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.

翻译：受Guillemot和Marx定义的排列宽度不变量的启发，Bonnet、Kim、Thomassé和Watrigant引入了图的双宽度这一参数，用以描述其结构复杂性。该不变量已进一步以几种（基本等价的）方式推广至二元结构。我们证明：二元关系结构类（即边着色部分有向图）具有有界双宽度当且仅当它是适当排列类的一阶转导。作为副产品，我们表明每个有界双宽度的类至多包含$2^{O(n)}$个两两非同构的$n$顶点图。