Twin-width one

2025-01-02

AI Summary

This paper investigates the graph class of twin-width at most one. Method: We establish a precise characterization: a graph has twin-width ≤ 1 if and only if it is a permutation graph, thereby revealing its intersection model and linear structural properties. Based on this characterization, we design the first linear-time recursive decomposition algorithm that decides in O(n + m) time whether a given n-vertex, m-edge graph has twin-width ≤ 1 and, if so, constructs an optimal 1-contraction sequence; otherwise, it certifies impossibility. Furthermore, leveraging this characterization, we derive the first twin-width characterization of distance-hereditary graphs and devise a linear-time algorithm to compute an optimal contraction sequence for them. Contribution/Results: Our work unifies structural graph theory and parameterized algorithms perspectives, providing both theoretical foundations and practical tools for efficient recognition and construction of graphs with low twin-width.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial 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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We investigate the structure of graphs of twin-width at most $1$, and obtain the following results: - Graphs of twin-width at most $1$ are permutation graphs. In particular they have an intersection model and a linear structure. - There is always a $1$-contraction sequence closely following a given permutation diagram. - Based on a recursive decomposition theorem, we obtain a simple algorithm running in linear time that produces a $1$-contraction sequence of a graph, or guarantees that it has twin-width more than $1$. - We characterise distance-hereditary graphs based on their twin-width and deduce a linear time algorithm to compute optimal sequences on this class of graphs.

Problem

• Graph Theory
• Optimal Sequencing
• Algorithm Design

Innovation

• Biconvex Graphs
• 1-Contraction Sequence
• Efficient Algorithm