On the thinness of trees
Flavia Bonomo-Braberman, Eric Brandwein, Carolina Luc\'ia Gonz\'alez,, Agust\'in Sansone

TL;DR
This paper introduces an efficient algorithm to compute the thinness of trees and uses it to analyze and improve bounds on the thinness of specific tree families, contributing to graph width parameter research.
Contribution
The paper presents a new $O(n\log n)$ algorithm for computing the thinness of trees and enhances bounds for certain tree families.
Findings
Efficient $O(n\log n)$ algorithm for tree thinness
Improved bounds on thinness for specific tree families
Constructive thin representation for trees
Abstract
The study of structural graph width parameters like tree-width, clique-width and rank-width has been ongoing during the last five decades, and their algorithmic use has also been increasing [Cygan et al., 2015]. New width parameters continue to be defined, for example, MIM-width in 2012, twin-width in 2020, and mixed-thinness, a generalization of thinness, in 2022. The concept of thinness of a graph was introduced in 2007 by Mannino, Oriolo, Ricci and Chandran, and it can be seen as a generalization of interval graphs, which are exactly the graphs with thinness equal to one. This concept is interesting because if a representation of a graph as a -thin graph is given for a constant value , then several known NP-complete problems can be solved in polynomial time. Some examples are the maximum weighted independent set problem, solved in the seminal paper by Mannino et al., and the…
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