[Paper Review] Lexicographic metric spaces: basic properties and the metric dimension

Juan A. Rodríguez‐Velázquez|arXiv (Cornell University)|Jun 28, 2018

Graph Labeling and Dimension Problems10 references3 citations

TL;DR

This paper introduces lexicographic metric spaces as a generalization of lexicographic product graphs, establishing foundational properties like completeness, boundedness, and separability. It derives a precise formula for the metric dimension of any lexicographic metric space, showing it equals the sum of the metric dimensions of its component gravitational metric spaces when the base space has positive nearness.

ABSTRACT

In this article, we introduce the concept of lexicographic metric space and, after discussing some basic properties of these metric spaces, such as completeness, boundedness, compactness and separability, we obtain a formula for the metric dimension of any lexicographic metric space.

Motivation & Objective

• To formalize and study lexicographic metric spaces as a generalization of lexicographic product graphs.
• To investigate fundamental topological properties such as completeness, boundedness, compactness, and separability in these spaces.
• To determine the metric dimension of lexicographic metric spaces and relate it to the structure of their component spaces.
• To introduce and analyze gravitational metric spaces as a related class of metric spaces arising in the construction.
• To provide a unified framework for computing metric dimension in lexicographic constructions, applicable to graphs and weighted metric spaces.

Proposed method

• Define a lexicographic distance ρ on the Cartesian product X×Y of two metric spaces M=(X,d_X) and M'=(Y,d_Y), where ρ((x,y),(x',y')) = d_X(x,x') if x≠x', and min{2η(x), d_Y(y,y')} if x=x'.
• Introduce the concept of nearness η(x) = inf{d(x,y) : y∈X\{x}} and η(M) = inf{η(x) : x∈X} to characterize discrete metric spaces.
• Define gravitational metric spaces M'_x as the copies of M' associated with each x∈X, equipped with a scaled metric to ensure boundedness.
• Use the property that D(M'') < η(M) to apply a key result (Corollary 21) that simplifies the metric dimension to |X|·dim(M'').
• Construct a bounded metric d* = η(M)·d_Y / (η(M) + d_Y) on M' to preserve metric dimension while ensuring D(M'') < η(M), enabling application of the dimension formula.
• Prove that the metric dimension of M∘M' is the sum of the metric dimensions of the gravitational metric spaces M'_x when M is twin-free and η(M)>0.

Experimental results

Research questions

• RQ1 How can lexicographic metric spaces be formally defined and what are their basic topological properties?
• RQ2 Under what conditions is a lexicographic metric space complete, bounded, compact, or separable?
• RQ3 What is the metric dimension of a lexicographic metric space in terms of its component spaces?
• RQ4 How do gravitational metric spaces relate to lexicographic metric spaces, and what role do they play in computing the metric dimension?
• RQ5 Can the metric dimension of unbounded or weighted metric spaces be preserved under lexicographic construction?

Key findings

• A lexicographic metric space M∘M' is complete if and only if η(M) > 0, as this ensures every Cauchy sequence is eventually constant.
• The metric dimension of M∘M' is infinite if |X| = ∞ or if any gravitational metric space M'_x has infinite metric dimension.
• If M is unbounded and η(M) > 0, then dim(M∘M') = ∞, regardless of M'.
• When M is twin-free and η(M) > 0, the metric dimension of M∘M' equals the sum of the metric dimensions of the gravitational metric spaces M'_x over all x∈X.
• If D(M') < η(M), then dim(M∘M') = |X|·dim(M'), enabling the use of bounded approximations to compute the dimension.
• By transforming the metric d_Y to d* = η(M)d_Y / (η(M) + d_Y), one obtains a bounded metric space M'' with dim(M'') = dim(M') and D(M'') < η(M), so dim(M∘M'') = |X|·dim(M') = |X|·dim(M'').