The metric dimension β(G) or dim(G) (Tomescu and Javid 2007, Ali et al. 2016) of a graph G is the smallest number of nodes required to identify all other nodes based on shortest path distances uniquely. More explicitly, following Foster-Greenwood and Uhl, let G be a finite connected graph with vertex set V. For vertices x, y element V, the graph distance d(x, y) is the length of the shortest path between x and y in G. Consider a subset of vertices W⊆V and refer to the vertices in W as "landmarks." Then W is called a resolving set if, for every pair of distinct vertices x, y element V - W, there exists a landmark w element W such that d(x, w)!=d(y, w). A resolving set of smallest possible size is called a metric basis for G, and the metric dimension of G is the size of a metric basis.
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