WALCOM 2019: WALCOM: Algorithms and Computation pp 188-198

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

• Y. B. Venkatakrishnan
• H. Naresh Kumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)

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.

Keywords

Double vertex-edge domination Chordal graph Proper interval graph NP-complete APX-complete

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