Twin-width one
🤖 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.
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Graph Theory
Optimal Sequencing
Algorithm Design
Innovation
Methods, ideas, or system contributions that make the work stand out.
Biconvex Graphs
1-Contraction Sequence
Efficient Algorithm
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