A cut locus for finite graphs and the farthest point mapping

We reflect upon an analogue of the cut locus, a notion classically studied in Differential Geometry, for finite graphs. The cut locus  C(x)C(x) of a vertex xx shall be the graph induced by the set of all vertices yy with the property that no shortest path between xx and zz, z≠yz≠y, contains yy. The cut locus coincides with the graph induced by the vertices realizing the local maxima of the distance function. The function FF mapping a vertex xx to F(x)F(x), the set of global maxima of the distance function from xx, is the farthest point mapping  . Among other things, we observe that if, for a vertex xx, C(x)C(x) is connected, then C(x)C(x) is the graph induced by F(x)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.