Metric basis and metric dimension of some infinite planar graphs

Let G = (V,E) be a nontrivial connected simple graph and d (a,b) be the distance between the vertices a and b in G. The metric dimension of a graph G, denoted by dim (G), refers to the smallest set of vertices required to uniquely identify every vertex in the graph G. A family of simple connected graphs say F _s, where s ∈ ℕ has a constant metric dimension, if dim (F _s) is finite and does not depend on the choice of s in F _s. In this paper, we consider two infinite families of planar graphs, say Q _s, where s ≥ 8 and Z _s, where s ≥ 6, and investigate their metric basis as well as the metric dimension. Additionally, we prove that the metric basis for these two graphs are independent.