Clique-width and the speed of hereditary properties

In this paper, we study the relationship between the number of n-vertex graphs in a hereditary class X, also known as the speed of the class X, and boundedness of the clique-width in this class. We show that if the speed of X is faster than n!cn for any c, then the clique-width of graphs in X is unbounded, while if the speed does not exceed the Bell number Bn, then the clique-width is bounded by a constant. The situation in the range between these two extremes is more complicated. This area contains both classes of bounded and unbounded clique-width. Moreover, we show that classes of graphs of unbounded clique-width may have slower speed than classes where the clique-width is bounded.