Twin-width and permutations
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Inspired by a width invariant defined on permutations by Guillemot and Marx, the twin-width invariant has been recently introduced by Bonnet, Kim, Thomassé, and Watrigant. 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, it shows that every class with bounded twin-width contains at most 2^O(n) pairwise non-isomorphic n-vertex graphs.
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