We prove that if C is a hereditary class of graphs that is polynomially χ-bounded, then the class of graphs that admit decompositions into pieces belonging to C along cuts of bounded rank is also polynomially χ-bounded. In particular, this implies that for every positive integer k, the class of graphs of cliquewidth at most k is polynomially χ-bounded.
@article{arxiv.1910.00697,
title = {Graphs of bounded cliquewidth are polynomially $\chi$-bounded},
author = {Marthe Bonamy and Michał Pilipczuk},
journal= {arXiv preprint arXiv:1910.00697},
year = {2020}
}