@online{DBLP:journals/corr/BishnuGP13,
TITLE = {{Linear Kernels for $k$-Tupel and Liar's Domination in Bounded Genus Graphs}},
AUTHOR = {Bishnu, Arijit and Ghosh, Arijit and Paul, Subhabrata},
LANGUAGE = {eng},
URL = {http://arxiv.org/abs/1309.5461},
EPRINT = {1309.5461},
EPRINTTYPE = {arXiv},
YEAR = {2014},
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.},
CONTENTS = {Title changed from "Parameterized complexity of k-tuple and liar's domination" to "Linear kernels for k-tuple and liar's domination in bounded genus graphs"},
}