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 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 NPcomplete [LC03, Sla09]. 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.