Minimum Metric Dimension of Iliac Networks

Abstract

Let \(M = \{v_1, v_2, \ldots, v_n\}\) be an ordered set of vertices in a graph \(G\). Then, \((d(u, v_1), d(u, v_2), \ldots, d(u, v_n))\) 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 the minimum metric dimension. This concept has wide applications in motion planning and robotics. In this paper, we solve the minimum metric dimension problem for Illiac networks.