Disconnected Complements of Steinhaus graphs

In 1976 Molluzzo asked for the conditions on a generating string of a Steinhaus graph that would guarantee that its complement would be connected. In this paper we give conditions that guarantee that the complement is not connected and a recursion, for the number of such strings. We also find tight upper and lower bounds for this recursion by examining the recursion dσ(2k) = dσ(k)+dσ(k−1) and dσ(2k+1) = 2dσ(k) + 1.