On The (k,t)-Metric Dimension Of Graphs

Identifier:  imarina:6345142

Handle: https://hdl.handle.net/20.500.11797/imarina6345142

Authors:  Estrada-Moren, A; Yero, IG; Rodríguez-Velázquez, JA

Abstract:
Let (X; d) be a metric space. A set S X is said to be a k-metric generator for X if and only if for any pair of dierent points u; v 2 X, there exist at least k points w1;w2; : : :wk 2 S such that d(u;wi) 6= d(v;wi); for all i 2 f1; : : : kg: Let Rk(X) be the set of metric generators for X. The k-metric dimension dimk(X) of (X; d) is dened as dimk(X) = inffjSj : S 2 Rk(X)g: Here, we discuss the k-metric dimension of (V; dt), where V is the set of vertices of a simple graph G and the metric dt : V V ! N [ f0g is dened by dt(x; y) = minfd(x; y); tg from the geodesic distance d in G and a positive integer t. The case t D(G), where D(G) denotes the diameter of G, corresponds to the original theory of k-metric dimension and the case t = 2 corresponds to the theory of k- adjacency dimension. Furthermore, this approach allows us to extend the theory of k-metric dimension to the general case of non-necessarily connected graphs. Finally, we analyse the computational complexity of determining the k-metric dimension of (V; dt) for the metric dt.