Bipartite graphs and best proximity pairs (Q2683534)

This paper introduces proximinal graphs, i.e, Definition 1.1. Let \((X,d)\in \boldsymbol{SM}\) (Semi-Metric Space). A set \(A\subset X\) is said to be \textit{proximinal} in \((X,d)\) if for every \(x\in X\), there exists \(a_0\in A\) such that \[ d(x,a_0) = \inf\left\{d(x,a):a\in A\right\}. \] It is observed that proximinal graphs are bipartite graphs that have vertex sets of arbitrary cardinality (Def. 1.7). This observation is accompanied by a detailed example. Among the many important results given in this paper, the following main result is given. Theorem 3.9. Let \(G\) be a bipartite graph with the fixed parts \(A\) and \(B\). Then the following statements are equivalent. \begin{itemize} \item[1.] Either \(G\) is nonempty and [graph] \(G'\) is the disjoint union of complete bipartite graphsm, or \(G\) is empty, but the [nonempty] sets \(A\) and \(B\) are infinite. \item[2.] \(G\) proximinal for an ultrametric space \((X,d)\) with \(X = A\cup B\). \item[3.] \(G\) is proximal for an ultrametic space. \end{itemize} A detailed proof spanning three pages is given for Theorem 3.9. The authors of this paper refer the reader to [\textit{V. V. Bilet} et al., Theory Appl. Graphs 9, No. 1, Article 8, 17 p. (2022; Zbl 07633287); \textit{T. D. Narang}, Demonstr. Math. 29, No. 2, 445--450 (1996; Zbl 0902.41019)]] for an in-depth view of complete bipartite graphs considered in the context of ultrametric spaces. This paper has 40 helpful references.