Computing the Metric Dimension of Gear Graphs

Muhammad Siddiqui,Imran Cheema,Zeeshan Saleem,Muhammad Imran,Muhammad Hussain,Ali Tabraiz,Hafiz Bilal,Shahid Imran

doi:10.3390/sym10060209

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 J2n,m be a m-level gear graph obtained by m-level wheel graph W2n,m &cong; mC2n + k1 by alternatively deleting n spokes of each copy of C2n and J3n be a generalized gear graph obtained by alternately deleting 2n spokes of the wheel graph W3n. In this paper, the metric dimension of certain gear graphs J2n,m and J3n generated by wheel has been computed. Also this study extends the previous result given by Tomescu et al. in 2007.

Highlights

Introduction and Preliminary ResultsIn a connected graph G (V, E), where V is the set of vertices and E is the set of edges
A resolving set of minimum cardinality is called a metric basis for G and the cardinality of a metric basis is said the metric dimension of G, denoted by dim( G ), see [3]
Let B be a basis of J2n,2, n ≥ 6 it contains at most one major gap induced by the vertices of cycles C2n,1 and C2n,2

Summary

Introduction and Preliminary Results

In a connected graph G (V, E), where V is the set of vertices and E is the set of edges. Let B be a basis of J2n,2 , n ≥ 6 it contains at most one major gap induced by the vertices of cycles C2n,1 and C2n,2. Let B be a basis of J2n,2 , n ≥ 6, any two neighboring gaps, one of which being a major gap induced by exactly one of two cycles C2n,1 or C2n,2 contain together at most six vertices. The existence of 2 − 3 major gap having four vertices is not possible if its neighboring minor gap is a 2 − 2 gap with three vertices If this case holds we consider the following path: u1∗ , u2 , u3∗ , u4 , u5∗ , u6 , u7∗ , u8 , where u4 ∈ B r u4∗ B = r (u5∗ B); a contradiction. The result is true for all positive integers m ≥ 3

The Metric Dimension of Generalized Gear Graph J3n

Conclusions