Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Alliance free and alliance cover sets   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.

31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
9
The categories of this paper are: nlin