The Stanley-Reisner ideal of the rook complex of polyominoes

We study the properties of the rook complex $\mathcal{R}$ of a polyomino $\mathcal{P}$ seen as independence complex of a graph $G$, and the associated Stanley--Reisner ideal $I_\mathcal{R}$. In particular, we characterize the polyominoes $\mathcal{P}$ having a pure rook complex, and the ones whose Stanley--Reisner ideal has linear resolution. Furthermore, we prove that for a class of polyominoes the Castelnuovo-Mumford regularity of $I_\mathcal{R}$ coincides with the induced matching number of $G$