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 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 exist 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 (Liao and Chang, 2003; Slater, 2009). 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. It can be verified that both the problems have a finite integer index and satisfy certain coverability property. Hence they admit linear kernel as per the meta-theorem in Bodlaender (2009), but the meta-theorem says nothing about the constant. In this paper, we present a direct proof of the existence of linear kernel with small constants for both the problems.