Almost Perfect Matrices and Graphs

if and only if the omega-projection and kappa-projection
of almost integral polytopes is again almost integral in the lower dimensional space.
We introduce the notions of omega-projection and kappa-projection
that map almost integral polytopes associated with almost perfect graphs G
with n nodes from the n-dimensional space into the (n - omega)-dimensional
space where omega is the maximum clique size in G. Several important
properties of these projections are established. We prove that the strong
perfect graph conjecture is wrong if an omega-projection and a related
kappa-projection of a (nontrivial) almost integral polytope produce different
polytopes in the (n - omega)-dimensional space.