Total outer connected vertex-edge domination

Abstract

A vertex [Formula: see text] of a graph is said to [Formula: see text]-[Formula: see text] dominate every edge incident with [Formula: see text], as well as every edge incident to vertices adjacent to [Formula: see text]. A subset [Formula: see text] is a total outer connected vertex-edge dominating set of a graph [Formula: see text] if every edge in [Formula: see text] is [Formula: see text]-[Formula: see text] dominated by a vertex in [Formula: see text], the subgraph induced by [Formula: see text] has no isolated vertices and the subgraph induced by [Formula: see text] 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 [Formula: see text]-Complete even for bipartite graphs. We prove that for every tree of order [Formula: see text] with [Formula: see text] leaves, [Formula: see text] 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.