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Consider the following condition: {equation}\\label{alliancecondition} \\delta_S(v)\\ge \\delta_{\\bar{S}}(v)+k, \\{equation} which states that a vertex $v$ has at least $k$ more neighbors in $S$ than it has in $\\bar{S}$. A set $S\\subseteq V$ that satisfies Condition (\\ref{alliancecondition}) for every vertex $v \\in S$ is called a \\emph{defensive} $k$-\\emph{alliance}; for every ver"},"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":"1112.2068","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-09T10:45:17Z","cross_cats_sorted":[],"title_canon_sha256":"bfd3bbceccea0cc5bffe5c362bd1949a9a5ffa9bcc00b8d5b7bb3c45e5c1b709","abstract_canon_sha256":"b58ba99cd5bf19f51becce5e2f69abf7bf79ddbd3c7792576157e33eec95c310"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:44.167836Z","signature_b64":"jRzsnS8OkPuXPIf4QMB9aYOoqy8Th+dFdFD781QJg0MW84GUlflPgi763ypGz8S1S9Sj4zBVM0AJYxLvx2RWBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de7326a4110e152cc165f307a04ddc888b55defa6e1f67ac5ebd8c9b2d6ff05a","last_reissued_at":"2026-05-18T04:06:44.167256Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:44.167256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Alliance free sets in Cartesian product graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ismael G. 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