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- W3037327413 abstract "We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and we analyze structural properties of polyominoes with $h$ holes and $g(h)$ tiles, characterizing their efficiency by a topological isoperimetric inequality that relates minimum perimeter, the area of the holes, and the structure of the dual graph of a polyomino. For $hleqslant 8$ the values of $g(h)$ were originally computed by Tomas Olivera e Silva in 2015, and for the sequence $h_l=(2^{2l}-1)/3$ by Kahle and Róldan-Roa in 2019, who also showed that asymptotically $g(h) approx 2h$. Here we also prove that the sequence of polyominoes constructed by Kahle and Róldan-Roa that have $h_l=(2^{2l}-1)/3$ holes and $g(h_l)$ tiles, are in fact unique up to isometry with respect to attaining these extremal topological properties; that is, having the minimal number of tiles for $h_l$ holes." @default.
- W3037327413 created "2020-07-02" @default.
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- W3037327413 date "2020-06-26" @default.
- W3037327413 modified "2023-10-14" @default.
- W3037327413 title "Extremal Topological and Geometric Problems for Polyominoes" @default.
- W3037327413 doi "https://doi.org/10.37236/9086" @default.
- W3037327413 hasPublicationYear "2020" @default.
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