{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,30]],"date-time":"2025-08-30T16:33:14Z","timestamp":1756571594015,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>\u00a0We 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 $h\\leqslant 8$ 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 R\u00f3ldan-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 R\u00f3ldan-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.<\/jats:p>","DOI":"10.37236\/9086","type":"journal-article","created":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T00:51:56Z","timestamp":1593132716000},"source":"Crossref","is-referenced-by-count":2,"title":["Extremal Topological and Geometric Problems for Polyominoes"],"prefix":"10.37236","volume":"27","author":[{"given":"Greg","family":"Malen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Erika Berenice","family":"Roldan-Roa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,6,26]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p56\/8113","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p56\/8113","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T00:52:13Z","timestamp":1593132733000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i2p56"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,26]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,4,3]]}},"URL":"https:\/\/doi.org\/10.37236\/9086","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,6,26]]},"article-number":"P2.56"}}