I'm looking for numerical graph invariants that are bounded by a constant either for a graph $G$ or its complement $\bar{G}$. (The complement graph $\bar{G}$ has the same set of vertices as $G$ but the edges are complemented.) More specifically I’m looking for what numerical “Invariant $X$” is out there for which the following statement is true:

“Invariant $X$ is bounded by a constant $c$ either for $G$ or $\bar{G}$”.

One example is diameter $d(G)$ of a graph $G$ where it is known that:

"Either $d(G) \leq 3$ or $d(\bar{G}) \leq 3$"

(Reference: F. Harary, R. W. Robinson: The diameter of a graph and its complement , The American Mathematical Monthly, Vol. 92, No. 3. (Mar., 1985), pp. 211-212”)

My question is what other such numerical graph invariants are out there? I looked at the list of invariants on wikipedia but there was no mention of such bound on their respective pages. Again, all I’m interested is that the condition holds for either $G$ or its complement $\bar{G}$ and not necessarily for both.