A vertex x in a digraph D is said to resolve a pair u, v of vertices of D if the distance from u to x does not equal the distance from v to x. A set S of vertices of D is a resolving set for D if every pair of vertices of D is resolved by some vertex of S. The smallest cardinality of a resolving set for D, denoted by dim(D), is called the metric dimension for D. Sharp upper and lower bounds for the metric dimension of the Cayley digraphs Cay( : ), where is the group Z n 1 ⊕ Z n 2 ⊕ · · · ⊕ Z n m and is the canonical set of generators, are established. The exact value for the metric dimension of Cay({(0, 1), (1, 0)} : Z n ⊕ Z m ) is found. Moreover, the metric dimension of the Cayley digraph of the dihedral group D n of order 2n with a minimum set of generators is established. The metric dimension of a (di)graph is formulated as an integer programme. The corresponding linear programming formulation naturally gives rise to a fractional version of the metric dimension of a (di)graph. The fractional dual implies an integer dual for the metric dimension of a (di)graph which is referred to as the metric independence of the (di)graph. The metric independence of a (di)graph is the maximum number of pairs of vertices such that no two pairs are resolved by the same vertex. The metric independence of the n-cube and the Cayley digraph Cay( : D n ), where is a minimum set of generators for D n , are established.
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
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