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arXiv · 2104.00412

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

Robert Brignall·Daniel Cocks
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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.

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BibTeXRIS

Robert Brignall, Daniel Cocks. 2021-04-01. Uncountably many minimal hereditary classes of graphs of unbounded clique-width. https://doi.org/10.37236/10483

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