Telephone center functions on trees

In Theorem 8.7 of [22] eight centralities on trees are presented that all coincide with the median. In this paper we explore a functional extension for three of these Centralities, viz. the Centroid, the Security Center, and The Telephone Center of a tree. In the functional extension model, instead of using the whole vertex set to determine `central’ vertices we allow any multiset of vertices to determine the central vertices. The centroid and security center allow straightforward functional extensions, and both coincide with the well-kown median function. The functional extension of the Telephone center is a different story, and we present three versions, each of which catches most but not all the features of the original Telephone center. These all have a close relationship with the median function. As a bonus we obtain a deeper insight in the median function on trees.