Structure of the matrix inverse

Hello everyone,

Given a square matrix, the transitive closure of its directed graph
gives the structure of the inverse of the matrix.

This does not seem to work for matrices that have one or more zeros
on the diagonal.  For example, the following two are inverses:

a =
     0     0     0     0     1
     1     0     0     0     0
     0     1     0     0     0
     0     0     1     0     0
     0     0     0     1     0

>> inv(a)
ans =
     0     1     0     0     0
     0     0     1     0     0
     0     0     0     1     0
     0     0     0     0     1
     1     0     0     0     0

Is there a graph description of how to determine the structure of
the inverse in these cases?  I suspect a bipartite graph representation
might be useful, since it can encode whether or not a diagonal
entry exists.

This has application to constructing sparse null-space bases.
Any help would be appreciated.

Thanks!
Edmond