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.

Computing Edge Metric Dimension of One-Pentagonal Carbon Nanocone

“…To show this, we have to prove that there exists no edge resolving set R E for CNC 5 [m] such that |R E | ≤ 2. Since 1-PCNC is not a path graph, so the possibility of a singleton edge resolving set for CNC 5 [m] is ruled out [32]. Next, suppose on the contrary that |R E | 2, such that R E {u l,1 , u z,j }.…”

“…Many authors have introduced and analyzed certain variations of resolving sets, such as local resolving set, partition resolving set, fault-tolerant resolving set, resolving dominating set, strong resolving set, independent resolving set, and so on. For further details the reader is referred to [1,6,10,21,22,32,33]. In addition to defining other variants of resolving sets in graphs, Kelenc et al [22], introduced a parameter used to uniquely distinguish graph edges and called it the edge metric dimension.…”

Edge Resolvability for Circular Ladder of Heptagons

“…Other Fields: Other important applications of metric dimension and of resolving sets can be pursued in computer science, strategies in mastermind games, mathematical disciplines, pharmacy, game theory, and signal processing where generally a moving spectator (or observer) in a network framework may be located by calculating the distance from the point to the collection of sonar stations, which have been appropriately situated in the network, for some of these aforementioned applications of metric dimension, see references in [8,10,11,12,13].…”

“…Convex polytopes assume a significant job both in different branches of arithmetic and in applied zones, most quite in linear programming. The metric dimension and the edge metric dimension of a few classes of convex polytopes have been considered in [13,14,15,16]. Next, we give some properties regarding the heptagonal circular ladder.…”

“…Heptagonal Circular Ladder: The Heptagonal circular ladder [13], denoted by Γ n , is the graph of a convex polytope with 4n vertices and 5n edges. It can be obtained from the P entagonal Circular Ladder P n [15] by introducing new vertices between the vertices of degree three in P n .…”