Ultrametrics and Complete Multipartite Graphs

Bilet, Viktoriia; Dovgoshey, Oleksiy; Kononov, Yu. D.

doi:10.20429/tag.2022.090108

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2024

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Cited by 8 publications

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“…This allows them to find the necessary and sufficient conditions for the uniqueness of the best proximity pairs and the best approximations. Some references concerning graphs and best proximity points are given in [1,4,[6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Proximinal sets and connectedness in graphs

Chaira¹,

Dovgoshey²

2023

Preprint

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Let G be a graph with a vertex set V . The graph G is path-proximinal if there are a semimetric d :and, for every p ∈ V , there is a path connecting A and B in G, and passing through p.It is shown that a graph is path-proximinal if and only if all its vertices are not isolated.It is also shown that a graph is simultaneously proximinal and path-proximinal for an ultrametric if and only if the degree of every its vertex is equal to 1.

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Section: Introductionmentioning

confidence: 99%

Proximinal sets and connectedness in graphs

Chaira¹,

Dovgoshey²

2023

Preprint

View full text Add to dashboard Cite

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Bipartite graphs and best proximity pairs

2022

Self Cite

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We say that a bipartite graph G(A, B) with the fixed parts A and B is proximinal if there is a semimetric space (X, d) such that A and B are disjoint proximinal subsets of X 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 if and only if 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 for an ultrametric space.

show abstract

Proximinal sets and connectedness in graphs

Chaira¹,

Dovgoshey

2023

J Math Sci

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Ultrametrics and Complete Multipartite Graphs

Cited by 8 publications

References 46 publications

Proximinal sets and connectedness in graphs

Proximinal sets and connectedness in graphs

Bipartite graphs and best proximity pairs

Proximinal sets and connectedness in graphs

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