2000 Fiscal Year Annual Research Report

離散幾何における組合せ論的手法

Research Project

Project/Area Number	10304008
Research Institution	NIHON UNIVERSITY
Principal Investigator	斎藤明日本大学, 文理学部, 教授 (90186924)
Co-Investigator(Kenkyū-buntansha)	金子篤司工学院大学, 工学部, 助教授 (30255608) 渡辺守倉敷芸術科学大学, 産業科学技術学部, 教授 (90068916) 前原濶琉球大学, 教育学部, 教授 (60044921) 太田克弘慶応義塾大学, 理工学部, 助教授 (40213722) 榎本彦衛慶応義塾大学, 理工学部, 教授 (00011669)
Keywords	離散幾何 / 組合せ論 / サイクル / グラフ / ハミルトン性 / ネットワークフロー / 直線描画 / Johnsonスキーム
Research Abstract	本研究は3年という研究期間の中で、離散幾何における組合せ論的手法を発展させ、方法論として確立させることを目的としていた。本年度はその最終年度に当たり、前年度までに離散幾何の諸問題から抽出した組合せ論的特徴を、それぞれ方法論として展開させることを目指した。以下にその研究成果の代表的なものを挙げる。 1.図形の直線描画問題が平面グラフのハミルトン性の側面から捉えられることを見出し、これをハミルトンサイクルの問題として特徴づけることに成功した。特に格子グラフがハミルトンサイクルを持つために必要な対角線の数を決定した。 2.離散幾何における図形の分割問題は、グラフの部分グラフへの分割問題に帰着されることが多い。本研究は完全グラフ、完全2部グラフのサイクル、bowtie、trefoilなどへの分割、及び一般のグラフのサイクル分解などの問題を解決し、それを離散幾何における方法論として応用した。 3.濃淡画像を白黒2値画像として表現する問題をグラフ理論におけるネットワークフローの問題に帰着させ、従来のものとは全く異なるアルゴリズムを開発した。 4.Johnsonスキームにおける与えられた直径を持つ最大集合を決定した。この議論は確率論的手法と組合せ論的手法を融合するもので、離散幾何における新たな手法となった。以上のように、本研究により離散幾何の組合せ論的側面に関する多くの重要な知見が得られ、それらを当初の目的の通り方法論として確立することに成功した。

Research Products
(29 results)

All Other

All Publications (29 results)

[Publications] A.Saito: "Closure and factor-critical graphs"Discrete Mathematics. 215. 217-220 (2000)
[Publications] H.Maehara: "Subdividing a graph toward a unit-distance graph in the plane"European J.Combinatorics. 21. 223-229 (2000)
[Publications] H.Maehara: "Can a convex polygon have a developable Face-cycle?"Theoretical Computer Science. 235. 267-270 (2000)
[Publications] H.Maehara: "Cutting a set of disks by a line with leaving many intact disks in both sides"J.Combinatorial Theory (A). 90. 235-240 (2000)
[Publications] H.Maehara: "On knotted necklaces of pearls and Simon's energies"Yokohama J.Mathematics. 47. 177-185 (2000)
[Publications] H.Maehara: "Piercing balls sitting on a table by a vertical line"European J.Combinatorics. 21. 509-517 (2000)
[Publications] H.Maehara: "Oriented graphs generated by random points on a cicle"J.Applied Probability. 37. 534-539 (2000)
[Publications] H.Maehara: "On the waiting time in a janken game"J.Applied Probability. 37. 601-605 (2000)
[Publications] M.Watanabe: "Cycle reversals in oriented plane quadrangulations and orthogonal plane partitions"J.Geometry. 68. 200-208 (2000)
[Publications] S.Matsunaga: "Graph decompositions and D_3-paths with a prescribed endvertex"Discrete Mathematics. 213. 87-104 (2000)
[Publications] H.Enomoto: "Induced graph theorem on magic valuations"Ars Combinatoria. 56. 25-32 (2000)
[Publications] H.Enomoto: "Toughness and the existence of k-factors IV"Discrete Mathematics. 216. 111-120 (2000)
[Publications] K.Ushio: "Star-factorization of symmetric complete bipartite multi-digraphs"Discrete Mathematics. 215. 293-297 (2000)
[Publications] K.Ushio: "Evenly partite bigraph-factorization of symmetric complete tripartite digraphs"Australasian J.Combinatorics. 21. 245-250 (2000)
[Publications] M.Tsuchiya: "On upper bound graphs with respect to operations on graphs"Theoretical Computer Science. 235. 219-223 (2000)
[Publications] M.Mori: "Higher order mixing property of piecewise linear transformations"Discrete and Continuous Dynamical Systems. 6. 915-934 (2000)
[Publications] T.Asano: "Algorithmic considerations on the computational complexities of digital line extraction problem"Trans.IEICE of Japan. J83-D-I. 80-89 (2000)
[Publications] T.Asano: "Effective use of geometric information for clustering and related topics"IEICE Trans.on Fundamentals. E83-D. 418-427 (2000)
[Publications] K.Ota: "On minimally (n,1)-connected graphs"J.Combinatorial Tgeory (B). 80. 156-171 (2000)
[Publications] S.Negami: "5-Connected planar triangulations quadrangulating other surfaces"Yokohama Math.J.. 47. 187-193 (2000)
[Publications] K.Motegi: "Toroidal surgery on periodic knots"Pacific J.Mathematics. 193. 381-396 (2000)
[Publications] A.Kaneko: "On spanning trees with restricted degrees"Information Preocssing Letters. 73. 163-165 (2000)
[Publications] H.Imai: "A path-exchage-type local search algorithm for vehicle routing and its efficient search strategy"J.Oper.Res.Soc.Japan. 43. 197-208 (2000)
[Publications] A.Nakamoto: "Diagonal flips in outer-torus triangulations"Discrete Mathematics. 216. 71-83 (2000)
[Publications] H.Enomoto: "Complete-factors and f-factors"Discrete Mathematics. 220. 239-242 (2000)
[Publications] H.Enomoto: "Partitions of a graph into paths with prescribed endvertices and lengths"J.Graph Theory. 34. 163-169 (2000)
[Publications] A.Kaneko: "Alternating hamiltonian cycles with minimum number of crossings in the plane"International J.Computational Geometry and Applications. 10. 73-78 (2000)
[Publications] A.Kaneko: "Straight line embeddings of rooted star forests in the plane"Discrete Applied Mathematics. 101. 167-175 (2000)
[Publications] A.Nakamoto: "Diagnonal flips in outer-Klein-bottle triangulations"Discrete Mathematics. 222. 41-50 (2000)

2000 Fiscal Year Annual Research Report

離散幾何における組合せ論的手法

Principal Investigator

斎藤 明 日本大学, 文理学部, 教授 (90186924)

Research Products

[Publications] A.Saito: "Closure and factor-critical graphs"Discrete Mathematics. 215. 217-220 (2000)

[Publications] H.Maehara: "Subdividing a graph toward a unit-distance graph in the plane"European J.Combinatorics. 21. 223-229 (2000)

[Publications] H.Maehara: "Can a convex polygon have a developable Face-cycle?"Theoretical Computer Science. 235. 267-270 (2000)

[Publications] H.Maehara: "Cutting a set of disks by a line with leaving many intact disks in both sides"J.Combinatorial Theory (A). 90. 235-240 (2000)

[Publications] H.Maehara: "On knotted necklaces of pearls and Simon's energies"Yokohama J.Mathematics. 47. 177-185 (2000)

[Publications] H.Maehara: "Piercing balls sitting on a table by a vertical line"European J.Combinatorics. 21. 509-517 (2000)

[Publications] H.Maehara: "Oriented graphs generated by random points on a cicle"J.Applied Probability. 37. 534-539 (2000)

[Publications] H.Maehara: "On the waiting time in a janken game"J.Applied Probability. 37. 601-605 (2000)

[Publications] M.Watanabe: "Cycle reversals in oriented plane quadrangulations and orthogonal plane partitions"J.Geometry. 68. 200-208 (2000)

[Publications] S.Matsunaga: "Graph decompositions and D_3-paths with a prescribed endvertex"Discrete Mathematics. 213. 87-104 (2000)

[Publications] H.Enomoto: "Induced graph theorem on magic valuations"Ars Combinatoria. 56. 25-32 (2000)

[Publications] H.Enomoto: "Toughness and the existence of k-factors IV"Discrete Mathematics. 216. 111-120 (2000)

[Publications] K.Ushio: "Star-factorization of symmetric complete bipartite multi-digraphs"Discrete Mathematics. 215. 293-297 (2000)

[Publications] K.Ushio: "Evenly partite bigraph-factorization of symmetric complete tripartite digraphs"Australasian J.Combinatorics. 21. 245-250 (2000)

[Publications] M.Tsuchiya: "On upper bound graphs with respect to operations on graphs"Theoretical Computer Science. 235. 219-223 (2000)

[Publications] M.Mori: "Higher order mixing property of piecewise linear transformations"Discrete and Continuous Dynamical Systems. 6. 915-934 (2000)

[Publications] T.Asano: "Algorithmic considerations on the computational complexities of digital line extraction problem"Trans.IEICE of Japan. J83-D-I. 80-89 (2000)

[Publications] T.Asano: "Effective use of geometric information for clustering and related topics"IEICE Trans.on Fundamentals. E83-D. 418-427 (2000)

[Publications] K.Ota: "On minimally (n,1)-connected graphs"J.Combinatorial Tgeory (B). 80. 156-171 (2000)

[Publications] S.Negami: "5-Connected planar triangulations quadrangulating other surfaces"Yokohama Math.J.. 47. 187-193 (2000)

[Publications] K.Motegi: "Toroidal surgery on periodic knots"Pacific J.Mathematics. 193. 381-396 (2000)

[Publications] A.Kaneko: "On spanning trees with restricted degrees"Information Preocssing Letters. 73. 163-165 (2000)

[Publications] H.Imai: "A path-exchage-type local search algorithm for vehicle routing and its efficient search strategy"J.Oper.Res.Soc.Japan. 43. 197-208 (2000)

[Publications] A.Nakamoto: "Diagonal flips in outer-torus triangulations"Discrete Mathematics. 216. 71-83 (2000)

[Publications] H.Enomoto: "Complete-factors and f-factors"Discrete Mathematics. 220. 239-242 (2000)

[Publications] H.Enomoto: "Partitions of a graph into paths with prescribed endvertices and lengths"J.Graph Theory. 34. 163-169 (2000)

[Publications] A.Kaneko: "Alternating hamiltonian cycles with minimum number of crossings in the plane"International J.Computational Geometry and Applications. 10. 73-78 (2000)

[Publications] A.Kaneko: "Straight line embeddings of rooted star forests in the plane"Discrete Applied Mathematics. 101. 167-175 (2000)

[Publications] A.Nakamoto: "Diagnonal flips in outer-Klein-bottle triangulations"Discrete Mathematics. 222. 41-50 (2000)

斎藤明日本大学, 文理学部, 教授 (90186924)