A cut locus for finite graphs and the farthest point mapping

Abstract

We reflect upon an analogue of the cut locus, a notion classically studied in Differential Geometry, for finite graphs. The cut locusC(x) of a vertex x shall be the graph induced by the set of all vertices y with the property that no shortest path between x and z, z≠y, contains y. The cut locus coincides with the graph induced by the vertices realizing the local maxima of the distance function. The function F mapping a vertex x to F(x), the set of global maxima of the distance function from x, is the farthest point mapping. Among other things, we observe that if, for a vertex x, C(x) is connected, then C(x) is the graph induced by F(x), and prove that the farthest point mapping has period 2. Elaborating on the analogy with Geometry, we study graphs satisfying Steinhaus’ condition, i.e. graphs for which the farthest point mapping is single-valued and involutive.