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We study different domination problems of attacking and non-attacking rooks
and queens on polyominoes and polycubes of all dimensions. Our main result
proves that maximal domination is NP-complete for non-attacking queens and for
non-attacking rooks on polycubes of dimension three and higher. We also analyse
these problems for polyominoes and convex polyominoes, conjecture the
complexity classes and provide a computer tool for investigation. We have also
computed new values for classical queen domination problems on chessboards
(square polyominoes). For our computations, we have translated the problem into
an integer linear programming instance. Finally, using this computational
implementation and the game engine Godot, we have developed a video game of
minimal domination of queens and rooks on randomly generated polyominoes.
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