On the Edge Resolvability of Double Generalized Petersen Graphs

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 mod 4 and β e DGP n , 1 = 3 when n ≡ 2 mod 4 .