  In 2007, Arkin et al. initiated a systematic study of the complexity of the
Hamiltonian cycle problem on square, triangular, or hexagonal grid graphs,
restricted to polygonal, thin, superthin, degree-bounded, or solid grid graphs.
They solved many combinations of these problems, proving them either
polynomially solvable or NP-complete, but left three combinations open. In this
paper, we prove two of these unsolved combinations to be NP-complete:
Hamiltonicity of Square Polygonal Grid Graphs and Hamiltonicity of Hexagonal
Thin Grid Graphs. We also consider a new restriction, where the grid graph is
both thin and polygonal, and prove that Hamiltonicity then becomes polynomially
solvable for square, triangular, and hexagonal grid graphs.
