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.

在图\（G =（V，E）\） ，顶点\（V \以V \）被说成已经-支配边缘入射v以及相邻的对这些事件的边缘边缘v。如果图的每个边都由D的至少两个顶点ve支配，则集合\（D \ subseteq V \）被称为双顶点边控制集。给定一个图G，双顶点边缘占优问题，即Min - DVEDS是找到G的最小双顶点边缘占优集。在本文中，我们表明对于弦图，Min - DVEDS的决策版本 是NP -complete。我们提出一种线性时间算法，以在适当的间隔图中找到最小的双顶点边控制集。我们还表明，对于具有n个顶点的图，除非NP \（\ subseteq \）DTIME，否则对于任何\（\ varepsilon> 0 \），Min - DVEDS 都不能在\（（1- -varepsilon ）\ ln n \）内近似。（\（n ^ {O（\ log \ log n）} \））。在积极的一面，我们表明，闽- DVEDS 可以通过倍近似 \（O（\ ln \ varDelta）\）。最后，我们证明对于最大度为5的图，Min - DVEDS 是APX -complete。