The Metric Dimension of Cayley Digraphs of Abelian Groups

Abstract

A vertex \(w\) in a di(graph) \(G\) is said to resolve a pair \(u, v\) of vertices of \(G\) if the distance from \(u\) to \(w\) does not equal the distance from \(v\) to \(w\). A set \(S\) of vertices of \(G\) is a resolving set for \(G\) if every pair of vertices of \(G\) is resolved by some vertex of \(S\). The smallest cardinality of a resolving set for \(G\), denoted by \(dim(G)\), is called the metric dimension for \(G\).

We show that if \(G\) is the Cayley digraph \(Cay(\Delta : \Gamma)\) where \(\Delta = \{ (1, 0, 0), (0, 1, 0), (0, 0, 1) \}\) and \(\Gamma =\mathbb{Z}_m \oplus \mathbb{Z}_n \oplus \mathbb{Z}_k\) with \(m \leq n \leq k\), then \(dim(G) = n\) if \(m < n\) and improve known upper bounds if \(m = n\). We use these results to establish improved upper bounds for the metric dimension of Cayley digraphs of abelian groups that are expressed as a direct product of four or more cyclic groups. Lower bounds for Cayley digraphs of groups that are multiple copies of \(\mathbb{Z}_n\) are established.