On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan

Abstract

Let G be a connected graph and let u, v ∈ V(G). For an ordered set W = {w₁, w₂, ..., w_n} of n distinct vertices in G, the representation of a vertex v of G with respect to W is the n-vector r(v∣W) = (d(v, w₁), d(v, w₂), ..., d(v, w_n)), where d(v, w_i) is the distance between v and w_i for 1 ≤ i ≤ n. The set W is a local metric set of G if r(u ∣ W) ≠ r(v ∣ W) for every pair u, v of adjacent vertices of G. The local metric set of G with minimum cardinality is called a local metric basis for G and its cardinality is called a local metric dimension, denoted by lmd(G). In this paper we determine the local metric dimension of a t-fold wheel graph, P_n ⊙ K_m graph, and generalized fan graph.