On the Characterization of a Minimal Resolving Set for Power of Paths

Abstract

For a simple connected graph G=(V,E), an ordered set W⊆V, is called a resolving set of G if for every pair of two distinct vertices u and v, there is an element w in W such that d(u,w)≠d(v,w). A metric basis of G is a resolving set of G with minimum cardinality. The metric dimension of G is the cardinality of a metric basis and it is denoted by β(G). In this article, we determine the metric dimension of power of finite paths and characterize all metric bases for the same.

Optimal Multi-Level Fault-Tolerant Resolving Sets of Circulant Graph C(n : 1, 2)

“…The exact value of β 0 (C(n : 1, 2)) (i.e., when L = {1, 2}) was determined by Javaid et al [10]. In [13], the authors determined the metric dimension of the power of finite paths. The increased connectivity of circulant networks makes them ideal for parallel computing and signal processing [14].…”

Metric Dimensions of Bicyclic Graphs

“…Applications of metric dimensions to various fields, e.g., navigation of robots [7], chemistry [8,9], coin-weighing, and mastermind game [10] have been presented in the literature. Further studies on metric dimension and metric basis were conducted in [11][12][13][14][15][16][17][18][19][20].…”

Optimal Fault-Tolerant Resolving Set of Power Paths

“…Basak et al [7] studied and envisioned the fault-tolerant metric dimension for the circulant graphs C n (1,2,3), which are basically the 3rd power of cycles. Saha et al in [9] calculated the metric dimension of the power of finite paths and also characterized all metric bases for the same. There are several interesting recent results on this topic; see [10][11][12].…”