On the Edge Resolvability of Double Generalized Petersen Graphs

Abstract

For a connected graph G=VG,EG, let v∈VG be a vertex and e=uw ∈EG be an edge. The distance between the vertex v and the edge e is given by dGe,v=mindGu,v,dGw,v. A vertex w∈VG distinguishes two edges e1,e2∈EG if dGw,e1≠dGw,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 βeG. As a main result, we investigate the minimum number of vertices which works as the edge metric generator of double generalized Petersen graphs, DGPn,1. We have proved that βeDGPn,1=4 when n≡0,1,3mod4 and βeDGPn,1=3 when n≡2mod4.