Sum-perfect graphs

Inspired by a famous characterization of perfect graphs due to Lovász, we define a graph G to be sum-perfect if for every induced subgraph H of G, $\alpha \left(H\right)+\omega \left(H\right)\ge \left|V\left(H\right)\right|$. (Here $\alpha$ and $\omega$ denote the stability number and clique number, respectively.) We give a set of 27 graphs and we prove that a graph G is sum-perfect if and only if G does not contain any of the graphs in the set as an induced subgraph.