Double vertex-edge domination in graphs: complexity and algorithms

Abstract

In a graph $G=(V,E)$, a vertex $v\in V$ is said to ve-dominate the edges incident on v as well as the edges adjacent to these incident edges on v. A set $D\subseteq V$ is called a double vertex-edge dominating set if every edge of the graph is ve-dominated by at least two vertices of D. Given a graph G, the double vertex-edge dominating problem, namely Min-DVEDS is to find a minimum double vertex-edge dominating set of G. In this paper, we show that the decision version of Min-DVEDS is NP-complete for chordal graphs. We present a linear time algorithm to find a minimum double vertex-edge dominating set in proper interval graphs. We also show that for a graph having n vertices, Min-DVEDS cannot be approximated within $(1 -\varepsilon ) \ln n$ for any $\varepsilon > 0$ unless NP $\subseteq$ DTIME($n^{O(\log \log n)}$). On positive side, we show that Min-DVEDS can be approximated by a factor of $O(\ln \varDelta )$. Finally, we show that Min-DVEDS is APX-complete for graphs with maximum degree 5.