Author, as appears in the article.: Alberto Rodriguez-Velazquez, Juan;
Department: Enginyeria Informàtica i Matemàtiques
URV's Author/s: Rodríguez Velázquez, Juan Alberto
Keywords: Universal lines Product graphs Metric spaces Lines in graphs Distance in graph
Abstract: In a metric space M = (X, d), a line induced by two distinct points x, x ' is an element of X, denoted by L-M{x, x'}, is the set of points given byL-M{x, x'} = {z is an element of X : d(x, x') = d(x, z) + d(z, x') or d(x, x') = |d(x, z) - d(z, x')|}.A line L-M{x, x'} gis universal whenever L-M{x, x'} = X.Chen and Chvatal [Disc. Appl. Math. 156 (2008), 2101-2108.] conjectured that in any finite metric space M = (X, d) either there is a universal line, or there are at least |X| different (nonuniversal) lines. A particular problem derived from this conjecture consists of investigating the properties of M that determine the existence of a universal line, and the problem remains interesting even if we can check that M has at least |X| different lines. Since the vertex set of any connected graph, equipped with the shortest path distance, is a metric space, the problem automatically becomes of interest in graph theory. In this paper, we address the problem of characterizing graphs that have universal lines. We consider several scenarios in which the study can be approached by analysing the existence of such lines in primary subgraphs. We first discuss the wide class of separable graphs, and then describe some particular cases, including those of block graphs, rooted product graphs and corona graphs. We also discuss important classes of nonseparable graphs, including Cartesian product graphs, join graphs and lexicographic product graphs.
Thematic Areas: Mathematics (miscellaneous) Mathematics Matemática / probabilidade e estatística
licence for use: https://creativecommons.org/licenses/by/3.0/es/
Author's mail: juanalberto.rodriguez@urv.cat
Author identifier: 0000-0002-9082-7647
Record's date: 2023-02-19
Papper version: info:eu-repo/semantics/acceptedVersion
Link to the original source: https://www.tandfonline.com/doi/abs/10.2989/16073606.2021.1950862
Licence document URL: http://repositori.urv.cat/ca/proteccio-de-dades/
Papper original source: Quaestiones Mathematicae. 45 (10): 1485-1500
APA: Alberto Rodriguez-Velazquez, Juan; (2022). UNIVERSAL LINES IN GRAPHS. Quaestiones Mathematicae, 45(10), 1485-1500. DOI: 10.2989/16073606.2021.1950862
Article's DOI: 10.2989/16073606.2021.1950862
Entity: Universitat Rovira i Virgili
Journal publication year: 2022
Publication Type: Journal Publications
Repositori URV