New resolvability parameters of graphs
Stephen T. Hedetniemi, Renu C. Laskar, Henry Martyn Mulder · AKCE International Journal of Graphs and Combinatorics · 2020
In this paper we introduce two concepts related to resolvability and the metric dimension of graphs. The kth dimension of a graph G is the maximum cardinality of a subset of vertices of G that is resolved by a set S of order k. Some first results are obtained. A pair of vertices u, v is totally resolved by a third vertex x if d(u,x)≠d(v,x). A total resolving set in G is a set S such that each pair of vertices of G is totally resolved be a vertex in S. The total metric dimension of a graph is the minimum cardinality of a total resolving set. We determine the total metric dimension of paths, cycles, and grids, and of the 3-cube, and the Petersen graph.