Given a graph \(G = (V,E)\), a vertex \(u \in V\) ve-dominates all edges incident to any vertex of \(N_G[u]\). A set \(S \subseteq V\) is a ve-dominating set if for all edges \(e\in E\), there exists a vertex \(u \in S\) such that u ve-dominates e. Lewis (Vertex-edge and edge-vertex parameters in graphs. Ph.D. thesis, Clemson, SC, USA, 2007) proposed a linear time algorithm for ve-domination problem for trees. In this paper, we have constructed an example where the algorithm proposed by Lewis, fails. We have proposed linear time algorithms for ve-domination and independent ve-domination problem in block graphs, which is a superclass of trees. We have also proposed a linear time algorithm for weighted ve-domination problem in trees. We have also proved that finding minimum ve-dominating set is NP-complete for undirected path graphs. Finally, we have characterized the trees with equal ve-domination and independent ve-domination number.

给定一个图\(G = (V,E)\)，顶点\(u \in V\) ve 支配与\(N_G[u]\) 的任何顶点相关的所有边。集合\(S \subseteq V\)是一个ve 支配集，如果对于所有边\(e\in E\)，都存在一个顶点\(u \in S\)使得u ve-dominates e. Lewis（图中的顶点-边和边-顶点参数。博士论文，克莱姆森，南卡罗来纳州，美国，2007 年）提出了一种用于树的 ve-domination 问题的线性时间算法。在本文中，我们构建了一个示例，其中 Lewis 提出的算法失败。我们已经提出了线性时间算法来解决块图中的 ve-domination 和独立 ve-domination 问题，这是树的超类。我们还为树中的加权 ve-domination 问题提出了一种线性时间算法。我们还证明，对于无向路径图，找到最小 ve 支配集是 NP 完全的。最后，我们对具有相等 ve-domination 和独立 ve-domination 数的树进行了表征。