Uniqueness of best proximity pairs and rigidity of semimetric spaces

For arbitrary semimetric space $$(X, d)$$ and disjoint proximinal subsets $$A$$ , $$B$$ of $$X$$ we define the graph as a bipartite with parts whose edges $$\{a, b\}$$ satisfy equality $$d(a, b) = {\text {dist}}(A, B)$$ . We characterize spaces graphs have at most one edge vertices degree $$1$$ only. This allows us to describe necessary sufficient conditions for uniqueness best proximity pairs approximations.