The recently announced Strong Perfect Graph Theorem states that the class ofperfect graphs coincides with the class of graphs containing no inducedodd cycle of length at least 5 or the complement of such a cycle. Agraph in this second class is called Berge. A bull is a graph with fivevertices x, a, b, c, d and five edges xa, xb, ab, ad, bc. A graph isbull-reducible if no vertex is in two bulls. In this paper we give asimple proof that every bull-reducible Berge graph is perfect. Althoughthis result follows directly from the Strong Perfect Graph Theorem, our proofleads to a recognition algorithm for this new class of perfect graphs whosecomplexity, O(n6), is much lower than that announced for perfect graphs.

We study the concept of an H-partition of the vertex set of agraph G, which includes all vertex partitioning problems intofour parts which we require to be nonempty with only externalconstraints according to the structure of a model graph H, withthe exception of two cases, one that has already been classifiedas polynomial, and the other one remains unclassified. In thecontext of more general vertex-partition problems, the problemsaddressed in this paper have these properties: non-list, 4-part,external constraints only (no internal constraints), each partnon-empty. We describe tools that yield for each problemconsidered in this paper a simple and low complexitypolynomial-time algorithm.