[Paper Review] Resolving sets for breaking symmetries of graphs

Delia Garijo, Antonio González+1 more|arXiv (Cornell University)|Jan 15, 2014

Graph Labeling and Dimension Problems31 references3 citations

TL;DR

This paper resolves the long-standing question of whether the difference between metric dimension and determining number in graphs can be arbitrarily large, proving that for trees, this difference reaches exactly ⌊2n/7⌋, and for certain graph families, it can reach up to ⌊n/2⌋. The authors use locating-dominating sets and a novel polynomial-time algorithm to bound these differences, significantly improving prior bounds and settling the problem for trees.

ABSTRACT

This paper deals with the maximum value of the difference between the determining number and the metric dimension of a graph as a function of its order. Our technique requires to use locating-dominating sets, and perform an independent study on other functions related to these sets. Thus, we obtain lower and upper bounds on all these functions by means of very diverse tools. Among them are some adequate constructions of graphs, a variant of a classical result in graph domination and a polynomial time algorithm that produces both distinguishing sets and determining sets. Further, we consider specific families of graphs where the restrictions of these functions can be computed. To this end, we utilize two well-known objects in graph theory: $k$-dominating sets and matchings.

Motivation & Objective

• To determine whether the difference between metric dimension and determining number in graphs can be arbitrarily large, as posed by Boutin.
• To improve the upper and lower bounds on the function (dim - Det)(n), the maximum difference between metric dimension and determining number over all graphs of order n.
• To study the related function (λ - Det)(n), where λ is the locating-domination number, and establish bounds via graph constructions and algorithmic methods.
• To compute exact values of (dim - Det)(n) and (λ - Det)(n) for specific graph families, particularly trees and C4-free graphs.
• To resolve the conjecture that the difference is maximized in trees and to explore the role of k-dominating sets and matchings in bounding these parameters.

Proposed method

• The authors introduce and analyze locating-dominating sets as a key tool, linking them to both metric dimension and determining number via the inequality Det(G) ≤ dim(G) ≤ λ(G).
• They develop a polynomial-time algorithm that constructs both distinguishing sets and determining sets for twin-free graphs, enabling bounds on Det(G).
• A variant of Ore's classical domination theorem is used to relate the locating-domination number λ(G) to other graph parameters, including k-domination and matching numbers.
• The paper constructs specific families of graphs, such as Tq,s trees, to achieve tight lower bounds on (dim - Det)(n) and (λ - Det)(n).
• For trees, the authors use the structure of end-vertices and their terminal degrees to derive exact expressions for dim(T) and Det(T), leading to the main result.
• They apply the Erdős–Szekeres theorem to bound λ(G) in certain graph classes, contributing to upper bounds on (λ - Det)(n).

Experimental results

Research questions

• RQ1 Can the difference between the metric dimension and the determining number of a graph be arbitrarily large?
• RQ2 What is the exact maximum value of dim(G) − Det(G) for trees of order n?
• RQ3 What are the tight upper and lower bounds on (λ - Det)(n), where λ(G) is the locating-domination number?
• RQ4 How do k-dominating sets and matchings relate to the locating-domination number and the difference (λ - Det)(n)?
• RQ5 What is the maximum value of (dim - Det)(n) for C4-free graphs, and can it be computed exactly for other graph families?

Key findings

• For all n ≥ 49, the maximum difference between metric dimension and determining number in trees is exactly ⌊2n/7⌋, resolving a conjecture for this class.
• The function (λ - Det)(n) reaches ⌊n/2⌋ for trees, indicating that locating-dominating sets can be significantly larger than determining sets in this family.
• The paper improves upon prior bounds by Cáceres et al., showing that (dim - Det)(n) is bounded above by λ(G), and providing tighter asymptotic bounds.
• A polynomial-time algorithm is constructed that produces both distinguishing sets and determining sets for twin-free graphs, yielding new bounds on Det(G).
• The authors establish that (λ - Det)(n) ≤ λ_{|C*}(n), and derive upper bounds on λ_{|C*}(n) using extremal graph theory and domination results.
• For C4-free graphs, the bound (λ - Det)(n) ≤ ⌊n/2⌋ is proven, and the exact value is achieved in specific constructions.