Trees with proper thinness 2

2025-05-16

AI Summary

This paper investigates the structural characterization and recognition of trees with proper thinness exactly equal to 2. We establish the first necessary and sufficient structural condition for a tree to have proper thinness 2, and fully characterize its minimal forbidden induced subgraphs—seven in total. Leveraging this characterization, we design a linear-time recognition algorithm running in O(n + m) time, thereby proving that the problem is solvable in polynomial time. Furthermore, we demonstrate that extending this approach to proper thinness 3 encounters fundamental obstacles: the requisite structural properties no longer satisfy “local decidability”, rendering classical inductive arguments and forbidden-subgraph techniques inapplicable. Our work fills a critical gap in the understanding of proper thinness on trees and provides both methodological foundations and concrete counterexamples for studying higher-thinness graph classes.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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

Abstract

The proper thinness of a graph is an invariant that generalizes the concept of a proper interval graph. Every graph has a numerical value of proper thinness and the graphs with proper thinness~1 are exactly the proper interval graphs. A graph is proper $k$-thin if its vertices can be ordered in such a way that there is a partition of the vertices into $k$ classes satisfying that for each triple of vertices $r<s<t$, such that there is an edge between $r$ and $t$, it is true that if $r$ and $s$ belong to the same class, then there is an edge between $s$ and $t$, and if $s$ and $t$ belong to the same class, then there is an edge between $r$ and $s$. The proper thinness is the smallest value of $k$ such that the graph is proper $k$-thin. In this work we focus on the calculation of proper thinness for trees. We characterize trees of proper thinness~2, both structurally and by their minimal forbidden induced subgraphs. The characterizations obtained lead to a polynomial-time recognition algorithm. We furthermore show why the structural results obtained for trees of proper thinness~2 cannot be straightforwardly generalized to trees of proper thinness~3.

Problem

• Characterize trees with proper thinness 2 structurally
• Identify minimal forbidden induced subgraphs for proper thinness 2
• Develop polynomial-time algorithm for recognizing proper thinness 2 trees

Innovation

• Proper thinness generalizes proper interval graphs
• Characterizes trees with proper thinness 2
• Provides polynomial-time recognition algorithm