Linear rankwidth meets stability

TL;DR

This paper explores the structural and model-theoretic properties of graph classes with bounded linear rankwidth, revealing their bounded chromatic number, stability conditions, and connections to transductions of pathwidth-bounded classes.

Contribution

It establishes new structural and logical properties of classes with bounded linear rankwidth, linking graph coloring, stability, and transduction concepts.

Findings

Graphs with linear rankwidth at most r are linearly χ-bounded.

For hereditary graph families, bounded rankwidth does not imply bounded coloring with certain properties.

Classes with bounded linear rankwidth are stable if and only if they exclude half-graphs or are transductions of bounded pathwidth classes.

Abstract

Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths. These results show a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on classes with bounded linear rankwidth and prove structural and model theoretic properties of these classes: 1) Graphs with linear rankwidth at most $r$ are linearly \mbox{$\chi$-bounded}. Actually, they have bounded $c$-chromatic number, meaning that they can be colored with $f(r)$ colors, each color inducing a cograph. 2) Based on a Ramsey-like argument, we prove for every proper hereditary family $\mathcal F$ of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing a subgraph in~$\mathcal F$. 3) For a class $\mathcal C$ with bounded linear rankwidth the following conditions are equivalent: a) $\mathcal C$~is~stable, b)~$\mathcal C$~excludes some half-graph as a semi-induced subgraph, c) $\mathcal C$ is a first-order transduction of a class with bounded pathwidth. These results open the perspective to study classes admitting low linear rankwidth covers.