On 2-metric resolvability in rotationally-symmetric graphs

2021 ◽  
pp. 1-9
Author(s):  
Humera Bashir ◽  
Zohaib Zahid ◽  
Agha Kashif ◽  
Sohail Zafar ◽  
Jia-Bao Liu
Keyword(s):  
Network Optimization ◽  
Intelligent Systems ◽  
Robot Navigation ◽  
Data Transformation ◽  
Metric Dimension ◽  
Intelligent Networks ◽  
Symmetric Graphs ◽  
Symmetric Plane ◽  
Rotationally Symmetric ◽  
Sensor Networking

The 2-metric resolvability is an extension of metric resolvability in graphs having several applications in intelligent systems for example network optimization, robot navigation and sensor networking. Rotationally symmetric graphs are important in intelligent networks due to uniform rate of data transformation to all nodes. In this article, 2-metric dimension of rotationally symmetric plane graphs Rn, Sn and Tn is computed and found to be independent of the number of vertices.

Metric dimension of heptagonal circular ladder

2020 ◽  
pp. 2050095
Author(s):  
Sunny Kumar Sharma ◽  
Vijay Kumar Bhat
Keyword(s):  
Open Problem ◽  
Geodesic Distance ◽  
Radial Symmetry ◽  
Metric Dimension ◽  
Plane Graphs ◽  
Path Graph ◽  
Symmetric Plane ◽  
Rotationally Symmetric ◽  
Circular Ladder ◽  
Multiple Edges

Let [Formula: see text] be an undirected (i.e., all the edges are bidirectional), simple (i.e., no loops and multiple edges are allowed), and connected (i.e., between every pair of nodes, there exists a path) graph. Let [Formula: see text] denotes the number of edges in the shortest path or geodesic distance between two vertices [Formula: see text]. The metric dimension (or the location number) of some families of plane graphs have been obtained in [M. Imran, S. A. Bokhary and A. Q. Baig, Families of rotationally-symmetric plane graphs with constant metric dimension, Southeast Asian Bull. Math. 36 (2012) 663–675] and an open problem regarding these graphs was raised that: Characterize those families of plane graphs [Formula: see text] which are obtained from the graph [Formula: see text] by adding new edges in [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, by answering this problem, we characterize some families of plane graphs [Formula: see text], which possesses the radial symmetry and has a constant metric dimension. We also prove that some families of plane graphs which are obtained from the plane graphs, [Formula: see text] by the addition of new edges in [Formula: see text] have the same metric dimension and vertices set as [Formula: see text], and only 3 nodes appropriately selected are sufficient to resolve all the nodes of these families of plane graphs.

scholarly journals On Mixed Metric Dimension of Rotationally Symmetric Graphs

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 11560-11569 ◽  
Author(s):  
Hassan Raza ◽  
Jia-Bao Liu ◽  
Shaojian Qu
Keyword(s):  
Metric Dimension ◽  
Symmetric Graphs ◽  
Rotationally Symmetric ◽  
Mixed Metric

On fault-tolerant partition dimension of graphs

2021 ◽  
Vol 40 (1) ◽  
pp. 1129-1135
Author(s):  
Kamran Azhar ◽  
Sohail Zafar ◽  
Agha Kashif ◽  
Zohaib Zahid
Keyword(s):  
Connected Graph ◽  
Intelligent Systems ◽  
Fault Tolerant ◽  
Robot Navigation ◽  
Natural Extension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Sensor Networking ◽  
Partition Dimension

Fault-tolerant resolving partition is natural extension of resolving partitions which have many applications in different areas of computer sciences for example sensor networking, intelligent systems, optimization and robot navigation. For a nontrivial connected graph G (V (G) , E (G)), the partition representation of vertex v with respect to an ordered partition Π = {Si : 1 ≤ i ≤ k} of V (G) is the k-vector r ( v | Π ) = ( d ( v , S i ) ) i = 1 k , where, d (v, Si) = min {d (v, x) |x ∈ Si}, for i ∈ {1, 2, …, k}. A partition Π is said to be fault-tolerant partition resolving set of G if r (u|Π) and r (v|Π) differ by at least two places for all u ≠ v ∈ V (G). A fault-tolerant partition resolving set of minimum cardinality is called the fault-tolerant partition basis of G and its cardinality the fault-tolerant partition dimension of G denoted by P ( G ) . In this article, we will compute fault-tolerant partition dimension of families of tadpole and necklace graphs.

scholarly journals On Metric Dimension of Some Rotationally Symmetric Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
M. Ali ◽  
M. T. Rahim ◽  
G. Ali ◽  
U. Ali
Keyword(s):  
Metric Dimension ◽  
Connected Graphs ◽  
Symmetric Graphs ◽  
Rotationally Symmetric

A family 𝒢 of connected graphs is a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in 𝒢. In this paper, we show that the graph An∗ and the graph Anp obtained from the antiprism graph have constant metric dimension.

scholarly journals A Combinatorial Approach to the Computation of the Fractional Edge Dimension of Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2364
Author(s):  
Nosheen Goshi ◽  
Sohail Zafar ◽  
Tabasam Rashid ◽  
Juan L. G. G. Guirao
Keyword(s):  
Computer Science ◽  
Intelligent Systems ◽  
Robot Navigation ◽  
Circulant Graph ◽  
Combinatorial Approach ◽  
Sensor Networking ◽  
Edge Transitive ◽  
Transitive Graphs

E. Yi recently introduced the fractional edge dimension of graphs. It has many applications in different areas of computer science such as in sensor networking, intelligent systems, optimization, and robot navigation. In this paper, the fractional edge dimension of vertex and edge transitive graphs is calculated. The class of hypercube graph Qn with an odd number of vertices n is discussed. We propose the combinatorial criterion for the calculation of the fractional edge dimension of a graph, and hence applied it to calculate the fractional edge dimension of the friendship graph Fk and the class of circulant graph Cn(1,2).

scholarly journals Power graphs and exchange property for resolving sets

2019 ◽  
Vol 17 (1) ◽  
pp. 1303-1309 ◽  
Author(s):  
Ghulam Abbas ◽  
Usman Ali ◽  
Mobeen Munir ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Shin Min Kang
Keyword(s):  
Present Article ◽  
Linear Algebra ◽  
Finite Groups ◽  
Robot Navigation ◽  
Simple Graph ◽  
Exchange Property ◽  
Metric Dimension ◽  
Sufficient Condition ◽  
Resolving Set ◽  
Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

scholarly journals On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 191 ◽  
Author(s):  
Shahid Imran ◽  
Muhammad Siddiqui ◽  
Muhammad Imran ◽  
Muhammad Hussain
Keyword(s):  
Connected Graph ◽  
The Other ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Four Dimensions ◽  
Symmetric Graphs ◽  
Metric Dimensions ◽  
Cycle Path

Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). In this paper, Cycle, Path, Harary graphs and their rooted product as well as their connectivity are studied and their metric dimension is calculated. It is proven that metric dimension of some graphs is unbounded while the other graphs are constant, having three or four dimensions in certain cases.

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