Twin-width one
Jungho Ahn, Hugo Jacob, Noleen K\"ohler, Christophe Paul, Amadeus, Reinald, Sebastian Wiederrecht

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 , and obtain the following results: - Graphs of twin-width at most are permutation graphs. In particular they have an intersection model and a linear structure. - There is always a -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 -contraction sequence of a graph, or guarantees that it has twin-width more than . - 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.
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