In this paper, we employ two types of implicit relations to define some new kind proximal contractions and study about their best proximity points. More precisely, use class functions $\mathcal{A}$ $\mathcal{A}'$ explore $\mathcal{A}$, $\mathcal{A}'$-contractions first second type strong $\mathcal{A}'$-contractions. We investigate the existence point results same. It is worth mentioning that we...

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

Conservation of proximity of a pair of genes across multiple genomes generally indicates that their functions could be linked. Here, we present a systematic evaluation using 42 complete microbial genomes from 25 phylogenetic groups to test the reliability of this observation in predicting function for genes. We find a relationship between the number of phylogenetic groups in which a gene pair i...

We present algorithms for five interdistance enumeration problems that take as input a set S of n points in IRd (for a fixed but arbitrary dimension d) and as output enumerate pairs of points in S satisfying various conditions. We present: an O(n log n + k) time and O(n) space algorithm that takes as additional input a distance δ and outputs all k pairs of points in S separated by a distance of...

We introduce and study a generalization of the well-known region of influence proximity drawings, called (ε1, ε2)-proximity drawings. Intuitively, given a definition of proximity and two real numbers ε1 ≥ 0 and ε2 ≥ 0, an (ε1, ε2)-proximity drawing of a graph is a planar straight-line drawing Γ such that: (i) for every pair of adjacent vertices u, v, their proximity region “shrunk” by the multi...

Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R, a (1+ǫ)-spanner with O(nǫ) edges is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The resul...