Problem is:
If $G$ graph is homogeneous, then $\alpha(G) \le{|V(G)| \over 2}$
By the Gallais theorem, for any $G$ graph $\alpha(G) + \beta(G) = |V|$, so from that it is enough to show that $$\alpha(G) \le \beta(G)$$After what I don't know where to go.
Any help or hint is appreciated! Thank you.