On the Algorithmic Complexity of Double Vertex-Edge Domination in Graphs

Abstract

Let \(G=(V,E)\) be a simple graph. A vertex \(v\in V\) ve-dominates every edge uv incident to v, as well as every edge adjacent to these incident edges. A set \(D\subseteq V\) is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of D. The double vertex-edge dominating problem is to find a minimum double vertex-edge dominating set of G. In this paper, we show that minimum double vertex-edge dominating problem is NP-complete for chordal graphs. A linear time algorithm to find the minimum double vertex-edge dominating set for proper interval graphs is proposed. We also show that the minimum double vertex-edge domination problem cannot be approximated within \((1-\varepsilon )\ln |V|\) for any \(\varepsilon >0\) unless NP\(\subseteq \) DTIME\((|V|^{O(\log \log |V|)})\). Finally, we prove that the minimum double vertex-edge domination problem is APX-complete for graphs with maximum degree 5.