The Stanley–Reisner ideal of the rook complex of polyominoes

We study the properties of the rook complex ${ℛ}$ of a polyomino ${𝒫}$ seen as independence complex of a graph $G$, and the associated Stanley–Reisner ideal $I_{{ℛ}}$. In particular, we characterize the polyominoes ${𝒫}$ 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 $I_{{ℛ}}$ coincides with the induced matching number of $G$.