DOUBLE VERTEX-EDGE DOMINATION IN TREES

A vertex v of a graph G = (V, E) is said to ye-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S subset of V is called a double vertex-edge dominating set if every edge of E is ye-dominated by at least two vertices of S. The minimum cardinality of a double vertex-edge dominating set of G is the double vertex-edge domination number gamma(dve) (G). In this paper, we provide an upper bound on the double vertex-edge domination number of trees in terms of the order n, the number of leaves and support vertices, and we characterize the trees attaining the upper bound. Finally, we design a polynomial time algorithm for computing the value of gamma(d)(ve)(T) for any trees. This gives an answer of an open problem posed in [4].