TY - JOUR

T1 - The dual diameter of triangulations

AU - Korman, Matias

AU - Langerman, Stefan

AU - Mulzer, Wolfgang

AU - Pilz, Alexander

AU - Saumell, Maria

AU - Vogtenhuber, Birgit

N1 - Funding Information:
Research for this work was supported by the ESF EUROCORES programme EuroGIGA–ComPoSe, Austrian Science Fund (FWF): I 648-N18 and grant EUI-EURC-2011-4306. M.K. was supported in part by MEXT KAKENHI Nos. 17K12635, 15H02665, and 24106007. S.L. is Directeur de Recherches du FRS-FNRS. W.M. is supported in part by DFG grants MU 3501/1 and MU 3501/2. Part of this work has been done while A.P. was recipient of a DOC-fellowship of the Austrian Academy of Sciences at the Institute for Software Technology, Graz University of Technology, Austria. M.S. was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports, and by the project NEXLIZ CZ.1.07/2.3.00/30.0038, which was co-financed by the European Social Fund and the state budget of the Czech Republic.
Funding Information:
Research for this work was supported by the ESF EUROCORES programme EuroGIGA–ComPoSe, Austrian Science Fund (FWF): I 648-N18 and grant EUI-EURC-2011-4306 . M.K. was supported in part by MEXT KAKENHI Nos. 17K12635 , 15H02665 , and 24106007 . S.L. is Directeur de Recherches du FRS-FNRS. W.M. is supported in part by DFG grants MU 3501/1 and MU 3501/2 . Part of this work has been done while A.P. was recipient of a DOC-fellowship of the Austrian Academy of Sciences at the Institute for Software Technology, Graz University of Technology, Austria. M.S. was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports , and by the project NEXLIZ CZ.1.07/2.3.00/30.0038 , which was co-financed by the European Social Fund and the state budget of the Czech Republic.
Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2018/3

Y1 - 2018/3

N2 - Let P be a simple polygon with n vertices. The dual graph T⁎ of a triangulation T of P is the graph whose vertices correspond to the bounded faces of T and whose edges connect those faces of T that share an edge. We consider triangulations of P that minimize or maximize the diameter of their dual graph. We show that both triangulations can be constructed in O(n3logn) time using dynamic programming. If P is convex, we show that any minimizing triangulation has dual diameter exactly 2⋅⌈log2(n/3)⌉ or 2⋅⌈log2(n/3)⌉−1, depending on n. Trivially, in this case any maximizing triangulation has dual diameter n−2. Furthermore, we investigate the relationship between the dual diameter and the number of ears (triangles with exactly two edges incident to the boundary of P) in a triangulation. For convex P, we show that there is always a triangulation that simultaneously minimizes the dual diameter and maximizes the number of ears. In contrast, we give examples of general simple polygons where every triangulation that maximizes the number of ears has dual diameter that is quadratic in the minimum possible value. We also consider the case of point sets in general position in the plane. We show that for any such set of n points there are triangulations with dual diameter in O(logn) and in Ω(n).

AB - Let P be a simple polygon with n vertices. The dual graph T⁎ of a triangulation T of P is the graph whose vertices correspond to the bounded faces of T and whose edges connect those faces of T that share an edge. We consider triangulations of P that minimize or maximize the diameter of their dual graph. We show that both triangulations can be constructed in O(n3logn) time using dynamic programming. If P is convex, we show that any minimizing triangulation has dual diameter exactly 2⋅⌈log2(n/3)⌉ or 2⋅⌈log2(n/3)⌉−1, depending on n. Trivially, in this case any maximizing triangulation has dual diameter n−2. Furthermore, we investigate the relationship between the dual diameter and the number of ears (triangles with exactly two edges incident to the boundary of P) in a triangulation. For convex P, we show that there is always a triangulation that simultaneously minimizes the dual diameter and maximizes the number of ears. In contrast, we give examples of general simple polygons where every triangulation that maximizes the number of ears has dual diameter that is quadratic in the minimum possible value. We also consider the case of point sets in general position in the plane. We show that for any such set of n points there are triangulations with dual diameter in O(logn) and in Ω(n).

KW - Diameter

KW - Dual graph

KW - Optimization

KW - Simple polygon

KW - Triangulation

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U2 - 10.1016/j.comgeo.2017.06.008

DO - 10.1016/j.comgeo.2017.06.008

M3 - Article

AN - SCOPUS:85023619687

VL - 68

SP - 243

EP - 252

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

ER -