2022 Fiscal Year Final Research Report
Equivalence of plane geometric graphs by transformations and their related topics
| Project/Area Number |
18K03390
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12030:Basic mathematics-related
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| Research Institution | Yokohama National University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | 幾何学的グラフ / 三角形分割 / 四角形分割 / 螺旋度 / edge flip |
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| Outline of Final Research Achievements |
For a point set S on the plane, consider a graph with vertex set S each of whose edge is a straight segment. In particular, if the outer cycle coincides with the convex hull of S and each finite face is quadrilateral, then the graph is a geometric quadrangulation on S. It is easy to see that if S satisfies an obvious condition, S admits a geometric quadrangulation, but it is not known whether two geometric quadrangulations can be transformed into each other by a local operation called an edge flip. On the other hand, the same problem for triangulations are studied by many people and we can find many research on this problem.
In this research, focusing on geometric quadrangulations on a point set S, we consider whether they can be transformed into each other by the local operations, and various problems related to it.
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| Free Research Field |
離散数学
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| Academic Significance and Societal Importance of the Research Achievements |
上記の問題設定に対して，本研究では，平面の必ずしも凸でない多角形Pの内部に直線分を加えて得られるPの幾何学的四角形分割を考える．どんなPも幾何学的四角形分割を持つとは限らないが，私たちはPの螺旋度という概念を定義し，Pの四角形分割可能性を螺旋度を用いて記述した．一方，変形による同値性の結果は得られたが，最良とはならなかった．
これらの問題は，組合せ的な設定では簡単に解決できるものであるが，幾何学的な制約が問題を難しくする．このように．組合せ論と幾何学の間には興味深い問題が多く存在し，これらの問題がその一例となっている．これらの解決により，両分野が大きく発展することが期待される．
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