{"paper":{"title":"Alliance free and alliance cover sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"I. G. Yero, J. A. Rodriguez-Velazquez, J. M. Sigarreta, S. Bermudo","submitted_at":"2006-02-20T14:15:55Z","abstract_excerpt":"A \\emph{defensive} (\\emph{offensive}) $k$-\\emph{alliance} in $\\Gamma=(V,E)$ is a set $S\\subseteq V$ such that every $v$ in $S$ (in the boundary of $S$) has at least $k$ more neighbors in $S$ than it has in $V\\setminus S$. A set $X\\subseteq V$ is \\emph{defensive} (\\emph{offensive}) $k$-\\emph{alliance free,} if for all defensive (offensive) $k$-alliance $S$, $S\\setminus X\\neq\\emptyset$, i.e., $X$ does not contain any defensive (offensive) $k$-alliance as a subset. A set $Y \\subseteq V$ is a \\emph{defensive} (\\emph{offensive}) $k$-\\emph{alliance cover}, if for all defensive (offensive) $k$-allian"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0602428","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"}