Metric dimension of heptagonal circular ladder

• Published in: Discrete mathematics, algorithms, and applications Vol. 13; no. 1; p. 2050095
• Main Authors: Sharma, Sunny Kumar, Bhat, Vijay Kumar
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Company 01.02.2021; World Scientific Publishing Co. Pte., Ltd
• Subjects: Apexes; Graph theory; Graphs; Nodes; Shortest-path problems; Symmetry; metric dimension; Locating set; nodes; convex polytopes; metric basis
• ISSN: 1793-8309; 1793-8317
• DOI: 10.1142/S1793830920500950

Let Ψ = Ψ ( , ) 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 d Ψ ( ϱ i , ϱ j ) denotes the number of edges in the shortest path or geodesic distance between two vertices ϱ i , ϱ j ∈ . 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 Φ which are obtained from the graph Ψ by adding new edges in Ψ such that β ( Ψ ) = β ( Φ ) and ( Φ ) = ( Ψ ) . In this paper, by answering this problem, we characterize some families of plane graphs Γ n , 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, Γ n by the addition of new edges in Γ n have the same metric dimension and vertices set as Γ n , and only 3 nodes appropriately selected are sufficient to resolve all the nodes of these families of plane graphs.