---
_id: '58'
abstract:
- lang: eng
  text: '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.'
article_processing_charge: No
arxiv: 1
author:
- first_name: Arseniy
  full_name: Akopyan, Arseniy
  id: 430D2C90-F248-11E8-B48F-1D18A9856A87
  last_name: Akopyan
  orcid: 0000-0002-2548-617X
- first_name: Erel
  full_name: Segal Halevi, Erel
  last_name: Segal Halevi
citation:
  ama: Akopyan A, Segal Halevi E. Counting blanks in polygonal arrangements. <i>SIAM
    Journal on Discrete Mathematics</i>. 2018;32(3):2242-2257. doi:<a href="https://doi.org/10.1137/16M110407X">10.1137/16M110407X</a>
  apa: Akopyan, A., &#38; Segal Halevi, E. (2018). Counting blanks in polygonal arrangements.
    <i>SIAM Journal on Discrete Mathematics</i>. Society for Industrial and Applied
    Mathematics. <a href="https://doi.org/10.1137/16M110407X">https://doi.org/10.1137/16M110407X</a>
  chicago: Akopyan, Arseniy, and Erel Segal Halevi. “Counting Blanks in Polygonal
    Arrangements.” <i>SIAM Journal on Discrete Mathematics</i>. Society for Industrial
    and Applied Mathematics, 2018. <a href="https://doi.org/10.1137/16M110407X">https://doi.org/10.1137/16M110407X</a>.
  ieee: A. Akopyan and E. Segal Halevi, “Counting blanks in polygonal arrangements,”
    <i>SIAM Journal on Discrete Mathematics</i>, vol. 32, no. 3. Society for Industrial
    and Applied Mathematics, pp. 2242–2257, 2018.
  ista: Akopyan A, Segal Halevi E. 2018. Counting blanks in polygonal arrangements.
    SIAM Journal on Discrete Mathematics. 32(3), 2242–2257.
  mla: Akopyan, Arseniy, and Erel Segal Halevi. “Counting Blanks in Polygonal Arrangements.”
    <i>SIAM Journal on Discrete Mathematics</i>, 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>.
  short: A. Akopyan, E. Segal Halevi, SIAM Journal on Discrete Mathematics 32 (2018)
    2242–2257.
date_created: 2018-12-11T11:44:24Z
date_published: 2018-09-06T00:00:00Z
date_updated: 2026-07-07T10:46:26Z
day: '06'
department:
- _id: HeEd
doi: 10.1137/16M110407X
ec_funded: 1
external_id:
  arxiv:
  - '1604.00960'
  isi:
  - '000450810500036'
intvolume: '        32'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1604.00960
month: '09'
oa: 1
oa_version: Preprint
page: 2242 - 2257
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
publication: SIAM Journal on Discrete Mathematics
publication_status: published
publisher: Society for Industrial and Applied Mathematics
publist_id: '7996'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Counting blanks in polygonal arrangements
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 32
year: '2018'
...