On the k-Metric Dimension of a Barbell Graph and a t-fold Wheel Graph

Abstract

Let G be a connected and simple graph with the vertex set V(G) and the edge set E(G). The set S ⊆ V (G) is called a k-metric generator for G if and only if for every two pairs different vertices u,v ∈ V(G), there are at least k vertices w1,w2, . . .,wk ∈ S such that d(u,wi) ≠ d(v,wi) for every i ∈ {1, 2, …, k}, with d(u,v) is the length of shortest uv path. A minimum k-metric generator is called a k-metric basis and its cardinality is called the k-metric dimension of G, denoted by dimk(G). A barbell graph Bn,n for n ≥ 3 is the simple graph obtained from two complete graph Kn connected by a bridge. A t-fold wheel graph Wnt for t ≥ 2 and n ≥ 3 is the simple graph that contain the central t vertex which are adjacent to each vertex in a cycle, but not adjacent to each other. In this paper, we determine the kmetric dimension of a barbell graph and a t-fold wheel graph.