On metric dimension of plane graphs with $$\frac{m}{2}$$ m 2 number of 10 sided faces

Sunny Kumar Sharma () and Vijay Kumar Bhat ()
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Sunny Kumar Sharma: Shri Mata Vaishno Devi University
Vijay Kumar Bhat: Shri Mata Vaishno Devi University

Journal of Combinatorial Optimization, 2022, vol. 44, issue 3, No 3, 1433-1458

Abstract: Abstract Let $$\Gamma =\Gamma (V, E)$$ Γ = Γ ( V , E ) be a simple (multiple edges and loops are not considered), connected (every pair of distinct vertices are joined by a path), and an undirected (all edges are bidirectional) graph, with the vertex set V and the edge set E. The length of the shortest path (geodesic distance) between two vertices p and q, denoted by d(p, q), is the minimum number of edges lying between the vertices p and q. The resolvability parameters for graph $$\Gamma $$ Γ are a relatively new advanced area in which the complete network is built so that each vertex or/and edge signifies a unique position. The challenge of characterizing families of planar graphs with constant and bounded metric dimensions is a widely studied topic. In this paper, we consider three new families of planar graphs viz., $$A_m$$ A m , $$B_m$$ B m , and $$C_m$$ C m (where $$m\ge 6$$ m ≥ 6 is always even natural), and study their metric dimensions. We prove that only 3 non-adjacent vertices are sufficient to resolve every pair of distinct vertices of $$A_m$$ A m , $$B_m$$ B m , and $$C_m$$ C m .

Keywords: Resolving set; Planar graphs; Metric basis; Metric dimension; Independent set; 05C12; 05C76; 05C90. (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10878-022-00899-2

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