  This paper is concerned with the question of whether geometric structures
such as cell complexes can be used to simultaneously describe the minimal free
resolutions of all powers of a monomial ideal. We provide a full answer in the
case of square-free monomial ideals of projective dimension one, by introducing
a combinatorial construction of a family of (cubical) cell complexes whose
1-skeletons are powers of a graph that supports the resolution of the ideal.
