On Minimum Metric Dimension of Circulant Networks

Let M = } ,..., , { 2 1 n v v v be an ordered set of vertices in a graph G. Then )) , ( ),..., , ( ), , ( ( 2 1 n v u d v u d v u d is called the M-coordinates of a vertex u of G. The set M is called a metric basis if the vertices of G have distinct M-coordinates. A minimum metric basis is a set M with minimum cardinality. The cardinality of a minimum metric basis of G is called minimum metric dimension and is denoted by β (G). This concept has wide applications in motion planning and in the field of robotics. In this paper we determine the minimum metric dimension of certain classes of circulant networks. We prove that for circulant graphs G(n; ± {1, 2}), β (G (n; ± {1, 2}) = 3, when n = 4l, 4l + 2, 4l + 3, 1 ≥ l and 2 < β (G (n; ± {1, 2}) ≤ 4, for 4l + 1, 1 ≥ l . We have similar results for circulant digraphs G(n; ± {1, 2, 3}) and certain subclasses of circulant graphs.