The graph of monomial ideals

There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring image. The definition involves Gröbner bases or the action of the algebraic torus image. We present algorithms for computing the (affine schemes representing) edges in this graph. We study the induced subgraphs on multigraded Hilbert schemes and on square-free monomial ideals. In the latter case, the edges correspond to generalized bistellar flips.