On the thinness of trees

2025-01-19

Discrete Applied Mathematics

AI Summary

This paper investigates “thinness,” a novel structural parameter for trees, aiming to establish its polynomial-time computability and structural characterization. Methodologically, the authors formally define thinness and conduct an in-depth analysis of its relationships with treewidth, pathwidth, and interval number, thereby uncovering, for the first time, an implicit dimensional hierarchy within the class of trees. They derive tight bounds for the thinness of any $n$-vertex tree: $1 leq mathrm{thin}(T) leq lfloor n/3 floor$, fully characterizing extremal cases such as stars and paths; moreover, they prove that deciding whether $mathrm{thin}(T) leq k$ is NP-complete for $k geq 3$. These results yield efficient algorithms for several NP-hard problems—including Maximum Weighted Independent Set and Capacitated Fixed-Coloring—on trees of bounded thinness, and provide new insights into interval graph embeddings and path decompositions.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Problem

• Tree Width
• Polynomial Time Algorithm
• Graph Theory

Innovation

• Efficient Algorithm
• Tree Thickness
• Polynomial Time Solutions