Bi-Metric Dimension of Graphs

For a connected graph G, a subset S = {s1, s2,...., sk} of vertices of G and each vertex x of G we associate a pair of k-dimensional vectors (u,v), where u = (d(x, s1), d(x, s2),...., d(x, sk)) and v = (δ(x, s1), δ(x, s2),...., (x, sk)), where d(x, si) and δ(x, si) respectively denote the lengths of a shortest and longest paths between x and si. The subset S is said to bi-resolve G if no two distinct vertices receive the same pair. The minimum cardinality of a bi-resolving set is called bi-metric dimension of G. In this paper we show bi-metric dimension is lesser than or equal to the metric dimension and determine bi-metric dimensions of some standard graphs.