Maximum independent queen set on polyominoes is NP-complete

Alexis Langlois-Rémillard; Mia Müßig

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arXiv · 2607.24668

Maximum independent queen set on polyominoes is NP-complete

Alexis Langlois-Rémillard·Mia Müßig

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Abstract

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-R\'emillard--M\"u{\ss}ig--Rold\'an. 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.

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Alexis Langlois-Rémillard, Mia Müßig. 2026-07-27. Maximum independent queen set on polyominoes is NP-complete. https://arxiv.org/abs/2607.24668

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