The partition dimension of Cayley digraphs

Abstract

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 Π = {P 1 , P 2 , . . . , P k } of V (G) is a resolving partition of G if every two vertices of G are resolved by P i 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 pd(D) dim(D) → 0 as |V (D)| → ∞. 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. (2000). 05C12, 05C20, 05C90. Mathematics Subject Classification

On the partition dimension of trees

“…The concepts of resolvability and location in graphs were described independently by Harary and Melter [9] and Slater [17], to define the same structure in a graph. After these papers were published several authors developed diverse theoretical works about this topic [2,3,4,5,6,7,8,9,10,14]. Slater described the usefulness of these ideas into long range aids to navigation [17].…”

Computing the local metric dimension of a graph from the local metric dimension of primary subgraphs

“…The concept of metric dimension of a connected graph was introduced independently by Harary and Metler in [14], where metric generators received the name of resolving sets. After these papers were published several authors developed diverse theoretical works about this topic, for instance, we cite [1,16,2,3,4,5,6,7,9,10,11,14,15,21,20,23,27,28,29]. Slater described the usefulness of these ideas into long range aids to navigation [30].…”

Neighbor-locating colorings in graphs

“…The partition Π is called a metric-locating partition, an ML-partition for short, if, for any pair of distinct vertices u, v ∈ V (G), r(u|Π) = r(v|Π). The partition dimension β p (G) of G is the minimum cardinality of an ML-partition of G. Metric-locating partitions were introduced in [14], and further studied in several papers: bounds [10], graph families [16,17,20,21,24,29,30,31,36,37,41,42] and graph operations [2,9,15,18,19,35,46,47].…”