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Free keywords:
Computer Science, Computational Complexity, cs.CC,Computer Science, Data Structures and Algorithms, cs.DS,
Abstract:
A set $D\subseteq V$ is called a $k$-tuple dominating set of a graph
$G=(V,E)$ if $\left| N_G[v] \cap D \right| \geq k$ for all $v \in V$, where
$N_G[v]$ denotes the closed neighborhood of $v$. A set $D \subseteq V$ is
called a liar's dominating set of a graph $G=(V,E)$ if (i) $\left| N_G[v] \cap
D \right| \geq 2$ for all $v\in V$ and (ii) for every pair of distinct vertices
$u, v\in V$, $\left| (N_G[u] \cup N_G[v]) \cap D \right| \geq 3$. Given a graph
$G$, the decision versions of $k$-Tuple Domination Problem and the Liar's
Domination Problem are to check whether there exists a $k$-tuple dominating set
and a liar's dominating set of $G$ of a given cardinality, respectively. These
two problems are known to be NP-complete \cite{LiaoChang2003, Slater2009}. In
this paper, we study the parameterized complexity of these problems. We show
that the $k$-Tuple Domination Problem and the Liar's Domination Problem are
$\mathsf{W}[2]$-hard for general graphs but they admit linear kernels for
graphs with bounded genus.
