Uncountably many minimal hereditary classes of graphs of unbounded clique-width

TL;DR

This paper constructs uncountably many minimal hereditary classes of graphs with unbounded clique-width using infinite words, proving a conjecture in the field and revealing complex structural properties of these classes.

Contribution

It introduces a new method to generate uncountably many minimal classes of unbounded clique-width via infinite words, and characterizes their minimality precisely.

Findings

Uncountably many pairwise distinct minimal classes of unbounded clique-width are constructed.

A characterization of minimal classes using a specific collection of infinite words is provided.

The paper disproves a part of a conjecture by showing the set of words includes non-almost periodic examples.

Abstract

Given an infinite word over the alphabet $\{0,1,2,3\}$, we define a class of bipartite hereditary graphs $\mathcal{G}^\alpha$, and show that $\mathcal{G}^\alpha$ has unbounded clique-width unless $\alpha$ contains at most finitely many non-zero letters. We also show that $\mathcal{G}^\alpha$ is minimal of unbounded clique-width if and only if $\alpha$ belongs to a precisely defined collection of words $\Gamma$. The set $\Gamma$ includes all almost periodic words containing at least one non-zero letter, which both enables us to exhibit uncountably many pairwise distinct minimal classes of unbounded clique width, and also proves one direction of a conjecture due to Collins, Foniok, Korpelainen, Lozin and Zamaraev. Finally, we show that the other direction of the conjecture is false, since $\Gamma$ also contains words that are \emph{not} almost periodic.