Graphs of twin-width at most 1 are exactly a subclass of permutation graphs, can be recognized in linear time, and distance-hereditary graphs have twin-width 0, 1, or 2 according to their split decomposition.
22 Emeric Gioan and Christophe Paul
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Twin-width one
Graphs of twin-width at most 1 are exactly a subclass of permutation graphs, can be recognized in linear time, and distance-hereditary graphs have twin-width 0, 1, or 2 according to their split decomposition.