Disjoint dominating and 2-dominating sets in graphs: Hardness and approximation results

Abstract

A set $D\subseteq V$ of a graph $G=(V,E)$ is a dominating set of $G$ if each vertex $v\in V\setminus D$ is adjacent to at least one vertex in $D,$ whereas a set $D_2\subseteq V$ is a 2-dominating (double dominating) set of $G$ if each vertex $v\in V\setminus D_2$ is adjacent to at least two vertices in $D_2.$ A graph $G$ is a $D{D}_2$-graph if there exists a pair ($D,D_2$) of dominating set and 2-dominating set of $G$ which are disjoint. In this paper, we give approximation algorithms for the problem of determining a minimal spanning $D{D}_2$-graph of minimum size (Min- $D{D}_2$) with an approximation ratio of 3; a minimal spanning $D{D}_2$-graph of maximum size (Max- $D{D}_2$) with an approximation ratio of 3; and for the problem of adding minimum number of edges to a graph $G$ to make it a $D{D}_2$-graph (Min-to- $D{D}_2$) with an $O(logn)$ approximation ratio. The above three results answer the open problems mentioned in the paper, Miotk et al. (2020). Furthermore, we prove that Min- $D{D}_2$ and Max- $D{D}_2$ are APX-complete for graphs with maximum degree $4.$ We also show that Min- $D{D}_2$ and Max- $D{D}_2$ are approximable within a factor of 1.8 and 1.5, respectively, for any 3-regular graph. Finally, we show the inapproximability result of Max-Min-to- $D{D}_2$ for bipartite graphs: this problem cannot be approximated within $n^{\frac{1}{6}-\varepsilon }$ for any $\varepsilon >0,$ unless P=NP.