Even pairs and the strong perfect graph conjecture

We will characterize all graphs that have the property that the graph and its complement are minimal even pair free. This characterization allows a new formulation of the Strong Perfect Graph Conjecture. The reader is assumed to be familiar with perfect graphs (see for example [2]). A hole is a cycle of length at least five. An odd hole is a hole that has an odd number of vertices. An (odd) anti-hole is the complement of an (odd) hole. Berge’s famous Strong Perfect Graph Conjecture may be formulated as follows [1]: Strong Perfect Graph Conjecture The only minimal imperfect graphs are the odd holes and the odd anti-holes. Two vertices in a graph are called an even pair if all induced paths between these two vertices have even length. Meyniel [6] has shown that minimal imperfect graphs are even pair free. A graph is called minimal even pair free if the graph does not contain an even pair but every induced subgraph contains an even pair or is a clique. We will show that proving that minimal imperfect graphs are minimal even pair free already implies the Strong Perfect Graph Conjecture. Proposition 1 Odd holes are minimal even pair free. Proof. Obviously odd holes do not contain an even pair. Every proper induced subgraph of an odd hole is either disconnected or a path of length at least two or a clique. Thus every proper induced subgraph of an odd hole contains an even pair or is a clique. 2 Proposition 2 Anti-holes are minimal even pair free. Proof. Let G be an anti-hole. Then every two non-adjacent vertices of G are connected by an induced path of length three and thus G does not contain an even pair. Let H be a proper induced subgraph of G that is not a clique. Then H contains a vertex of degree one and therefore H contains a vertex x that is connected to all but one vertex in H . All induced paths between x and its non-neighbor have length two and thus H contains an even pair. 2 We are now able to give a characterization of all the graphs G such that G and G are minimal even pair free: