  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 whose
proximinal graphs have at most one edge and the semimetric spaces whose
proximinal graphs have the vertices with degree at most $1$ only. This allows
us to describe the necessary and sufficient conditions for uniqueness of the
best proximity pairs and best approximations.
