Title: Classes of graphs with small rank decompositions are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>χ</mml:mi></mml:math>-bounded
Abstract: 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.