Metric basis and metric dimension of some infinite planar graphs

Let [Formula: see text] be a nontrivial connected simple graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The metric dimension of a graph [Formula: see text], denoted by dim([Formula: see text]), refers to the smallest set of vertices required to uniquely identify every vertex in the graph [Formula: see text]. A family of simple connected graphs say [Formula: see text], where [Formula: see text] has a constant metric dimension, if dim[Formula: see text] is finite and does not depend on the choice of [Formula: see text] in [Formula: see text]. In this paper, we consider two infinite families of planar graphs, say [Formula: see text], where [Formula: see text] and [Formula: see text], where [Formula: see text], and investigate their metric basis as well as the metric dimension. Additionally, we prove that the metric basis for these two graphs are independent.