The equivalence relation that HJRW defines in the comments (that is twin vertices may be adjacent) is related to the [cochromatic number](http://en.wikipedia.org/wiki/Cocoloring) $z(G)$ of a graph $G$.  The cochromatic number of $G$ is the minimum number of colours needed to colour $V(G)$ such that each colour class induces a clique or stable set.  Thus, the number of such equivalence classes gives an upper bound on the cochromatic number.