Flip-width: Unifying Sparsity and Twin-width Theories on Dense Graphs

Core Concepts

New flip-width parameters unify Sparsity and Twin-width theories, providing a common language for studying fundamental graph notions in dense settings.

Abstract

The content introduces flip-width parameters that generalize treewidth, degeneracy, and generalized coloring numbers for sparse graphs. It proposes bounded flip-width as a dense counterpart to classes of bounded expansion and nowhere denseness. The paper unifies Sparsity Theory with Twin-width Theory, offering insights into winning strategies and combinatorial obstructions in the context of dense graphs.

Introduction

Focus on extending sparse graph parameters to dense settings.
Central role of treewidth, degeneracy, and generalized coloring numbers.

Overview

Introduces flip-width parameters for dense graphs.
Relates flip-width to key graph parameters like treewidth and twin-width.

Preliminaries

Notation, Sparsity theory basics, Vapnik-Chervonenkis dimension overview.

Cop-width

Introduction of cop-width parameters characterizing various graph properties.

Flip-width

Introduction of flip-width parameters as a generalization for dense graphs.

Flip-width in Weakly Sparse Classes

Relationship between radius-one flip-width and degeneracy.

Flip-Width of Ordered Graphs and Twin-Width

Study on the relationship between twin-width and flip-width.

Closure under Transductions

Preservation of flip-width under transductions explained.

Definable Flip-Width

Introduction of definable variant of flip-width with approximation algorithm.

Almost Bounded Flip-Width

Introduction of classes with almost bounded flip-width as counterparts to nowhere dense classes.

Discussion

Obstructions to small flip width, model checking implications discussed.

Stats

A class has bounded expansion if each generalized coloring number is bounded by a constant on all graphs in the class.

Quotes

"We propose a new family of graph parameters called flip width..."
"Classes with almost bounded flip width coincide with monadically dependent classes..."

Key Insights Distilled From

Flip-width

by Szym... at arxiv.org 03-26-2024

https://arxiv.org/pdf/2302.00352.pdf

Deeper Inquiries

What implications do almost bounded flip widths have on algorithmic graph theory

Almost bounded flip widths have significant implications on algorithmic graph theory. They provide a new framework for understanding the structural properties of graph classes, particularly in dense graphs. By characterizing classes with almost bounded flip-width, researchers can identify hereditary graph classes for which the model-checking problem is fixed-parameter tractable. This has profound implications for developing efficient algorithms to analyze and manipulate graphs within these specific classes.

How do the findings in this paper impact current approaches to structural graph analysis

The findings in this paper revolutionize current approaches to structural graph analysis by introducing a unified theory that combines Sparsity and Twin-Width theories. This unification provides a common language for studying fundamental notions such as weak coloring numbers, twin-width, and other combinatorial properties across both sparse and dense graphs. By establishing connections between different parameters like degeneracy, treewidth, clique-width, and generalized coloring numbers through flip-width parameters, researchers can now approach structural graph analysis more holistically.

What are the potential applications of unifying Sparsity and Twin-Width theories in practical graph problems

The unification of Sparsity Theory and Twin-Width Theory offers practical applications in various graph problems. It allows researchers to study complex structures in both sparse and dense graphs using a consistent framework based on flip-width parameters. This unified theory enables the development of more efficient algorithms for analyzing diverse types of graphs by providing insights into their structural characteristics. Additionally, it opens up avenues for exploring new relationships between different graph parameters and their impact on algorithmic complexity in solving practical graph-related challenges.