  A technique called graphical condensation is used to prove various
combinatorial identities among numbers of (perfect) matchings of planar
bipartite graphs and tilings of regions. Graphical condensation involves
superimposing matchings of a graph onto matchings of a smaller subgraph, and
then re-partitioning the united matching (actually a multigraph) into matchings
of two other subgraphs, in one of two possible ways. This technique can be used
to enumerate perfect matchings of a wide variety of bipartite planar graphs.
Applications include domino tilings of Aztec diamonds and rectangles, diabolo
tilings of fortresses, plane partitions, and transpose complement plane
partitions.
