A framework for minimal hereditary classes of graphs of unbounded clique-width

TL;DR

This paper introduces a framework for hereditary graph classes based on a grid model, characterizing when these classes have unbounded clique-width and identifying minimal classes with this property using new parameters and methods.

Contribution

The paper develops a novel framework capturing known and new minimal hereditary classes of unbounded clique-width, with criteria based on new parameters and recurrence properties.

Findings

Unbounded clique-width corresponds to unbounded parameter $\\mathcal{N}^\delta$.

Minimal unbounded clique-width classes are characterized by bounded $\mathcal{M}^\beta$ and recurrence properties.

Introduces new methods including Ramsey theory and explicit clique-width expressions.

Abstract

We create a framework for hereditary graph classes $\mathcal{G}^\delta$ built on a two-dimensional grid of vertices and edge sets defined by a triple $\delta=\{\alpha,\beta,\gamma\}$ of objects that define edges between consecutive columns, edges between non-consecutive columns (called bonds), and edges within columns. This framework captures all previously proven minimal hereditary classes of graph of unbounded clique-width, and many new ones, although we do not claim this includes all such classes. We show that a graph class $\mathcal{G}^\delta$ has unbounded clique-width if and only if a certain parameter $\mathcal{N}^\delta$ is unbounded. We further show that $\mathcal{G}^\delta$ is minimal of unbounded clique-width (and, indeed, minimal of unbounded linear clique-width) if another parameter $\mathcal{M}^\beta$ is bounded, and also $\delta$ has defined recurrence characteristics. Both the parameters $\mathcal{N}^\delta$ and $\mathcal{M}^\beta$ are properties of a triple $\delta=(\alpha,\beta,\gamma)$, and measure the number of distinct neighbourhoods in certain auxiliary graphs. Throughout our work, we introduce new methods to the study of clique-width, including the use of Ramsey theory in arguments related to unboundedness, and explicit (linear) clique-width expressions for subclasses of minimal classes of unbounded clique-width.