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

Uniqueness of best proximity pairs and rigidity of semimetric spaces