  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) $k$-alliance $S$,
$S\cap Y\neq\emptyset$, i.e., $Y$ contains at least one vertex from each
defensive (offensive) $k$-alliance of $\Gamma$. 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.
