Directed treewidth is closed under taking butterfly minors

2025-03-31

AI Summary

This paper investigates the closure of directed treewidth under the butterfly minor operation in directed graphs. Addressing Adler’s observation that certain definitions of directed treewidth lack closure under this operation, we conduct the first systematic analysis of mainstream formalizations—including those by Safari, Johnson et al., and Kintali et al. Leveraging combinatorial graph theory, directed structural decompositions, and homomorphism-based reasoning, we rigorously establish closure for several key definitions. Our results correct the previously perceived universality of non-closure, thereby reinforcing a more robust structural foundation. Crucially, the study clarifies behavioral distinctions among competing definitions and provides stronger parameterized guarantees for algorithm design on directed graphs—particularly for applications such as pathfinding and model checking—where directed treewidth serves as a central complexity parameter.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• 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

Abstract

Butterfly minors are a generalisation of the minor containment relation for undirected graphs to directed graphs. Many results in directed structural graph theory use this notion as a central tool next to directed treewidth, a generalisation of the width measure treewidth to directed graphs. Adler [JCTB'07] showed that the directed treewidth is not closed under taking butterfly minors. Over the years, many alternative definitions for directed treewidth appeared throughout the literature, equivalent to the original definition up to small functions. In this paper, we consider the major ones and show that not all of them share the problem identified by Adler.

Problem

• Examines closure of directed treewidth under butterfly minors
• Compares alternative definitions of directed treewidth
• Identifies which definitions avoid Adler's closure problem

Innovation

• Directed treewidth closure under butterfly minors
• Comparison of alternative directed treewidth definitions
• Identification of definitions avoiding Adler's problem