Results on vertex-edge and independent vertex-edge domination

Given a graph G= (V, E) , a vertex u∈ Vve-dominates all edges incident to any vertex of NG[u]. A set S⊆ V is a ve-dominating set if for all edges e∈ E, there exists a vertex u∈ 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.