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.

@article{paperbot2745,
  title = {Twin-width and permutations},
  author = {Édouard Bonnet and Jaroslav Nešetřil and Patrice Ossona de Mendez and Sebastian Siebertz and Stéphan Thomassé},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 20, Issue 3},
  year = {2024},
  doi = {10.46298/lmcs-20(3:4)2024}
}