Metric Dimension of Enhanced Hypercube Networks

Abstract

Let $M=\{v_1,v_2,\dots ,v_n\}$ be an ordered set of vertices in a graph $G$. Then $(d(u,v_1),d(u,v_2),\dots ,d(u,v_n))$ 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 minimum metric dimension. This concept has wide applications in motion planning and in the field of robotics. In this paper we provide bounds for minimum metric dimension of certain classes of enhanced hypercube networks.