velazquez","submitted_at":"2021-05-11T17:02:41Z","abstract_excerpt":"Let $w=(w_0,w_1, \\dots,w_l)$ be a vector of nonnegative integers such that $ w_0\\ge 1$. 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Given a $w$-dominating function $f$ and any pair of adjacent vertices $v, u\\in V(G)$ with $f(v)=0$ and $f(u)>0$, the function $f_{u\\rightarrow v}$ is defined by $f_{u\\rightarrow v}(v)=1$, $f_{u\\rightarr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.05199","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2105.05199/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":"2105.05199","created_at":"2026-07-05T02:39:29.401085+00:00"},{"alias_kind":"arxiv_version","alias_value":"2105.05199v1","created_at":"2026-07-05T02:39:29.401085+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.05199","created_at":"2026-07-05T02:39:29.401085+00:00"},{"alias_kind":"pith_short_12","alias_value":"46CWNLMSREEC","created_at":"2026-07-05T02:39:29.401085+00:00"},{"alias_kind":"pith_short_16","alias_value":"46CWNLMSREECLUU3","created_at":"2026-07-05T02:39:29.401085+00:00"},{"alias_kind":"pith_short_8","alias_value":"46CWNLMS","created_at":"2026-07-05T02:39:29.401085+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/46CWNLMSREECLUU3PJB6JV4AGL","json":"https://pith.science/pith/46CWNLMSREECLUU3PJB6JV4AGL.json","graph_json":"https://pith.science/api/pith-number/46CWNLMSREECLUU3PJB6JV4AGL/graph.json","events_json":"https://pith.science/api/pith-number/46CWNLMSREECLUU3PJB6JV4AGL/events.json","paper":"https://pith.science/paper/46CWNLMS"},"agent_actions":{"view_html":"https://pith.science/pith/46CWNLMSREECLUU3PJB6JV4AGL","download_json":"https://pith.science/pith/46CWNLMSREECLUU3PJB6JV4AGL.json","view_paper":"https://pith.science/paper/46CWNLMS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2105.05199&json=true","fetch_graph":"https://pith.science/api/pith-number/46CWNLMSREECLUU3PJB6JV4AGL/graph.json","fetch_events":"https://pith.science/api/pith-number/46CWNLMSREECLUU3PJB6JV4AGL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/46CWNLMSREECLUU3PJB6JV4AGL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/46CWNLMSREECLUU3PJB6JV4AGL/action/storage_attestation","attest_author":"https://pith.science/pith/46CWNLMSREECLUU3PJB6JV4AGL/action/author_attestation","sign_citation":"https://pith.science/pith/46CWNLMSREECLUU3PJB6JV4AGL/action/citation_signature","submit_replication":"https://pith.science/pith/46CWNLMSREECLUU3PJB6JV4AGL/action/replication_record"}},"created_at":"2026-07-05T02:39:29.401085+00:00","updated_at":"2026-07-05T02:39:29.401085+00:00"}