Bipartite graphs and best proximity pairs

Abstract

In the present paper we introduce a proximinal graph as a bipartite graph G A,B with parts A, B for which (A, B) is a disjoint proximinal pair in a semimetric space (M, d) 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 if and only if G is isomorphic to a nonempty proximinal graph G A,B such that (A, B) is a proximinal pairs in an ultrametric space.

Uniqueness of best proximity pairs and rigidity of semimetric spaces

“…Then the following statements are equivalent: (i) Either G is not a null graph and G is a disjoint union of complete bipartite graphs, or E(G) = ∅, but the sets A and B are infinite. (ii) G is proximinal for an ultrametric space (X, d) with X = A ∪ B. Theorem 3.5 was proved in [9]. Proposition 3.6.…”

“…Investigations of proximinal graphs for semimetric and ultrametric spaces were started in [9]. The present paper mainly examines the conditions for the uniqueness of best proximity pairs and, in particular, of best approximations in semimetric spaces.…”