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.

Let Γ = Γ(V, E) be a simple connected graph, where V and E represent the set of vertices and edges respectively. The distance between a vertex z and an edge ε = pq, denoted by d(z, ε), is defined as d(z, ε) = min{d(p, z), d(q, z)}, where d(p, z) represents the length of a shortest p−z path in Γ. A subset Y e ⊆ V of ordered distinct vertices is refer to be an edge resolving set (ERS) for Γ, if for any pair of different edges ε 1 and ε 2 in E, we have d(z, ε 2 ) = d(z, ε 1 ) for at least one vertex z in Y e . An edge metric basis for Γ is the ERS with the smallest cardinality and this smallest cardinality is called the edge metric dimension (EMD) of Γ, represented by dim e (Γ). A molecular (chemical) graph is a simple connected graph, where the vertices represent the compound's atoms and the edges represent bonds between the atoms. In this paper, we determine the edge metric basis and EMD of the complex molecular graph of a one-heptagonal carbon nanocone (HCN 7 (q)). We prove that only three non-adjacent vertices are the minimum requirement for the identification of all the edges in HCN 7 (q), uniquely.INDEX TERMS Connected graph, edge metric basis, edge metric dimension, independent set, oneheptagonal carbon nanocone, resolving set.

The problem of characterizing the classes of plane graphs with the bounded metric dimension, edge metric dimension, and fault-tolerant metric dimension is of great interest nowadays. In this paper, we study the metric dimension, the fault-tolerant metric dimension, and the edge metric dimension of a two-fold heptagonal-nonagonal circular ladder (denoted by [Formula: see text]). We show that the metric dimension and the edge metric dimension of [Formula: see text] are the same. We also study its fault-tolerant metric dimension and prove that the metric basis and the edge metric basis sets are independent.

A vertex w ∈ V H distinguishes (or resolves) two elements (edges or vertices) a , z ∈ V H ∪ E H if d w , a ≠ d w , z . A set W m of vertices in a nontrivial connected graph H is said to be a mixed resolving set for H if every two different elements (edges and vertices) of H are distinguished by at least one vertex of W m . The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by m  dim H . The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We specifically analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.

Minimum resolving sets (edge or vertex) have become an integral part of molecular topology and combinatorial chemistry. Resolving sets for a specific network provide crucial information required for the identification of each item contained in the network, uniquely. The distance between an edge e = cz and a vertex u is defined by d(e, u) = min{d(c, u), d(z, u)}. If d(e1, u) ≠ d(e2, u), then we say that the vertex u resolves (distinguishes) two edges e1 and e2 in a connected graph G. A subset of vertices RE in G is said to be an edge resolving set for G, if for every two distinct edges e1 and e2 in G we have d(e1, u) ≠ d(e2, u) for at least one vertex u ∈ RE. An edge metric basis for G is an edge resolving set with minimum cardinality and this cardinality is called the edge metric dimension edim(G) of G. In this article, we determine the edge metric dimension of one-pentagonal carbon nanocone (1-PCNC). We also show that the edge resolving set for 1-PCNC is independent.