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View all- Diwan ARoy B(2017)On Colouring Point Visibility GraphsAlgorithms and Discrete Applied Mathematics10.1007/978-3-319-53007-9_14(156-165)Online publication date: 26-Jan-2017
On planar point sets with the pentagon property
Pages 81 - 90
Abstract
Motivated by recent papers of Eppstein, Abel et al. and Barat et al., and by a question of Wood, we investigate properties of planar point sets with no 5-hole (no empty convex pentagon). We answer a question of Wood by showing that the visibility graph of a finite point set with no 5-hole may contain a clique of arbitrary size. This is in contrast with the previous examples of sets with no 5-hole, including the example of the (finite) square lattice. In our construction we use several equivalent local characterizations of (locally) finite planar point sets with no 5-hole which may be of independent interest. Our construction relies on a construction scheme which allows to derive new, non-trivial examples of sets with no 5-hole.
References
[1]
Z. Abel, B. Ballinger, P. Bose, S. Collette, V. Dujmović, F. Hurtado, S. D. Kominers, S. Langerman, A. Pór, and D. R. Wood. Every large point set contains many collinear points or an empty pentagon. Graphs and Combinatorics, 27(1):47--60, 2011.
[2]
O. Aichholzer. {Empty}{colored} k-gons -- Recent results on some Erdos--Szekeres type problems. In Proc. XIII Encuentros de Geometría Computacional, pages 43--52. Zaragoza, Spain, 2009.
[3]
J. Barát, V. Dujmović, G. Joret, M. S. Payne, L. Scharf, D. Schymura, P. Valtr, and D. R. Wood. Empty pentagons in point sets with collinearities, submitted.
[4]
V. Dujmović, D. Eppstein, M. Suderman, and D. R. Wood. Drawings of planar graphs with few slopes and segments. Computational Geometry: Theory and Applications, 38:194--212, 2007.
[5]
D. Eppstein. Dilation-free planar graphs. http://www.ics.uci.edu/ eppstein/junkyard/dilation-free, 1997.
[6]
D. Eppstein. Happy endings for flip graphs. Journal of Computational Geometry, 1(1):3--28, 2010.
[7]
P. Erdos. On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl., 103:99--108, 1975.
[8]
T. Gerken. Empty convex hexagons in planar point sets. Discrete Comput. Geom., 39:239--272, 2008.
[9]
H. Harborth. Konvexe Fünfecke in ebenen Punktmengen. Elem. Math., 33:116--118, 1978.
[10]
J. D. Horton. Sets with no empty convex 7-gons. Canadian Math. Bull., 26:482--484, 1983.
[11]
J. Kára, A. Pór, and D. R. Wood. On the chromatic number of the visibility graph of a set of points in the plane. Discrete Comput. Geom., 34(3):497--506, 2005.
[12]
C. M. Nicolás. The empty hexagon theorem. Discrete Comput. Geom., 38(2):389--397, 2007.
[13]
S. Rabinowitz. Consequences of the pentagon property. Geombinatorics, 14:208--220, 2005.
[14]
L. A. Steen and J. A. Seebach. Counterexamples in topology. Dover publications, Incorporated, 1995.
[15]
P. Valtr. On empty hexagons. Goodman, Jacob E. (ed.) et al., Surveys on discrete and computational geometry. Twenty years later. AMS-IMS-SIAM summer research conference, Snowbird, UT, USA, June 18--22, 2006. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 453, 433--441., 2008.
[16]
D. R. Wood. Point sets with no empty pentagon. problem posted at Open Problem Garden, section Geometry, http://www.openproblemgarden.org/op/point_sets_with_no_empty_pentagon, 2010.
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Published In
June 2013
472 pages
ISBN:9781450320313
DOI:10.1145/2462356
- General Chairs:
- Guilherme D. da Fonseca,
- Thomas Lewiner,
- Luis Peñaranda,
- Program Chairs:
- Timothy Chan,
- Rolf Klein
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Published: 17 June 2013
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View all- Diwan ARoy B(2017)On Colouring Point Visibility GraphsAlgorithms and Discrete Applied Mathematics10.1007/978-3-319-53007-9_14(156-165)Online publication date: 26-Jan-2017
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