Total vertex-edge domination
Razika Boutrig, Mustapha Chellali · International Journal of Computer Mathematics · 2017
A vertex v of a graph G=(V,E) is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S⊆V 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 γve(G). 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 γvet(G). In this paper we initiate the study of total vertex-edge domination. We show that determining the number γvet(G) 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 γvet(T)≤(n−ℓ+s)/2. Moreover, we characterize the trees attaining this upper bound. Finally, we establish a necessary condition for graphs G such that γvet(G)=2γve(G) and we provide a characterization of all trees T with γvet(T)=2γve(T).