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.