Proximinal sets and connectedness in graphs

Abstract

Let G be a graph with a vertex set V . The graph G is path-proximinal if there are a semimetric d :and, for every p ∈ V , there is a path connecting A and B in G, and passing through p.It is shown that a graph is path-proximinal if and only if all its vertices are not isolated.It is also shown that a graph is simultaneously proximinal and path-proximinal for an ultrametric if and only if the degree of every its vertex is equal to 1.