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

Classes of graphs with small rank decompositions are chi-bounded