On (Kq,k)(Kq,k) vertex stable graphs with minimum size

A graph GG is a (Kq,k)(Kq,k) vertex stable graph if it contains a KqKq after deleting any subset of kk vertices. We give a characterization of (Kq,k)(Kq,k) vertex stable graphs with minimum size for q=3,4,5q=3,4,5.