  A binary modified de Bruijn sequence is an infinite and periodic binary
sequence derived by removing a zero from the longest run of zeros in a binary
de Bruijn sequence. The minimal polynomial of the modified sequence is its
unique least-degree characteristic polynomial. Leveraging on a recent
characterization, we devise a novel general approach to determine the minimal
polynomial. We translate the characterization into a problem of identifying a
Hamiltonian cycle in a specially constructed graph. Along the way, we
demonstrate the usefullness of computational tools from the cycle joining
method in the modified setup.
