The CkCk-extended graft construction

Gutjahr, Welzl and Woeginger found polynomial-time algorithms for a number of digraph homomorphism problems. These algorithms are based on the X¯-enumeration, theCkCk-extended X¯-enumeration and the X¯-graft construction. In this note, we show how the last two methods can be combined to obtain new polynomial-time algorithms, which also work for list homomorphisms. In the process, we are able to extend results of Bang-Jensen and Hell, dealing with homomorphisms to bipartite tournaments, to list homomorphisms.