Twin-width and permutations

Édouard Bonnet, Sebastian Siebertz, P. Ossona de Mendez, J. Nešetřil, Stéphan Thomassé

Abstract: Inspired by a width invariant on permutations defined by Guillemot and Marx, Bonnet, Kim, Thomass'e, and Watrigant introduced the twin-width of graphs, which is a parameter describing its structural complexity. This invariant has been further extended to binary structures, in several (basically equivalent) ways. We prove that a class of binary relational structures (that is: edge-colored partially directed graphs) has bounded twin-width if and only if it is a first-order transduction of a~proper permutation class. As a by-product, we show that every class with bounded twin-width contains at most

2^{O (n)}

pairwise non-isomorphic

n

-vertex graphs.

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