The metric dimension of Cayley digraphs

Abstract

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.

The partition dimension of Cayley digraphs

“…Figure 2 shows a resolving 3-partition Π = {P 1 , P 2 , P 3 } of V (D) (the vertices of D are labelled with their representations with respect to this 3-partition); so pd(D) ≤ 3. In [9] it was shown that dim(D) = 3, and hence, by Proposition 2.1, pd(D) ≥ 3.…”

“…The following result was established in [9]. is adjacent to a different vertex of P 2 in the underlying simple graph.…”

“…a symmetric pair of arcs) for 1 ≤ i ≤ n. Note that C 1 and C 2 are oriented in opposite directions. In [9] it was shown that dim(G = Cay({R 360 n , A} : D n )) = n, for n ≥ 3. For 3 ≤ n ≤ 11, it can be shown that pd(G) = 3.…”

“…In [9] we presented sharp bounds on the metric dimension of certain types of Cayley digraphs. In this paper, we reexamine these graphs with respect to the partition dimension.…”

The Metric Dimension of a Graph Composition Products With Star

“…Some classes of graph whose metric dimension have been known are cycles [6], trees [5,8,10], stars [5,8,10], wheels [2,3,15], complete multipartite [5,14], unicylic graphs [12], Cayley graphs [7], and regular graphs [1].…”