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The metric dimension of Cayley digraphs of abelian groups. (English) Zbl 1189.05055

A vertex \(w\) of a (di)graph \(G\) resolves vertices \(u\) and \(v\) if the distance of \(w\) and \(u\) is different from the distance of \(w\) and \(v\). A set \(S\) of vertices of \(G\) is a resolving set for \(G\) if any vertex of \(G\) is resolved by some vertex in \(S\). The smallest cardinality of a resolving set is called metric dimension of \(G\), denoted by \(\dim(G)\). The authors showed that for Cayley digraph \(\text{Cay}(\Delta ,\Gamma )\), where \(\Gamma =Z_m\oplus Z_n\oplus Z_k\) with \(m\leq n\leq k\) and \(\Delta =\{(1,0,0),(0,1,0),(0,0,1)\}\), the dimension \(\dim(G)=n\) if \(m<n\) and improved known upper bounds if \(m=n\). Then they used these results to improve upper bounds on \(\dim(G)\) for groups which are direct product of four or more cyclic groups. Moreover, lower bounds for these groups are given.
Reviewer: Jan Hora (Praha)

MSC:

05C12 Distance in graphs
05C20 Directed graphs (digraphs), tournaments
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