  We consider random perfect matchings on a general class of contracting
bipartite graphs by letting certain edge weights be 0 on the contracting
square-hexagon lattice in a periodic way. We obtain a deterministic limit shape
in the scaling limit. The results can also be applied to prove the existence of
multiple disconnected liquid regions for all the contracting square-hexagon
lattices with certain edge weights, extending the results proved in [13] for
contracting square-hexagon lattices where the number of square rows in each
period is either 0 or 1.
