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 is a semimetric \(d \colon V \times V \to [0, \infty[\) and disjoint proximinal subsets of the semimetric space \((V, d)\) such that \(V = A \cup B\). The vertices \(u\), \(v \in V\) are adjacent iff \[ d(u, v) \leqslant \inf \{d(x, y) \colon x \in A, y \in B\}, \] and, for every \(p \in V\), 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 \(1\).