{"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. Yero, Juan A. Rodriguez-Velazquez, Sergio Bermudo","submitted_at":"2011-12-09T10:45:17Z","abstract_excerpt":"Let $G=(V,E)$ be a graph. For a non-empty subset of vertices $S\\subseteq V$, and vertex $v\\in V$, let $\\delta_S(v)=|\\{u\\in S:uv\\in E\\}|$ denote the cardinality of the set of neighbors of $v$ in $S$, and let $\\bar{S}=V-S$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.2068","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"}