The median function of a block graph: Axiomatic characterizations

Abstract

A profile π=(x1,x2,…,xk) of length k in a connected graph G is a sequence of vertices in G with possible repetitions of vertices. A median x of a profile π in G is a vertex that minimizes the remoteness value, that is, the sum of the distances from x to the elements xi in π is minimized. The median function output the set of medians (denoted as Med(π)) of π, for every profile π in G. It is one of the basic models for the location of a desirable facility in a network. The median function on graphs is well studied. For instance, it has been characterized axiomatically by simple axioms on trees, hypercubes, median graphs, cocktail-party graphs, and complete graphs minus a matching. In this paper, we study the median sets and median function of block graphs, an immediate generalization of trees. We determine the median sets of all types of profiles and we obtain two sets of axiomatic characterizations of the median function on block graphs.