Bipartite graphs and best proximity pairs

TL;DR

This paper characterizes bipartite graphs that can be represented as proximinal subsets in semimetric spaces, linking graph structure to metric properties and ultrametric spaces.

Contribution

It provides necessary and sufficient conditions for bipartite graphs to be proximinal graphs in semimetric and ultrametric spaces, including a complete characterization.

Findings

A bipartite graph is not isomorphic to any proximinal graph iff it is finite and empty.

The subgraph of non-isolated vertices in a bipartite graph is a disjoint union of complete bipartite graphs iff it is a proximinal graph for an ultrametric space.

Abstract

We say that a bipartite graph $G(A, B)$ with fixed parts $A$, $B$ is proximinal if there is a semimetric space $(X, d)$ such that $A$ and $B$ are disjoint proximinal subsets of $X$ and all edges $\{a, b\}$ satisfy the equality $d(a, b) = \operatorname{dist}(A, B)$. It is proved that a bipartite graph $G$ is not isomorphic to any proximinal graph iff $G$ is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph $G$ is a disjoint union of complete bipartite graphs iff $G$ is isomorphic to a nonempty proximinal graph for an ultrametric space.