Total domination and vertex-edge domination in tres

Abstract

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 if every edge of E is ve-dominated by at least one vertex of S. The minimum cardinality of a vertex-edge dominating set of G is the vertex-edge domination number γve(G) . In this paper we prove (γt(T)−l+1)/2 ≤ γve(T) ≤(γt(T)+l−1)/2 and characterize trees attaining each of these bounds.