minimum perimeter
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10.37236/9086 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Greg Malen ◽  
Erika Berenice Roldan-Roa

 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 $h\leqslant 8$ 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 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.



2020 ◽  
Vol 104 (560) ◽  
pp. 338-339
Author(s):  
Ángel Plaza
Keyword(s):  


2019 ◽  
Vol 63 (2) ◽  
pp. 483-505
Author(s):  
Mikkel Abrahamsen ◽  
Mark de Berg ◽  
Kevin Buchin ◽  
Mehran Mehr ◽  
Ali D. Mehrabi
Keyword(s):  


2017 ◽  
Vol 97 (2) ◽  
pp. 2339-2358 ◽  
Author(s):  
Wahiba Larbi-Mezeghrane ◽  
Louiza Bouallouche-Medjkoune ◽  
Ali Larbi


Author(s):  
Sándor P. Fekete ◽  
Andreas Haas ◽  
Michael Hemmer ◽  
Michael Hoffmann ◽  
Irina Kostitsyna ◽  
...  
Keyword(s):  


2015 ◽  
pp. 213-233
Author(s):  
Jin Akiyama ◽  
Kiyoko Matsunaga
Keyword(s):  




2012 ◽  
Vol 04 (04) ◽  
pp. 1250044 ◽  
Author(s):  
ADRIAN DUMITRESCU

In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. In the path variant, we seek a shortest path that visits each region. We present several linear-time approximation algorithms with improved ratios for these problems for two cases of neighborhoods that are (infinite) lines, and respectively, (half-infinite) rays. Along the way we derive a tight bound on the minimum perimeter of a rectangle enclosing an open curve of length L.



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