On the Determining Number and the Metric Dimension of Graphs

Abstract

This paper initiates a study on the problem of computing the difference between the metric dimension and the determining number of graphs. We provide new proofs and results on the determining number of trees and Cartesian products of graphs, and establish some lower bounds on the difference between the two parameters.

Base size, metric dimension and other invariants of groups and graphs

“…We conclude this subsection by showing how these generalize results in the graph theory literature, which are expressed in terms of Cartesian products of graphs. For instance, [20, Theorem 1] and [28,Theorem 4] both give restatements of Proposition 2.11 in the case of Cartesian products.…”

“…More specifically, can the gap between the two parameters be made arbitrarily large? This question is asked by Boutin [19] and (implicitly) by Vince [93], while the paper by Cáceres et al [28] is devoted to investigating it. In the same vein, we can ask: for which graphs are the two parameters equal?…”

“…Cáceres et al [28] considered both the metric dimension of the square lattice graphs H(2, n) and the base size of its automorphism group S n S 2 (see Subsections 2.4 and 3.6), obtaining exact values in each case. This gives the following.…”

“…Example 4.6. Cáceres et al [28] construct an infinite family of trees T n by attaching n paths, of lengths 1, 2, . .…”

Metric Dimension for Random Graphs

“…The metric dimension of the cartesian product of graphs was investigated by Cáceres, Hernando et al [7], and the relationship between β(G) and the determination number of G (the smallest size of a set S such that every automorphism of G is uniquely determined by its action on S) was studied by Cáceres, Garijo et al [6]. Also, Bailey and Cameron [2] studied the metric dimension of groups, and the relationship of the problem of determining β(G) to the graph isomorphism problem.…”