Abstract
ABSTRACTA vertex v of a graph is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set is a vertex-edge dominating set (or simply, a ve-dominating set) if every edge of E is ve-dominated by at least one vertex of S. The minimum cardinality of a ve-dominating set of G is the vertex-edge domination number A ve-dominating set is said to be total if its induced subgraph has no isolated vertices. The minimum cardinality of a total ve-dominating set of G is the total vertex-edge domination number In this paper we initiate the study of total vertex-edge domination. We show that determining the number for bipartite graphs is NP-complete. Then we show that if T is a tree different from a star with order n, ℓ leaves and s support vertices, then Moreover, we characterize the trees attaining this upper bound. Finally, we establish a necessary condition for graphs G such that and we provide a characterization of all trees T with .
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