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

Daligault, Rao and Thomassé proposed in 2010 a fascinating conjecture connecting two seem-ingly unrelated notions: clique-width and well-quasi-ordering. They asked if the clique-width ofgraphs in a hereditary class which is well-quasi-ordered under labelled induced subgraphs is boundedby a constant. This is equivalent to asking whether every hereditary class of unbounded clique-widthhas a labelled infinite antichain. We believe the answer to this question is positive and propose astronger conjecture stating that every minimal hereditary class of graphs of unbounded clique-widthhas a canonical labelled infinite antichain. To date, only two hereditary classes are known to beminimal with respect to clique-width and each of them is known to contain a canonical antichain. Inthe present paper, we discover two more minimal hereditary classes of unbounded clique-width andshow that both of them contain canonical antichains.