Proximinal sets and connectedness in graphs (Q6113259)

Let \(G(V,E)\) be a graph where \(V\neq \emptyset\) denotes the set of vertices and \(E\) denotes the set of edges. A bipartite graph \(G(A, B)\) with fixed parts \(A\) and \(B\) is said to be proximinal if there exists a semimetric space \((X, d)\) such that \(A\) and \(B\) are disjoint proximinal subsets of \(X\), and vertices \(a\in A\) and \(b\in B\) are adjacent if and only if \(d(a, b) = \operatorname{dist}(A, B)\). In [\textit{K. Chaira} et al., J. Math. Sci., New York 264, No. 4, 369--388 (2022; Zbl 1511.54016)], it was proved besides the main structure of bipartite graph that a bipartite graph \(G\) is not isomorphic to any proximinal graph if and only if \(G\) is finite and empty. In the paper [\textit{O. Dovgoshey} and \textit{R. Shanin}, J. Fixed Point Theory Appl. 25, No. 1, Paper No. 34, 31 p. (2023; Zbl 1505.05049)], it is characterized that the semimetric spaces whose proximinal graphs have either at most one edge or vertices of degree at most one only. The graph \(G\) is called path-proximinal graph if there is a semimetric \(d: V\times V\to [0, \infty[\) and disjoint proximinal subsets of the semimetric space \((V, d)\) such that \(V=A\cup B\). The vertices \(u, v\in V\) are adjacent if and only if \[ d(u, v) \leq \inf \{d(x, y): x\in A, y\in B\}, \] and, for every \(p\in V\), there is a path connecting \(A\) and \(B\) in \(G\), and passing through \(p\). In this paper, authors continue to study the interaction between proximity and graphs by introducing path-proximinal graphs.