The Complexity of Domino Tiling Problems

In this paper, we give a combinatorial formalization that describes a wide range of related problems in the theory of tiling derived from the popular game of dominos. We show that determining if a given finite region of the plane can be tiled with a given set of dominos is NP-Complete. The new graph theoretical methods we introduce allow this result to be easily generalized to a variety of similar tiling problems. Several subproblems of the graph theoretical version of the problem are explored in depth. We then apply these methods to infinite tiling problems to lay the foundation for a more general study of undecidable tiling problems.