On Vertex-Edge and Independent Vertex-Edge Domination

Given a graph, a vertex ve-dominates all edges incident to any vertex of. A set is a ve-dominating set if for all edges, there exists a vertex such that u ve-dominates e. Lewis [Ph.D. thesis, 2007] proposed a linear time algorithm for ve-domination problem for trees. In this paper, first we have constructed an example where the proposed algorithm fails. Then we have proposed a linear time algorithm for ve-domination problem in block graphs, which is a superclass of 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.