Total outer connected vertex-edge domination

A vertex v of a graph is said to vertex-edge dominate every edge incident with v, as well as every edge incident to vertices adjacent to v. A subset D subset of V is a total outer connected vertex-edge dominating set of a graph G if every edge in G is vertex-edge dominated by a vertex in D, the subgraph induced by D has no isolated vertices and the subgraph induced by V(G)\D is connected. We initiate the study of total outer connected vertex-edge domination in graphs. We show that the decision problem for total outer-connected vertex-edge domination problem is NP-Complete even for bipartite graphs. We prove that for every tree of order n >= 3 with l leaves, gamma(oc)(tve)(T) >= 2(n -l + 1)/3 and characterize the trees attaining the lower bound. We also study the effect of edge removal, edge addition and edge subdivision on total outer connected vertex-edge domination number of a graph.