The k-partition problem is as follows: Given a graph G and a positive integer k, partition the vertices of G into at most k parts $A_1, A_2, \ldots , A_k$, where it may be specified that $A_i$ induces a stable set, a clique, or an arbitrary subgraph, and pairs $A_i, A_j (i \neq j)$ be completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list k-partition problem generalizes the k-partition problem by specifying for each vertex x, a list $L(x)$ of parts in which it is allowed to be placed. Many well-known graph problems can be formulated as list k-partition problems: e.g., 3-colorability, clique cutset, stable cutset, homogeneous set, skew partition, and 2-clique cutset. We classify, with the exception of two polynomially equivalent problems, each list 4-partition problem as either solvable in polynomial time or NP-complete. In doing so, we provide polynomial-time algorithms for many problems whose polynomial-time solvability was open, including the list 2-clique cutset problem. This also allows us to classify each list generalized 2-clique cutset problem and list generalized skew partition problem as solvable in polynomial time or NP-complete.

The class of weakly triangulated comparability graphs and their complements are generalizations of interval graphs and chordal comparability graphs. We show that problems on these classes of graphs can be solved efficiently by transforming them into problems on chordal bipartite graphs. We show that recognition and independent set on weakly triangulated comparability graphs can be solved in O(n 2 ) time in this manner, and that the number of weakly triangulated comparability graphs is $2^{\Theta ( n {{\log}^2} n)}$.\ We also give algorithms to compute transitive closure and transitive reduction in O(n 2 loglogn) time if the underlying undirected graph of the transitive closure is a weakly triangulated comparability graph.