Combinatorial properties of weighted point sets in the plane

• Project/Area Number: 24540144
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Tokai University
• Principal Investigator: Sakai Toshinori 東海大学, 高輪教養教育センター, 教授 (20267842)
• Research Collaborator: URRUTIA Jorge メキシコ国立自治大学, 数学研究所, 専任上級研究員レベルC ; HURTADO Ferran カタルーニャ工科大学(スペイン), 応用数学科II, 教授
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 重み付き点集合 / 単調多角形 / 単調無交差道 / 島 / 国際研究者交流（メキシコ） / 国際情報交換（スペイン） / 国際研究者交流 / メキシコ：スペイン

Outline of Final Research Achievements

Let P be a set of n points in general position in the plane, with weights 1, 2, ..., n. A monotonic path is a path connecting points of P along which the weights of its vertices monotonically increase or decrease. A monotonic polygon is a simple polygon whose perimeter consists of a non-crossing monotonic path and an edge. A subset S of P is called an island if S consists of the points of P contained in the convex hull of S. We studied the problems concerning the maximum number of vertices contained in monotonic polygons or non-crossing monotonic paths; the maximum number of the sums of weights of vertices of monotonic polygons; the number of faces contained in a decomposition of the convex hull of P into monotonic polygons; the existence of an island with weight close to a given number; the existence of disjoint islands each of which has weight close to a given number; and so on. We also obtained some results concerning the islands of weighted point sets in d-dimensional space.

Research Products

• [Int'l Joint Research] メキシコ国立自治大学数学研究所(メキシコ)
• [Int'l Joint Research] カタルーニャ工科大学応用数学Ⅱ学科(スペイン)
• [Journal Article] Blocking the k-holes of point sets in the plane (2015)
  • Author(s): Javier Cano, Alfredo Garcia, Ferran Hurtado, Toshinori Sakai, Javier Tejel, Jorge Urrutia
  • Journal Title: Graphs and Combinatorics Volume: 31 Issue: 5 Pages: 1271-1287
  • DOI: 10.1007/s00373-014-1488-z
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On balanced 4-holes in bichromatic point sets (2015)
  • Author(s): Sergey Bereg, Jose-Miguel Diaz-Banez, Ruy Fabila-Monroy, Pablo Perez-Lantero, Adriana Ramirez-Vigueras, Toshinori Sakai, Jorge Urrutia, Inmaculada Ventura
  • Journal Title: Computational Geometry: Theory and Applications Volume: 48 Issue: 3 Pages: 169-179
  • DOI: 10.1016/j.comgeo.2014.09.004
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] On the heaviest increasing or decreasing subsequence of a permutation, and paths and matchings on weighted point sets (2012)
  • Author(s): Toshinori Sakai, Jorge Urrutia
  • Journal Title: Computational Geometry Volume: LNCS 7579 Pages: 175-184
  • Peer Reviewed
• [Presentation] Some results on weighted point sets: weights of islands and lengths of non-crossing monotonic paths (2016)
  • Author(s): Toshinori Sakai, Jorge Urrutia
  • Organizer: XXXI Coloquio Victor Neumann-Lara de Teoria de las Graficas, Combinatoria y sus Aplicaciones
  • Place of Presentation: Universidad de Guanajuato
  • Year and Date: 2016-02-29
  • Invited
• [Presentation] On weights of islands of weighted point sets (2015)
  • Author(s): Toshinori Sakai, Jorge Urrutia
  • Organizer: The 18th Japan Conference on Discrete and Computational Geometry and Graphs
  • Place of Presentation: 京都大学
  • Year and Date: 2015-09-15
  • Int'l Joint Research
• [Presentation] Non-crossing monotonic paths in labeled point sets on the plane (2015)
  • Author(s): Toshinori Sakai, Jorge Urrutia
  • Organizer: EuroCG 2015 (The European Workshop on Computational Geometry)
  • Place of Presentation: リュブリャナ大学
  • Year and Date: 2015-03-17
• [Presentation] Ramsey numbers for empty convex polygons (2015)
  • Author(s): Crevel Bautista-Santiago, Javier Cano, Ruy Fabila-Monroy, Carlos Hidalgo Toscano, Clemens Huemer, Jesus Leanos, Toshinori Sakai, Jorge Urrutia
  • Organizer: EuroCG 2015 (The European Workshop on Computational Geometry)
  • Place of Presentation: リュブリャナ大学
  • Year and Date: 2015-03-17
• [Presentation] Long monotonic paths in weighted point sets (2014)
  • Author(s): Toshinori Sakai
  • Organizer: JCDCGG 2014 (The 17th Japan Conference on Discrete and Computational Geometry and Graphs)
  • Place of Presentation: 東京理科大学
  • Year and Date: 2014-09-15
• [Presentation] Long monotonic paths and polygons in weighted point sets (2013)
  • Author(s): Toshinori Sakai
  • Organizer: The 16th Japan Conference on Discrete and Computational Geometry and Graphs
  • Place of Presentation: 東京理科大学
• [Presentation] Problems for sets of weighted points in the plane (2012)
  • Author(s): Toshinori Sakai
  • Organizer: Thai-Japan Joint Conference on Computational Geometry and Graphs 2012 (TJJCCGG 2012)
  • Place of Presentation: Srinakharinwirot University （タイ）
  • Invited
• [Presentation] A point contained in many edge-disjoint triangles of a point set (2012)
  • Author(s): Javier Cano Vila, Luis Felipe Barba, Jorge Urrutia, 酒井利訓
  • Organizer: １日離散数学研究集会
  • Place of Presentation: 東京理科大学