The difference between the metric dimension and the determining number of a graph

Abstract

a b s t r a c tWe study the maximum value of the difference between the metric dimension and the determining number of a graph as a function of its order. We develop a technique that uses functions related to locating-dominating sets to obtain lower and upper bounds on that maximum, and exact computations when restricting to some specific families of graphs. Our approach requires very diverse tools and connections with well-known objects in graph theory; among them: a classical result in graph domination by Ore, a Ramsey-type result by Erd} os and Szekeres, a polynomial time algorithm to compute distinguishing sets and determining sets of twin-free graphs, k-dominating sets, and matchings.

Domination and location in twin-free digraphs

“…In this section, we propose a general method to obtain locating-dominating sets of twin-free digraphs, based on special dominating sets. This method was used in [6] for the case of undirected graphs (a similar argument was also used in [9]). It was adapted to digraphs in [12] for quasi-twin-free digraphs, and here we extend it to all twin-free digraphs.…”

“…Such a theorem does not hold for the location-domination number of undirected graphs, for example complete graphs and stars of order n have location-domination number n − 1, see [19]. Nevertheless, Garijo, González and Márquez have conjectured in [9] that in the absence of twins, the upper bound of Ore's theorem also holds for the location-domination number of undirected graphs; they also proved that an upper bound of roughly two thirds the order holds in this context (see [4,5,6] for further developments on this matter).…”

“…In Section 4, we devise a general technique to obtain small locating-dominating sets in twin-free digraphs. This technique is a refinement of those used in [6,9,12]. We use this technique in Section 5 to show that every source-free and twin-free digraph of order n has a locating-dominating set of size at most 4n 5 .…”

Location-domination in line graphs

“…Garijo et al [15] proved that for any n ≥ 14, the maximum value of the location-domination number of a connected twin-free graph is at least ⌊ n 2 ⌋. Thus, together with this fact, the statement of Conjecture 2 implies the statement of Conjecture 1.…”

“…(b) ( [15]) G has independence number at least n 2 . (c) ( [15]) G has clique number at least ⌈ n 2 ⌉ + 1. (d) ( [13]) G is a split graph.…”