Counting blanks in polygonal arrangements

article Counting blanks in polygonal arrangements published yes Arseniy Akopyan author 430D2C90-F248-11E8-B48F-1D18A9856A870000-0002-2548-617X Erel Segal Halevi author HeEd department International IST Postdoc Fellowship Programme project Inside a two-dimensional region (``cake""), there are m nonoverlapping tiles of a certain kind (``toppings""). We want to expand the toppings while keeping them nonoverlapping, and possibly add some blank pieces of the same ``certain kind,"" such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axis-parallel rectangles. (4) The cake is an axis-parallel rectilinear polygon and the toppings are axis-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks. Society for Industrial and Applied Mathematics2018 eng SIAM Journal on Discrete Mathematics 1604.00960 00045081050003610.1137/16M110407X 3232242 - 2257 Akopyan A, Segal Halevi E. 2018. Counting blanks in polygonal arrangements. SIAM Journal on Discrete Mathematics. 32(3), 2242–2257. A. Akopyan, E. Segal Halevi, SIAM Journal on Discrete Mathematics 32 (2018) 2242–2257. Akopyan A, Segal Halevi E. Counting blanks in polygonal arrangements. SIAM Journal on Discrete Mathematics. 2018;32(3):2242-2257. doi:<a href="https://doi.org/10.1137/16M110407X">10.1137/16M110407X</a> A. Akopyan and E. Segal Halevi, “Counting blanks in polygonal arrangements,” SIAM Journal on Discrete Mathematics, vol. 32, no. 3. Society for Industrial and Applied Mathematics, pp. 2242–2257, 2018. Akopyan, Arseniy, and Erel Segal Halevi. “Counting Blanks in Polygonal Arrangements.” SIAM Journal on Discrete Mathematics, vol. 32, no. 3, Society for Industrial and Applied Mathematics, 2018, pp. 2242–57, doi:<a href="https://doi.org/10.1137/16M110407X">10.1137/16M110407X</a>. Akopyan, A., & Segal Halevi, E. (2018). Counting blanks in polygonal arrangements. SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics. <a href="https://doi.org/10.1137/16M110407X">https://doi.org/10.1137/16M110407X</a> Akopyan, Arseniy, and Erel Segal Halevi. “Counting Blanks in Polygonal Arrangements.” SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics, 2018. <a href="https://doi.org/10.1137/16M110407X">https://doi.org/10.1137/16M110407X</a>. 582018-12-11T11:44:24Z2026-07-07T10:46:26Z