Classes of graphs with small rank decompositions are χχ-bounded

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

► Tree decomposition of small rank over a χχ-bounded class implies χχ-boundedness.
► Graphs of bounded rank-width are χχ-bounded.
► 1-joins preserve χχ-boundedness.