Minimal Classes of Graphs of Unbounded Clique-width and Well-quasi-ordering

A. Atminas, R. Brignall, V. Lozin, J. Stacho

Published 2015-03-05Version 1

Daligault, Rao and Thomass\'e proposed in 2010 a fascinating conjecture connecting two seemingly unrelated notions: clique-width and well-quasi-ordering. They asked if the clique-width of graphs in a hereditary class which is well-quasi-ordered under labelled induced subgraphs is bounded by a constant. This is equivalent to asking whether every hereditary class of unbounded clique-width has a labelled infinite antichain. We believe the answer to this question is positive and propose a stronger conjecture stating that every minimal hereditary class of graphs of unbounded clique-width has a canonical labelled infinite antichain. To date, only two hereditary classes are known to be minimal with respect to clique-width and each of them is known to contain a canonical antichain. In the present paper, we discover two more minimal hereditary classes of unbounded clique-width and show that both of them contain canonical antichains.