On the Complexity of Identifying Strongly Regular Graphs

In this note, we show that Graph Isomorphism (GI) is not $\textsf{AC}^{0}$-reducible to several problems, including the Latin Square Isotopy problem and isomorphism testing of several families of Steiner designs. As a corollary, we obtain that GI is not $\textsf{AC}^{0}$-reducible to isomorphism testing of Latin square graphs and strongly regular graphs arising from special cases of Steiner $2$-designs. We accomplish this by showing that the generator-enumeration technique for each of these problems can be implemented in $\beta_{2}\textsf{FOLL}$, which cannot compute Parity (Chattopadhyay, Tor\'an, & Wagner, $\textit{ACM Trans. Comp. Theory}$, 2013).