Algorithms for the Metric Dimension of a Simple Graph

Abstract

Let G = (V, E) be a connected, simple graph with n vertices and m edges. Let v1, v2 \(\in\) V, d(v1, v2) is the number of edges in the shortest path from v1 to v2. A vertex v is said to distinguish two vertices x and y if d(v, x) and d(v, y) are different. D(v) as the set of all vertex pairs which are distinguished by v. A subset of V, S is a metric generator of the graph G if every pair of vertices from V is distinguished by some element of S. Trivially, the whole vertex set V is a metric generator of G. A metric generator with minimum cardinality is called a metric basis of the graph G. The cardinality of metric basis is called the metric dimension of G. In this paper, we develop algorithms to find the metric dimension and a metric basis of a simple graph. These algorithms have the worst-case complexity of O(nm).