Twin-width and Transductions of Proper k-Mixed-Thin Graphs

TL;DR

This paper introduces proper k-mixed-thin graphs, shows they have bounded twin-width linear in k, and establishes a transduction relation to posets of bounded width, expanding the understanding of graph classes with efficiently bounded twin-width.

Contribution

It defines proper k-mixed-thin graphs, proves their twin-width is linear in k, and relates them to posets of bounded width through transductions, extending the class of graphs with known bounded twin-width.

Findings

Proper k-mixed-thin graphs have twin-width linear in k.

A subclass of k-mixed-thin graphs is transduction-equivalent to posets of width k'.

The red potential method is used to establish twin-width bounds.

Abstract

The new graph parameter twin-width, introduced by Bonnet, Kim, Thomass e and Watrigant in 2020, allows for 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, classes of efficiently bounded 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 slight subclass of k-mixed-thin graphs is transduction-equivalent to posets of width k' such that there is a quadratic-polynomial relation between k and k'. In addition to that, we also give an abstract overview of the so-called red potential method which we use to prove our twin-width bounds.