  We use the subgraph replacement method to investigate new properties of the
tilings of regions on the square lattice with diagonals drawn in. In
particular, we show that the centrally symmetric tilings of a generalization of
the Aztec diamond are always enumerated by a simple product formula. This
result generalizes the previous work of Ciucu (1997) and Yang (1992) about
symmetric tilings of the Aztec diamond. We also use our method to prove a
closed-form product formula for the number of centrally symmetric tilings of a
quasi-hexagon.
