List Partitions

List partitions generalize list colourings and list homomorphisms. Each symmetric matrix M over 0; 1; deenes a list partition problem. Diierent choices of the matrix M lead to many well-known graph the-oretic problems including the problem of recognizing split graphs and their generalizations, nding homogeneous sets, joins, clique cutsets, stable cutsets, skew cutsets and so on. We develop tools which allow us to classify the complexity of many list partition problems and, in particular, yield the complete classiication for small matrices M. Along the way, we obtain a variety of speciic results including: generalizations of Lovv asz's communication bound on the number of clique-versus-stable-set separators; polynomial-time algorithms to recognize generalized split graphs; a polynomial algorithm for the list version of the Clique Cutset Problem; and the rst subexponential algorithm for the Skew Cutset Problem of Chvv atal. We also show that the dichotomy (NP-complete versus polynomial-time solvable), conjectured for certain graph homomorphism problems would, if true, imply a slightly weaker dichotomy (NP-complete versus quasipolynomial) for our list partition problems 1. A preliminary version of this paper has appeared in 22].