We introduce a parametrized family of graph problems that includes several well-known graph partition problems as special czses. We develop tools which allow us to classify the complexity of many problems in this family, and in particular lead us to a complete classification for small values of the parameters. Along the way, we obtain a variety of specific results including the following: a generalization of a communication bound on the number of clique-versus-independentset separators; polynomial-time algorithms to recognize generalized split graphs; and, a quasi-polynomial algorithm for the Skew Cutset Problem that essentially resolves an open problem posed by Chv&tal.The last two problems have interesting connections to the Strong Perfect Graph Conjecture of Berge. We also observe that the dichotomy (NPcomplete versus polynomial-time solvable) conjectured for certain graph homomorphism problems, would, if true, imply a slightly weaker dichotomy (NP-complete versus quasipolynomial) for our graph partition problems.

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We study the concept of an H-partition of the vertex set of a graph G, which includes all vertex partitioning problems into four parts which we require to be nonempty with only external constraints according to the structure of a model graph H, with the exception of two cases, one that has already been classified as polynomial, and the other one remains unclassified. In the context of more general vertex-partition problems, the problems addressed in this paper have these properties: non-list, 4-part, external constraints only (no internal constraints), each part non-empty. We describe tools that yield for each problem considered in this paper a simple and low complexity polynomial-time algorithm. Mathematics Subject Classification. 05C85, 68R10.Keywords and phrases. Structural graph theory, computational difficulty of problems, analysis of algorithms and problem complexity, perfect graphs, skew partition. * This research was partially supported by CNPq, FAPERJ, CAPES (Brazil)/COFECUB (France), project 359/01/03.

It is well known that a clique with k + 1 vertices is the only minimal obstruction to k-colourability of chordal graphs. A similar result is known for the existence of a cover by cliques. Both of these problems are in fact partition problems, restricted to chordal graphs. The first seeks partitions into k independent sets, and the second is equivalent to finding partitions into cliques. In an earlier paper we proved that a chordal graph can be partitioned into k independent sets and cliques if and only if it does not contain an induced disjoint union of + 1 cliques of size k + 1. (A linear time algorithm for finding such partitions can be derived from the proof.)In this paper we expand our focus and consider more general partitions of chordal graphs. For each symmetric matrix M over 0, 1, * , the M-partition problem seeks a partition of the input graph into independent sets, cliques, or arbitrary sets, with certain pairs of sets being required to have no edges, or to have all edges joining them, as encoded in the matrix M. Moreover, the vertices of the input chordal graph can be equipped with lists, restricting the parts to which a vertex can be placed. Such (list) partitions generalize (list) colourings and (list) homomorphisms, and arise frequently in the theory of graph perfection. We show that many M-partition problems that are NP-complete in general become solvable in polynomial time for chordal graphs, even in the presence of lists. On the other hand, we show that there are M-partition problems (without lists) that remain NP-complete for chordal graphs. It is not known whether or not each list M-partition problem is NP-complete or polynomial, but it has been shown that each is NP-complete or quasi-polynomial (n O(log n) ). For chordal graphs even this 'quasi-dichotomy' is not known, but we do identify large families of matrices M for which dichotomy, or at least quasi-dichotomy, holds.We also discuss forbidden subgraph characterizations of graphs admitting an M-partition. Such characterizations have recently been investigated for partitions of perfect graphs, and we focus on highlighting the improvements one can obtain for the class of chordal, rather than just perfect, graphs.

We consider the following generalization of split graphs: A graph is said to be a (k, f.)-graph if its vertex set can be partitioned into k independent sets and f. cliques. (Split graphs are obtained by setting k = l = 1). Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by (k, f.)-graphs in general. (For instance, being a (k, O)-graph is equivalent to being k-colourable.) However, if we keep the assuIhption of chordality, nice algorithms and characterization theorems are possible. Indeed, our main result is a forbidden subgraph characterization of chordal (k, f.)-graphs. We also give an O(n(m + n)) recognition algorithm for chordal (k, f.)-graphs. When k = 1, our algorithm runs in time O(m + n). In particular, we obtain a new simple and efficient greedy algorithm for the recognition of split graphs, from which it is easy to derive the well known forbidden Preprint submitted to Elsevier Science Another way to think of our main result is the following min-max property of chordal graphs: the maximum number of independent (i.e., disjoint and nonadjacent) Kr's equals the minumum number of cliques that meet alI Kr's.