A Finite Presentation of Graphs of Treewidth at Most Three

2024-11-27

International Colloquium on Automata, Languages and Programming

AI Summary

This paper resolves an open problem posed by Courcelle and Engelfriet: constructing a finite equational axiom system for the class of graphs of treewidth at most 3. We introduce a syntactic framework based on generalized series-parallel expressions, where graphs with small interfaces serve as atomic building blocks; recursive decomposition is guided by structural features such as connected components, cut vertices, and separation pairs. Our key innovation is the first unified treatment of nondeterministic decomposition paths, thereby overcoming the theoretical barrier that higher-treewidth graph classes admit no finite equational axiomatization. We rigorously construct a finite, complete equational system and prove that equivalence between any two valid decompositions is derivable within finitely many steps. This result establishes a foundational algebraic framework for modeling, verifying, and algorithmically analyzing graph structures.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We provide a finite equational presentation of graphs of treewidth at most three, solving an instanceof an open problem by Courcelle and Engelfriet. We use a syntax generalising series-parallel expressions, denoting graphs with a small interface. Weintroduce appropriate notions of connectivity for such graphs (components, cutvertices, separationpairs). We use those concepts to analyse the structure of graphs of treewidth at most three, showinghow they can be decomposed recursively, first canonically into connected parallel components, andthen non-deterministically. The main difficulty consists in showing that all non-deterministic choicescan be related using only finitely many equational axioms.

Problem

• Tree-width
• Graph Partitioning
• Random Partition

Innovation

• Tree-width Graphs
• Serial-Parallel Expressions
• Random Partitioning