A unified approach to polynomial algorithms on graphs of bounded (bi-)rank-width

In this paper we develop new algorithmic machinery for solving hard problems on graphs of bounded rank-width and on digraphs of bounded bi-rank-width in polynomial (XP, to be precise) time. These include, particularly, graph colouring and chromatic polynomial problems, the Hamiltonian path and c-min-leaf outbranching, the directed cut, and more generally MSOL-partitioning problems on digraphs. Our focus on a formally clean and unified approach for the considered algorithmic problems is in contrast with many previous published XP algorithms running on graphs of bounded clique-width, which mostly used ad hoc techniques and ideas. The new contributions include faster algorithms for computing the chromatic number and the chromatic polynomial on graphs of bounded rank-width, and new algorithms for solving the defective colouring, the min-leaf outbranching, and the directed cut problems.