Linear kernels for k-tuple and liar's domination in bounded genus graphs · arXivDesk
1309.5461Sep 21, 2013Title 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"
Linear kernels for k-tuple and liar's domination in bounded genus graphs
A set D⊆V is called a k-tuple dominating set of a graph G=(V,E) if ∣NG[v]∩D∣≥k
Nearby in the stack
for all
v∈V
, where
NG[v]
denotes the closed neighborhood of
v
. A set
D⊆V
is called a liar's dominating set of a graph
G=(V,E)
if (i)
∣NG[v]∩D∣≥2
for all
v∈V
and (ii) for every pair of distinct vertices
u,v∈V
,
∣(NG[u]∪NG[v])∩D∣≥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 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
W[2]
-hard for general graphs but they admit linear kernels for graphs with bounded genus.