Minimal Doubly Resolving Sets and the Strong Metric Dimension of Hamming Graphs

The metric dimension problem, introduced independently by Slater [22] and Harary [8], has been widely investigated [1,3,5-7,9-13]. It arises in many diverse areas including network discovery and verification [2], geographical routing protocols [18], the robot navigation, connected joints in graphs, chemistry, etc. Given a simple connected undirected graph G = (VG,EG), where VG = {1, 2, ..., n}, |EG| = m, d(u, v) denotes the distance between vertices u and v, i.e. the length of a shortest u − v path. A vertex x of graph G is said to resolve two vertices u and v of G if d(u, x) 6= d(v, x). A subset B = {x1, x2, ..., xk} of VG is a resolving set of G if every two distinct vertices of G are resolved by some vertex of B. Given a vertex t, the k-tuple r(t, B) = (d(t, x1), d(t, x2), ..., d(t, xk)) is called the vector of metric coordinates of t with respect to B. A metric basis of G is a resolving set of the minimum cardinality. The metric dimension of G, denoted by β(G), is the cardinality of its metric basis.