The six bidecomposable graphs

A graph G is said to be decomposable if G can be decomposed into a cartesian product of two nontrivial graphs. G is bidecomposable if not only G but also its complement G is decomposable. We prove that there are only six bidecomposable graphs; 2K(2), C₄, Q ₃, K(2) ×(K(2) + K(2)) , K(3) × K(3).