Bounds on the vertex-edge domination number of a tree

A vertex-edge dominating set of a graph G is a set D of vertices of G such that every edge of G is incident with a vertex of D or a vertex adjacent to a vertex of D. The vertex-edge domination number of a graph G, denoted by gamma(ve)(T), is the minimum cardinality of a vertex-edge dominating set of G. We prove that for every tree T of order n >= 3 with l leaves and s support vertices, we have (n - l - s + 3)/4 <= gamma(ve)(T) <= n/3, and we characterize the trees attaining each of the bounds. (C) 2014 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.