Maximum independent queen set on polyominoes is NP-complete

Finding a set of vertices in a graph with no edges between them, INDSET, is a well-known NP-complete problem.

The queen graph of a chessboard is constructed by taking vertices as the tiles of the chessboard and drawing edges between two tiles if a queen can move from one to the other.

We call INDQUEENS the independent set problem on a queen graph where the chessboard is a polyomino.

We prove that INDQUEENS on polyominoes is NP-complete, proving a conjecture of Langlois-Rémillard--Müßig--Roldán.

As our reduction is parsimonious, we can further prove that it is #P-complete.

We furthermore prove that INDROOKS on polyominoes is #P-complete, despite being solvable in polynomial time.