On the Edge Resolvability of Double Generalized Petersen Graphs
Tanveer Iqbal, Muhammad Rafiq, Muhammad Naeem Azhar, Muhammad Salman, Imran Khalid
Source 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 d G ( e , v ) = min{ d G ( u , v ), d G ( w , v )}. A vertex w ∈ V ( G ) distinguishes two edges e 1 , e 2 ∈ E ( G ) if d G ( w , e 1 ) ≠ d G ( w , e 2 ). 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 β e ( 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 β e (DGP( n ,1)) = 4 when n ≡ 0,1,3(mod4) and β e (DGP( n , 1)) = 3 when n ≡ 2(mod4).
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