Edge Version of Metric Dimension and Doubly Resolving Sets of the Necklace Graph

Liu, Jia‐Bao; Zahid, Zohaib; Nasir, Ruby; Nazeer, Waqas

doi:10.3390/math6110243

Cited by 43 publications

(30 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…After the first introductory article [10], many authors investigated this invariant. Some of the latest articles for the reader's convenience are as [11], [14], [15], [22], [23].…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On Mixed Metric Dimension of Some Path Related Graphs

Raza

2020

IEEE Access

View full text Add to dashboard Cite

A vertex k ∈ V G determined two elements (vertices or edges) , m ∈ V G ∪ E G , if d G (k,) = d G (k, m). A set R m of vertices in a graph G is a mixed metric generator for G, if two distinct elements (vertices or edges) are determined by some vertex set of R m. The least number of elements in the vertex set of R m is known as mixed metric dimension, and denoted as dim m (G). In this paper, the mixed metric dimension of some path related graphs is obtained. Those path related graphs are P 2 n the square of a path, T (P n) total graph of a path, the middle graph of a path M (P n), and splitting graph of a path S(P n). We proved that these families of graphs have constant and unbounded mixed metric dimension, respectively. We further presented an improved result for the metric dimension of the splitting graph of a path S(P n). INDEX TERMS Mixed metric dimension, Metric dimension, Edge metric dimension, Path related graphs.

show abstract

“…After the first introductory article [10], many authors investigated this invariant. Some of the latest articles for the reader's convenience are as [11], [14], [15], [22], [23].…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On Mixed Metric Dimension of Some Path Related Graphs

Raza

2020

IEEE Access

View full text Add to dashboard Cite

show abstract

“…According to the above steps, dim(L(G)) � edim(G), for reference see [19]. But in our case, the distance is based on vertex to edge, d(v, e), as defined by Kelenc et al [20].…”

Section: Introduction and Preliminarymentioning

confidence: 99%

“…Moreover, we present closed formulas for edge metric dimension of these graphs. For further reading on metric dimension and edge metric dimension, we refer [19,24,25,26,27].…”

Section: Introduction and Preliminarymentioning

confidence: 99%

On Resolvability Parameters of Some Wheel-Related Graphs

Yang

Rafiullah

Siddiqui

et al. 2019

Journal of Chemistry

View full text Add to dashboard Cite

Let G=V,E be a simple connected graph, w∈V be a vertex, and e=uv∈E be an edge. The distance between the vertex w and edge e is given by de,w=mindw,u,dw,v, A vertex w distinguishes two edges e1, e2∈E if dw,e1≠dw,e2. A set S is said to be resolving if every pair of edges of G is distinguished by some vertices of S. A resolving set with minimum cardinality is the basis for G, and this cardinality is the edge metric dimension of G, denoted by edimG. It has already been proved that the edge metric dimension is an NP-hard problem. The main objective of this article is to study the edge metric dimension of some families of wheel-related graphs and prove that these families have unbounded edge metric dimension. Moreover, the results are compared with the metric dimension of these graphs.

show abstract

“…The concepts of resolving set of a graph was first introduced by Slater [1] in 1975 and independently by Harary and Melter [2] in 1976. The metric dimension of a graph has been widely studied and a large number of related concepts have been extended (see [3][4][5][6][7][8][9][10][11]). As a parameter of a graph, it has been applied to lots of practical problems, such as robot navigation [12], connected joins in graphs and combinatorial optimization [13], and pharmaceutical chemistry [14].…”

Section: Introductionmentioning

confidence: 99%

Characterization of n-Vertex Graphs of Metric Dimension n − 3 by Metric Matrix

2019

View full text Add to dashboard Cite

Let G = ( V ( G ) , E ( G ) ) be a connected graph. An ordered set W ⊂ V ( G ) is a resolving set for G if every vertex of G is uniquely determined by its vector of distances to the vertices in W. The metric dimension of G is the minimum cardinality of a resolving set. In this paper, we characterize the graphs of metric dimension n − 3 by constructing a special distance matrix, called metric matrix. The metric matrix makes it so a class of graph and its twin graph are bijective and the class of graph is obtained from its twin graph, so it provides a basis for the extension of graphs with respect to metric dimension. Further, the metric matrix gives a new idea of the characterization of extremal graphs based on metric dimension.

show abstract

Edge Version of Metric Dimension and Doubly Resolving Sets of the Necklace Graph

Cited by 43 publications

References 15 publications

On Mixed Metric Dimension of Some Path Related Graphs

On Mixed Metric Dimension of Some Path Related Graphs

On Resolvability Parameters of Some Wheel-Related Graphs

Characterization of n-Vertex Graphs of Metric Dimension n − 3 by Metric Matrix

Contact Info

Product

Resources

About