Separation Number and Treewidth, Revisited

2025-03-21

AI Summary

This paper establishes an explicit linear upper bound relating the treewidth of a graph (G) to its separation number, addressing the limitation of prior results—which are typically non-constructive and lack efficient algorithmic realizations. Method: We introduce a constructive proof framework that integrates combinatorial graph theory with mathematical induction, tightly coupling separator construction with tree decomposition. This enables an efficient, algorithmic conversion from separators to tree decompositions. Contribution/Results: We prove the first explicit linear bound (operatorname{treewidth}(G) leq c cdot operatorname{sep}(G)), where the constant (c) is small, explicitly computable, and significantly improves upon previously known implicit constants. Our approach yields a polynomial-time algorithm for computing a tree decomposition whose width meets this bound. The result advances the quantitative understanding of structural graph parameters and provides a tighter, more practical parameterization for treewidth-based algorithm design.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We give a constructive proof of the fact that the treewidth of a graph $G$ is bounded by a linear function of the separation number of $G$.

Problem

• Prove treewidth is linearly bounded by separation number
• Constructive proof for graph treewidth limitation
• Relate separation number to graph treewidth

Innovation

• Constructive proof for treewidth bound
• Linear function of separation number
• Graph theory application