Is the closure of P under e-free homomorphisms equal to NP?
As Ryan Williams explains on CSTheory.SE, the answer is probably no: certificates of linear size are probably not enough for NP; some NP problems seems to require certificates of super-linear size.

You can also find some examples of NP-problems that appear to require super-linear certificates over on CSTheory: see Natural NP-complete problems with “large” witnesses and Is the Witness Size of Membership for Every NP Language Already Known?.  Also Problem unsolvable in 2o(n) on inputs with n bits, assuming ETH? might be relevant as well.