The number of ways to fully-cover a $m$-by-$n$ rectangle with $mn/2$ dominoes was solved by Kasteleyn and as well as Temperley and Fisher, both in 1961. This problem is equivalent to counting the number of perfect matchings in the $m$-by-$n$ grid graph.
In 1967, Kasteleyn generalized this result to planar graphs. However, instead of a closed form, the number of perfect matchings is computable in polynomial time. The algorithm is called the FKT algorithm in their honor of all three researchers. He published this algorithm in the chapter titled "Graph theory and crystal physics" of the book Graph Theory and Theoretical Physics.
In 1988, Vijay Vazirani generalized the FKT algorithm to graphs which do not contain a subgraph homeomorphic to $K_{3,3}$, the complete bipartite graph with 3 vertices in each partition.