Bounds on the vertex–edge domination number of a tree

Abstract

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 γve(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⩽γve(T)⩽n/3, and we characterize the trees attaining each of the bounds.