Twin-width and Transductions of Proper k-Mixed-Thin Graphs
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The new graph parameter twin-width, recently introduced by Bonnet, Kim, Thomassé and Watrigant, admits an FPT algorithm for testing all FO properties of graphs. This makes classes of efficiently bounded twin-width attractive from the algorithmic point of view. In particular, such classes (of small twin-width) include proper interval graphs, and (as digraphs) posets of width k. Inspired by an existing generalization of interval graphs into so-called k-thin graphs, we define a new class of proper k-mixed-thin graphs which largely generalizes proper interval graphs. We prove that proper k-mixed-thin graphs have twin-width linear in k, and that a certain subclass of k-mixed-thin graphs is transduction-equivalent to posets of width k' such that there is a quadratic relation between k and k'.
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