Bipartite graphs and best proximity pairs
Karim Chaira, O. Dovgoshey, Samih Lazaiz · Ukrainian Mathematical Bulletin · 2022
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) = 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.