The partition dimension of Cayley digraphs

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Let G be a (di)graph and S a set of vertices of G. We say S resolves two vertices u and v of G if d(u, S) ≠ d(v, S). A partition $$ \prod $$ = {P1, P2,..., P k } of V (G) is a resolving partition of G if every two vertices of G are resolved by Pi for some i (1≤ i ≤ k). The smallest cardinality of a resolving partition of G, denoted by pd(G), is called the partition dimension of G. A vertex r of G resolves a pair u, v of vertices of G if d(u, r) ≠ d(v, r). A set R of vertices of G is a resolving set for G if every two vertices of G are resolved by some vertex of R. The smallest cardinality of a resolving set of vertices, denoted by dim(G), is called the metric dimension of G. We begin by disproving a conjecture made by Chartrand, Salehi and Zhang regarding the partition dimension of products of graphs. The partition dimension of Cayley digraphs of abelian groups with a specific minimal set of generators is shown to be at most one more than the number of generators with equality for one or two generators. It is known that pd(G) ≤ dim(G)+1. It is pointed out that for every positive integer M there are Cayley digraphs D for which dim(D) . pd(D) ≥ M, and that there are classes of Cayley digraphs D such that $$ \frac{{{\text{pd}}(D)}} {{{\text{dim}}(D)}} \to 0\,{\text{as}}\,{\text{ | }}V(D){\text{ | }} \to \infty $$ Moreover, it is shown that the partition dimension of the Cayley digraph for the dihedral group of order 2n (n ≤ 3) with a minimum set of generators is 3. We conclude by introducing a more general class of problems for which the problems of finding the metric dimension and partition dimension of a (di)graph are the two extremes and provide an interpretation of the transition between these two invariants.