Classes of graphs with small rank decompositions are -bounded

A class of graphs G is χ -bounded if the chromatic number of graphs in G is bounded by a function of the clique number. We show that if a class G is χ -bounded, then every class of graphs admitting a decomposition along cuts of small rank to graphs from G is χ -bounded. As a corollary, we obtain that every class of graphs with bounded rank-width (or equivalently, clique-width) is χ -bounded.