• DOI:10.37236/10483
• Corpus ID: 232478625

Uncountably Many Minimal Hereditary Classes of Graphs of Unbounded Clique-Width

@article{Brignall2022UncountablyMM,
  title={Uncountably Many Minimal Hereditary Classes of Graphs of Unbounded Clique-Width},
  author={Robert Brignall and Daniel G. Cocks},
  journal={Electron. J. Comb.},
  year={2022},
  volume={29}
}

• R. Brignall, D. Cocks
• Published 1 April 2021
• Mathematics
• Electron. J. Comb.

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 not almost periodic.