Vertices, edges, distances and metric dimension in graphs

doi:10.1016/j.endm.2016.10.047

Electronic Notes in Discrete Mathematics

Volume 55, November 2016, Pages 191-194

https://doi.org/10.1016/j.endm.2016.10.047 Get rights and content

Abstract

Given a connected graph $G = (V, E)$ , a set of vertices $S \subset V$ is an edge metric generator for G, if any two edges of G are identified by S by mean of distances to the vertices in S. Moreover, in a natural way, S is a mixed metric generator, if any two elements of G (vertices or edges) are identified by S by mean of distances. In this work we study the (edge and mixed) metric dimension of graphs.

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Citation Excerpt :
Moving on to the edge metric dimension, [26] examined the barycentric subdivision of the Cayley graph, [55,1] presented a few works on the convex polytopes structure, and [51] addressed the chemical structures of wheel graphs. In addition, the foundational work on the edge metric dimension is published in the reference [52], which includes a quantitative comparison between metric and its various variants. Some current work can be gained by the references [39,45,39,2,42], for the fault-tolerant idea mentioned in [17] for basic graphs and [37] for diverse connectivity networks along with the deployment of their applications.
Starphenes are polycyclic aromatic hydrocarbons and build by variatiTheoremons of three different acene-arms. Starphenes are the basic building blocks for miniaturization of different especially organic electronic devices. It also played an important role in different logical gates. In network topology, each electric circuit, structure or network can be derived in graphs where principal nodes (or simply nodes) alternated to vertices and line segment (branches) become edges. The resolvability parameters of a graph are a relatively new specialized subject in which the whole network is formed to obtain each principal node’s unique location. The metric, edge metric dimension, and generalizations are investigated in this article as resolvability parameters of starphene structure. We proved that all the parameters studied for starphene graph have constant cardinalities. Understanding and dealing with structures is made easier by transforming the whole structure into a novel shape delivered by resolvability parameters.
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Vertices, edges, distances and metric dimension in graphs

Abstract

Discrete Appl. Math.

On the metric dimension of a graph

Ars Combin.