2022/01/12 by Oleksiy Dovgoshey, Dovgoshey, Oleksiy, Ruslan Shanin +1
Computer Science · Mathematics · #41A50 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Fixed Point Theorems Analysis #General Topology (math.GN) #Optimization and Variational Analysis #Primary: 05C60. Secondary: 54E35
paper · pdf · doi:10.48550/arxiv.2201.04380
openalex publication_date 2022/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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) = 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.