Tree decompositions with small width, spread, order and degree

2025-09-01

AI Summary

This paper addresses the multi-objective optimization of tree decompositions for arbitrary graphs: simultaneously bounding treewidth, vertex participation frequency (i.e., number of bags containing each vertex), total number of bags, and maximum degree of the decomposition’s index tree—while keeping width close to optimal. We propose a combinatorial, structure-driven construction that achieves linear bounds on all key parameters for the first time. Specifically, for any graph (G) with treewidth at most (k), our method yields a tree decomposition of width at most (72k+1), at most (max{|V(G)|/(2k),,1}) bags, and index tree maximum degree at most 12; moreover, each vertex (v) appears in at most (deg(v)+1) bags. This improves Ding–Oporowski’s prior exponential bound to a linear one and strongly confirms their conjecture, thereby substantially enhancing the practical utility of tree decompositions in algorithm design and structural graph theory.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Tree-decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. The main property of tree-decompositions is the width (the maximum size of a bag $-1$). We show that every graph has a tree-decomposition with near-optimal width, plus several additional properties of interest. In particular every graph $G$ with treewidth at most $k$ has a tree-decomposition with width at most $72k+1$, where each vertex $v$ appears in at most $ ext{deg}_G(v)+1$ bags, the number of bags is at most $max{frac{|V(G)|}{2k},1}$, and the tree indexing the decomposition has maximum degree at most 12. This improves exponential bounds to linear in a result of Ding and Oporowski [1995], and establishes a conjecture of theirs in a strong sense.

Problem

• Improve tree-decomposition width bounds from exponential to linear
• Limit vertex appearance in bags to degree plus one
• Bound tree index degree and number of bags

Innovation

• Tree-decomposition with width at most 72k+1
• Each vertex appears in deg(v)+1 bags maximum
• Decomposition tree has maximum degree at most 12