Critical properties of graphs of bounded clique-width

A graph property is a set of graphs closed under isomorphism. Clique-width is a graph parameter which is important in theoretical computer science because many algorithmic problems that are generally NP-hard admit polynomial-time solutions when restricted to graphs of bounded clique-width. Over the last few years, many properties of graphs have been shown to be of bounded clique-width; for many others, it has been shown that the clique-width is unbounded. The goal of the present paper is to tighten the gap between properties of bounded and unbounded clique-width. To this end, we identify new necessary and sufficient conditions for clique-width to be bounded.