Medians in median graphs

Abstract

A median of a family of vertices in a graph is any vertex whose distance-sum to that family is minimum. In the framework of metric spaces the problem of minimizing a distance-sum is often referred to as the Fermat problem. On the other hand, medians have been studied from a purely order-theoretic or combinatorial point of view (for instance, in statistics, or in Jordan’s work [12] on trees). The aim of this paper is to investigate the mutual relationship of the metric and the ordinal/ combinatorial approaches to the median problem in the class of median graphs. A connected graph is a median graph if any three vertices admit a unique median (see Avann [l]). Note that trees and the covering graphs of distributive lattices are median graphs. Very little is known about medians in arbitrary graphs (cf. Slater [20]); so far, only trees (Zelinka [22], and many others) and the covering graphs of distributive lattices (Barbut [4]) have been considered. In both cases we get that (i) the medians of any family form an interval (a path in a tree, an order-theoretic interval in a distributive lattice), and (ii) medians of odd numbered families are unique (see Slater [19] for trees, and Barbut [4] for distributive lattices). These results point to the fact that (i) and (ii) must be true for any median graph. After recalling some basic definitions and facts concerning median graphs and median semilattices (for further information, see Bandelt and Hedlikova [3]), we establish (i) and (ii) for arbitrary median graphs. Our results are based on theorems of Avann, Sholander, and Barbut. In trees medians have nice local properties (cf. [7]). Indeed, median sets are related to mass centers (Zelinka [22]) and security centers (Slater [18]). In Section 3 this is extended to median graphs. The study of medians applies to social choice theory (see Barbut 151, and Barthelemy and Monjardet [8]). The median procedure is strongly related to the simple majority rule: the median of a family (A,, . . . , Azk+ ,) of subsets of a set X may be written as U n Ai (Barbut’s formula).