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Alliance free and alliance cover sets

J. A. Rodriguez-Velazquez·J. M. Sigarreta·I. G. Yero·S. Bermudo
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Abstract

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.

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BibTeXRIS

J. A. Rodriguez-Velazquez, J. M. Sigarreta, I. G. Yero, S. Bermudo. 2008-12-08. Alliance free and alliance cover sets. https://doi.org/10.1007/s10114-011-0056-1

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