Twin-width one

TL;DR

This paper characterizes graphs with twin-width at most 1, showing they are permutation graphs, and provides linear-time algorithms for recognizing and constructing contraction sequences for these graphs.

Contribution

It establishes the structure of twin-width 1 graphs, links them to permutation graphs, and offers efficient algorithms for their analysis.

Findings

Graphs of twin-width at most 1 are permutation graphs.

A linear-time algorithm constructs 1-contraction sequences or identifies higher twin-width.

Distance-hereditary graphs are characterized by their twin-width, enabling optimal sequence computation.

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.