Graph isomorphism problem with invertible adjacency matrices 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?
Isn't this graph just a collection of cycles? So we only need to compare that all the lengths in the two graphs match (which can be done by sorting the lengths).