# The Hamiltonian problem on Polyominoes

A polyomino is a connected subset of $\mathbb{Z}^2$ - a set of squares joined along their edges such that the resulting form is connected (or, more shortly, a generalized form of Tetris cube). A polyomino can be easily thought of as graph with vertices being elements of $\mathbb{Z}^2$ and edges between two vertices of hamming distance 1 (i.e. with one coordinate equal and the other different by 1).

The Hamiltonian path/cycle problem is the problem of determining for a given graph whether it contains a path/cycle that visits every vertex exactly once. It is known to be NP-complete for general graphs and even for planar graphs. My question is whether it's still NP-complete when restricting the graphs to be polyominoes (and if it is NP-complete, how it is shown).

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