联盟自由集和联盟覆盖集
Alliance free and alliance cover sets
摘要：
$Γ=(V,E)$中的\emph{defensive}（\emph{offensive}）$k$ - \emph{alliance}是一组$S\subseteq V$，使得$S$中的每个$v$（在$S$的边界中）在$S$中具有比其在$V\setminus S$中至少多$k$个邻居。如果对于所有防御（进攻）$k$ -联盟$S$，$S\setminus X\neq\emptyset$，则集合$X\subseteq V$是\emph{defensive}（\emph{offensive}）$k$-\emph{alliance free,}，即，$X$不包含任何防御（进攻）$k$ -联盟作为子集。集合$Y \subseteq V$是\emph{defensive}（\emph{offensive}）\emph{alliance cover}，如果对于所有防御（进攻）$k$ -联盟$S$，$S\cap Y\neq\emptyset$，即，$Y$包含至少一个来自每个防御（进攻）$k$ -$Γ$联盟的顶点。本文给出了防御（进攻）$k$ -联盟自由集和防御（进攻）$k$ -联盟覆盖集的一些数学性质，包括防御（进攻）$k$ -联盟自由（覆盖）集的基数的紧界.
英文摘要：
A \emph{defensive} (\emph{offensive}) $k$-\emph{alliance} in $Γ=(V,E)$ 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) $k$-alliance $S$, $S\cap Y\neq\emptyset$, i.e., $Y$ contains at least one vertex from each defensive (offensive) $k$-alliance of $Γ$. In this paper we show several mathematical properties of defensive (offensive) $k$-alliance free sets and defensive (offensive) $k$-alliance cover sets, including tight bounds on the cardinality of defensive (offensive) $k$-alliance free (cover) sets.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
