1-Embeddings on closed surfaces from the viewpoint of re-embeddings

• Project/Area Number: 24740056
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Niigata University
• Principal Investigator: Suzuki Yusuke 新潟大学, 自然科学系, 准教授 (10390402)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 1-平面グラフ / 1-交差埋め込み / 閉曲面 / 三角形分割 / 再埋蔵 / 最適1-平面グラフ / 四角形分割 / 二部グラフ / 外1-平面的グラフ / 線形時間アルゴリズム / グラフ

Outline of Final Research Achievements

A graph is said to be 1-planar (or 1-embedding of a closed surface) if the graph can be drawn on the sphere (or the surface) so that each edge has at most one crossing point with another edge. In this research, we discuss 1-planar graphs (or 1-embeddings of closed surfaces) from the viewpoint of “re-embeddings”of those graphs, and obtained some results around this topic, which are clearly different from past results. In particular, in the joint work with Prof. Eades in Australia, we had shown that the problem of testing maximal 1-planarity of a graph can be solved in linear time, if a rotation system is given. Furthermore, we could prove that there exists no optimal 1-planar graph which can triangulate a non-orientable surface.

Research Products

• [Journal Article] Relationship among triangulations, quadrangulations and optimal 1-planar graphs (2015)
  • Author(s): Kenta Noguchi, Yusuke Suzuki
  • Journal Title: Graphs and Combinatorics Volume: 印刷中 Issue: 6 Pages: 1965-1972
  • DOI: 10.1007/s00373-015-1568-8
  • Peer Reviewed
• [Journal Article] A Linear-Time Algorithm for Testing Outer-1-Planarity (2015)
  • Author(s): Seok-Hee Hong, Peter Eades, Naoki Katoh, Giuseppe Liotta, Pascal Schweitzer, and Yusuke Suzuki
  • Journal Title: Algorithmica Volume: 72 Issue: 4 Pages: 1033-1054
  • DOI: 10.1007/s00453-014-9890-8
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] A Linear Time Algorithm for Testing Maximal 1-planarity of Graphs with a Rotation System (2013)
  • Author(s): Seok-Hee Hong, Peter Eades, Naoki Katoh, Giuseppe Liotta, Pascal Schweitzer, Yusuke Suzuki
  • Journal Title: Theoretical Computer Science Volume: 513 Pages: 65-76
  • DOI: 10.1016/j.tcs.2013.09.029
  • Peer Reviewed
• [Journal Article] On properties of maximal 1-planar graphs (2012)
  • Author(s): D. Hudac, T. Madaras, Y. Suzuki
  • Journal Title: Discuss. Math. Graph Theory Volume: 32 Issue: 4 Pages: 737-747
  • DOI: 10.7151/dmgt.1639
  • Peer Reviewed
• [Journal Article] Coloring Eulerian triangulations of the Klein bottle (2012)
  • Author(s): D. Kral, B. Mohar, A. Nakamoto, O. Pangrac, Y. Suzuki
  • Journal Title: Graphs Combin. Volume: 28 Issue: 4 Pages: 499-530
  • DOI: 10.1007/s00373-011-1063-9
  • Peer Reviewed
• [Presentation] 最適1-交差埋め込みのK_7-マイナーについて (2015)
  • Author(s): 鈴木 有祐
  • Organizer: 第27回位相幾何学的グラフ理論研究集会
  • Place of Presentation: 横浜国立大学（みなとみらいキャンパス）
  • Year and Date: 2015-11-14
• [Presentation] K_7-Minors in optimal 1-planar graphs (2015)
  • Author(s): Yusuke Suzuki
  • Organizer: The Second Japan-Sino Symposium on Graph Theory, Combinatorics and Their Applications
  • Place of Presentation: Morito Memorial Hall(Tokyo University of Science)
  • Year and Date: 2015-11-02
  • Int'l Joint Research
• [Presentation] Recent works on optimal 1-planar graphs (2015)
  • Author(s): Yusuke Suzuki
  • Organizer: The 9th KIAS Combinatorics Workshop
  • Place of Presentation: Broom Vista (Korea)
  • Year and Date: 2015-09-12
  • Int'l Joint Research / Invited
• [Presentation] K_7-Minors in optimal 1-planar graphs (2015)
  • Author(s): 鈴木 有祐
  • Organizer: 離散数学とその応用研究集会2015
  • Place of Presentation: 熊本大学工学部百周年記念館
  • Year and Date: 2015-08-23
• [Presentation] K_7-Minors in optimal 1-planar graphs (2015)
  • Author(s): Yusuke Suzuki
  • Organizer: 8th Slovenian Conference on Graph Theory
  • Place of Presentation: Ramada Resort (Kranjska Gora, Slovenia)
  • Year and Date: 2015-06-26
  • Int'l Joint Research
• [Presentation] 二部グラフ的1-planarグラフの辺数 (2015)
  • Author(s): 鈴木有祐
  • Organizer: TGTセミナー
  • Place of Presentation: 横浜国立大学
  • Year and Date: 2015-02-06
• [Presentation] Cube-contractions in 3-connected quadrangulations on surfaces
  • Author(s): Yusuke Suzuki
  • Organizer: The Sixth Workshop Graph Embeddings and Maps on Surfaces
  • Place of Presentation: Smolenice castle, Slovakia
• [Presentation] Cube-contractions in 3-connected quadrangulations
  • Author(s): 鈴木 有祐
  • Organizer: 離散数学とその応用研究集会2013
  • Place of Presentation: 山形市保健センター・視聴覚室
• [Presentation] Cube-contractions in 3-connected quadrangulations
  • Author(s): Yusuke Suzuki
  • Organizer: The 25th Workshop on Topological Graph Theory in Yokohama
  • Place of Presentation: Yokohama National University
• [Presentation] 多面体的四角形分割に対する減少操作について
  • Author(s): 鈴木 有祐
  • Organizer: 応用数学合同研究集会
  • Place of Presentation: 龍谷大学 瀬田キャンパス １号館
• [Presentation] 閉曲面上のグラフの生成定理について
  • Author(s): 鈴木 有祐
  • Organizer: 第10回組合せ論若手研究集会
  • Place of Presentation: 慶應義塾大学矢上キャンパス 14棟 創想館2階14-203番教室
  • Invited
• [Presentation] 多面体的四角形分割に対する縮小操作について
  • Author(s): 鈴木 有祐
  • Organizer: 日本数学会2014年度会
  • Place of Presentation: 学習院大学 目白キャンパス
• [Presentation] 再埋蔵的視点から見たグラフの1-交差埋め込み
  • Author(s): 鈴木 有祐
  • Organizer: 日本数学会・秋季総合分科会・分科会特別講演
  • Place of Presentation: 九州大学
  • Invited
• [Remarks] YUSUKE SUZUKI
  • URL: http://mathweb.sc.niigata-u.ac.jp/~y-suzuki/