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On the Edge Resolvability of Double Generalized Petersen Graphs

Author

Listed:
  • Tanveer Iqbal
  • Muhammad Rafiq
  • Muhammad Naeem Azhar
  • Muhammad Salman
  • Imran Khalid

Abstract

For a connected graph G = (V(G), E(G)), let v ∈ V(G) be a vertex and e = uw ∈ E(G) be an edge. The distance between the vertex v and the edge e is given by dG(e, v) = min{dG(u, v), dG(w, v)}. A vertex w ∈ V(G) distinguishes two edges e1, e2 ∈ E(G) if dG(w, e1) ≠ dG(w, e2). A well‐known graph invariant related to resolvability of graph edges, namely, the edge resolving set, is studied for a family of 3‐regular graphs. A set S of vertices in a connected graph G is an edge metric generator for G if every two edges of G are distinguished by some vertex of S. The smallest cardinality of an edge metric generator for G is called the edge metric dimension and is denoted by βe(G). As a main result, we investigate the minimum number of vertices which works as the edge metric generator of double generalized Petersen graphs, DGP(n, 1). We have proved that βe(DGP(n,1)) = 4 when n ≡ 0,1,3(mod4) and βe(DGP(n, 1)) = 3 when n ≡ 2(mod4).

Suggested Citation

  • Tanveer Iqbal & Muhammad Rafiq & Muhammad Naeem Azhar & Muhammad Salman & Imran Khalid, 2022. "On the Edge Resolvability of Double Generalized Petersen Graphs," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).
    • Handle: RePEc:wly:jjmath:v:2022:y:2022:i:1:n:6490698
    DOI: 10.1155/2022/6490698
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    References listed on IDEAS

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    1. Knor, Martin & Majstorović, Snježana & Masa Toshi, Aoden Teo & Škrekovski, Riste & Yero, Ismael G., 2021. "Graphs with the edge metric dimension smaller than the metric dimension," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    2. Ali N. A. Koam & Ali Ahmad & Muhammad Ibrahim & Muhammad Azeem, 2021. "Edge Metric and Fault-Tolerant Edge Metric Dimension of Hollow Coronoid," Mathematics, MDPI, vol. 9(12), pages 1-14, June.
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