A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one.The "strong perfect graph conjecture" (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornuéjols and Vušković -that every Berge graph either falls into one of a few basic classes, or admits one of a few kinds of separation (designed so that a minimum counterexample to Berge's conjecture cannot have either of these properties).In this paper we prove both of these conjectures.

A graph is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. In this paper we give an algorithm to test if a graph G is Berge, with running time O(|V (G)| 9 ). This is independent of the recent proof of the strong perfect graph conjecture. (2000): 05C17 Mathematics Subject Classification

A graph is claw-free if no vertex has three pairwise nonadjacent neighbours. At first sight, there seem to be a great variety of types of claw-free graphs. For instance, there are line graphs, the graph of the icosahedron, complements of triangle-free graphs, and the Schläfli graph (an amazingly highly-symmetric graph with 27 vertices), and more; for instance, if we arrange vertices in a circle, choose some intervals from the circle, and make the vertices in each interval adjacent to each other, the graph we produce is claw-free. There are several other such examples, which we regard as "basic" claw-free graphs.Nevertheless, it is possible to prove a complete structure theorem for clawfree graphs. We have shown that every connected claw-free graph can be obtained from one of the basic claw-free graphs by simple expansion operations. In this paper we explain the precise statement of the theorem, sketch the proof, and give a few applications.

A graph is claw-free if no vertex has three pairwise nonadjacent neighbours. In earlier papers of this series we proved that every claw-free graph either belongs to one of several basic classes that we described explicitly, or admits one of a few kinds of decomposition. In this paper we convert this "decomposition" theorem into a theorem describing the global structure of claw-free graphs.

The independence polynomial of a graph G is the polynomial A x |A| , summed over all independent subsets A ⊆ V (G). We prove that if G is clawfree, then all the roots of its independence polynomial are real. This extends a theorem of Heilmann and Lieb [O.J. Heilmann, E.H. Lieb, Theory of monomer-dimer systems, Comm. Math. Phys. 25 (1972) 190-232], answering a question posed by Hamidoune [Y.O. Hamidoune, On the numbers of independent k-sets in a clawfree graph, J. Combin. Theory Ser. B 50 (1990) 241-244] and Stanley [R.P. Stanley, Graph colorings and related symmetric functions: Ideas and applications, Discrete Math. 193 (1998) 267-286].