Classes of Planar Graphs with Constant Edge Metric Dimension. (5th April 2021) - Wei, Changcheng; Salman, Muhammad; Shahzaib, Syed; Rehman, Masood Ur; Fang, Juanyan
Abstract : The number of edges in a shortest walk (without repetition of vertices) from one vertex to another vertex of a connected graph G is known as the distance between them. For a vertex x and an edge e = a b in G, the minimum number from distances of x with a and b is said to be the distance between x and e . A vertex x is said to distinguish (resolves) two distinct edges e 1 and e 2 if the distance between x and e 1 is different from the distance between x and e 2 . A set X 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 in X . The number of vertices in such a smallest set X is known as the edge metric dimension of G . In this article, we solve the edge metric dimension problem for certain classes of planar graphs.