An improved upper bound of edge-vertex domination number of a tree

An edge e∈E(G) dominates a vertex v∈V(G) if e is incident with v or e is incident with a vertex adjacent to v. An edge-vertex dominating set of a graph G is a set D of edges of G such that every vertex of G is edge-vertex dominated by an edge of D. The edge-vertex domination number of a graph G is the minimum cardinality of an edge-vertex dominating set of G. A subset D⊂V(G) is a total dominating set of G if every vertex of G has a neighbor in D. The total domination number of G is the minimum cardinality of a total dominating set of G. We prove that for every nontrivial tree T of order n, with s support vertices we have γev(T)≤(γt(T)+s−1)/2, and we characterize the trees attaining this upper bound.