# Linear kernels for k-tuple and liar's domination in bounded genus graphs

## Article Type

Research Article

## Publication Title

Discrete Applied Mathematics

## Abstract

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.

## First Page

67

## Last Page

77

## DOI

10.1016/j.dam.2016.06.008

## Publication Date

11-20-2017

## Recommended Citation

Bishnu, Arijit; Ghosh, Arijit; and Paul, Subhabrata, "Linear kernels for k-tuple and liar's domination in bounded genus graphs" (2017). *Journal Articles*. 2341.

https://digitalcommons.isical.ac.in/journal-articles/2341

## Comments

Open Access, Bronze, Green