Computing the Metric Dimension of Gear Graphs

Abstract

Let G = (V, E) be a connected graph and d(u, v) denote the distance between the vertices u and v 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). Let J 2n,m be a m-level gear graph obtained by m-level wheel graph W 2n,m ∼ = mC 2n + k 1 by alternatively deleting n spokes of each copy of C 2n and J 3n be a generalized gear graph obtained by alternately deleting 2n spokes of the wheel graph W 3n. In this paper, the metric dimension of certain gear graphs J 2n,m and J 3n generated by wheel has been computed. Also this study extends the previous result given by Tomescu et al. in 2007.

On the Metric Dimension of Arithmetic Graph of a Composite Number

“…For example, this parameter is studied in Cayley digraphs [15], wheels [16], unicyclic graphs [17], Cartesian products [18] and trees [2,3]. Moreover, Imran et al studied the metric dimension of gear graphs [19] and symmetric graphs obtained by rooted product [20], Hussain et al [21] studied the metric dimension of 2D lattice of alpha-boron nanotubes, Bailey et al [11] studied the metric dimension of Johnson and Kneser graphs, Ahmad et al [22] studied the metric dimension of generalized Petersen graphs, Min Feng et al [23] studied the metric dimension of the power graph of a finite group. Recently, Gerold and Frank [24] studied the metric dimension of Z n × Z n × Z n and proved that dim(Z n × Z n × Z n ) = 3n 2 where Z n is the set of modulo classes of a natural number n ≥ 2.…”

Removing Twins in Graphs to Break Symmetries

“…These sets are a generalization of resolving sets, independently introduced by Slater [5] and Harary and Melter [6], motivated by the problem of identifying the location of an intruder in a network, by means of distances. Resolving sets and some related sets were recently studied in [7][8][9][10][11][12]. Determining sets and resolving sets were jointly studied (see [13,14]).…”

On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

“…Thus, a graph theoretic interpretation of this problem is to provide representations for the vertices of a graph in such a way that distinct vertices have distinct representations. This is the subject of the papers [1,6,[14][15][16][17].…”