Metric dimension of heptagonal circular ladder

Abstract

Let [Formula: see text] be an undirected (i.e., all the edges are bidirectional), simple (i.e., no loops and multiple edges are allowed), and connected (i.e., between every pair of nodes, there exists a path) graph. Let [Formula: see text] denotes the number of edges in the shortest path or geodesic distance between two vertices [Formula: see text]. The metric dimension (or the location number) of some families of plane graphs have been obtained in [M. Imran, S. A. Bokhary and A. Q. Baig, Families of rotationally-symmetric plane graphs with constant metric dimension, Southeast Asian Bull. Math. 36 (2012) 663–675] and an open problem regarding these graphs was raised that: Characterize those families of plane graphs [Formula: see text] which are obtained from the graph [Formula: see text] by adding new edges in [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, by answering this problem, we characterize some families of plane graphs [Formula: see text], which possesses the radial symmetry and has a constant metric dimension. We also prove that some families of plane graphs which are obtained from the plane graphs, [Formula: see text] by the addition of new edges in [Formula: see text] have the same metric dimension and vertices set as [Formula: see text], and only 3 nodes appropriately selected are sufficient to resolve all the nodes of these families of plane graphs.