Proximinal sets and connectedness in graphs

Let G be a graph with a vertex set V . The graph G is path-proximinal if there is a semimetric d : V × V → [0, ∞[ and disjoint proximinal subsets of the semimetric space (V, d) such that V = A ∪ B. The vertices u, v ∈ V are adjacent iff duv⩽infdxy:x∈Ay∈B,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ d\left(u,v\right)\leqslant \operatorname{inf}\left\{d\left(x,y\right):x\in A,y\in B\right\}, $$\end{document} and, for every p ∈ V , there is a path connecting A and B in G, and passing through p. It has been shown that a graph is path-proximinal if and only if all of its vertices are not isolated. It has also been shown that a graph is simultaneously proximinal and path-proximinal for an ultrametric if and only if the degree of each of its vertices is equal to 1.