This question is supplementary to the question asked here.
One of the answers give a class of graphs for which the adjacency matrices are invertible which is defined as follows.
Given a permutation $\pi$ of a finite set $V$, form its cycle graph $G$ as follows: the vertex set is $V$ and the edges are pairs $(v,w)$ for which $\pi(v)=w$. (This is a simple directed graph.) The adjacency matrix will in fact be the permutation matrix corresponding to $\pi$, which is invertible.
Let's consider a class of graph isomorphism problems where it is promised that both of the input graphs will be from the class defined above.
Can the graph isomorphism problems from this class be solved efficiently?