From Vertex-Telecenters to Subtree-Telecenters

Abstract

Let T be a tree and v a vertex in T. It is well-known that the branch-weight of v is defined as the maximum number of vertices in the components of T−v and that a vertex of T with the minimum branch-weight is called a vertex-centroid of T. Mitchell (Discrete Math. 24:277–280, 1978) introduced a type of a central vertex called the telephone center or the vertex-telecenter of a tree and showed that v is a vertex-centroid of T if and only if it is a vertex-telecenter of T. In this paper we introduce the notions of the subtree-centroid and the subtree-telecenter of a tree which are natural extensions of the vertex-centroid and the vertex-telecenter, and generalize two theorems of Mitchell (Discrete Math. 24:277–280, 1978) in the extended framework of subtree-centroids and subtree-telecenters. As a consequence of these generalized results we also obtain an efficient solution method which computes a subtree-telecenter of a tree.