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• W2073023540 abstract "Previous article Next article Nonconvex Representations of Plane GraphsGiuseppe Di Battista, Fabrizio Frati, and Maurizio PatrignaniGiuseppe Di Battista, Fabrizio Frati, and Maurizio Patrignanihttps://doi.org/10.1137/090748640PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractWe show that every plane graph admits a planar straight-line drawing in which all faces with more than three vertices are nonconvex polygons.1. I. Bárány and G. Rote , Strictly convex drawings of planar graphs , Doc. Math. , 11 ( 2006 ) pp. 369 -- 391 . ISIGoogle Scholar2. N. Bonichon, S. Felsner and M. Mosbah , Convex drawings of 3-connected plane graphs , Algorithmica , 47 ( 2007 ), pp. 399 -- 420 . CrossrefISIGoogle Scholar3. N. Chiba, T. Yamanouchi and T. Nishizeki , Linear algorithms for convex drawings of planar graphs, in Progress in Graph Theory, J. A. Bondy and U. S. R. Murty, eds., Academic Press , New York , 1984 , pp. 153 -- 173 . Google Scholar4. M. Chrobak, M. T. Goodrich and R. Tamassia , Convex drawings of graphs in two and three dimensions, in Proceedings of the Twelfth Annual Symposium on Computational Geometry, ACM , New York , 1996 , pp. 319 -- 328 . Google Scholar5. M. Chrobak and G. Kant , Convex grid drawings of 3-connected planar graphs , Internat. J. Comput. Geom. Appl. , 7 ( 1997 ), pp. 211 -- 223 . CrossrefISIGoogle Scholar6. H. de Fraysseix, J. Pach and R. Pollack , How to draw a planar graph on a grid , Combinatorica , 10 ( 1990 ), pp. 41 -- 51 . CrossrefISIGoogle Scholar7. G. Di Battista, F. Frati and M. Patrignani , Non-convex representations of graphs, in Graph Drawing (GD'08), Lecture Notes in Comput. Sci. 5417, I. G. Tollis and M. Patrignani, eds., Springer-Verlag, Berlin , Heidelberg , 2009 , pp. 390 -- 395 . Google Scholar8. G. Di Battista, R. Tamassia and I. G. Tollis , Area requirement and symmetry display of planar upward drawings , Discrete Comput. Geom. , 7 ( 1992 ), pp. 381 -- 401 . CrossrefISIGoogle Scholar9. G. Di Battista, R. Tamassia and L. Vismara , Incremental convex planarity testing , Inform. Comput. , 169 ( 2001 ), pp. 94 -- 126 . CrossrefISIGoogle Scholar10. I. Fáry , On straight line representation of planar graphs , Acta Univ. Szeged. Sect. Sci. Math. , 11 ( 1948 ), pp. 229 -- 233 . Google Scholar11. F. Frati , On minimum area planar upward drawings of directed trees and other families of directed acyclic graphs , Internat. J. Comput. Geom. Appl. , 18 ( 2008 ), pp. 251 -- 271 . CrossrefISIGoogle Scholar12. R. Haas, D. Orden, G. Rote, F. Santos, B. Servatius, H. Servatius, D. L. Souvaine, I. Streinu and W. Whiteley , Planar minimally rigid graphs and pseudo-triangulations , Comput. Geom. , 31 ( 2005 ), pp. 31 -- 61 . CrossrefISIGoogle Scholar13. S. Hong and H. Nagamochi , Convex drawings of graphs with non-convex boundary constraints , Discrete Appl. Math. , 156 ( 2008 ), pp. 2368 -- 2380 . CrossrefISIGoogle Scholar14. S. Hong and H. Nagamochi , An algorithm for constructing star-shaped drawings of plane graphs , Comput. Geom. , 43 ( 2010 ), pp. 191 -- 206 . CrossrefISIGoogle Scholar15. G. Kant , Drawing planar graphs using the canonical ordering , Algorithmica , 16 ( 1996 ), pp. 4 -- 32 . CrossrefISIGoogle Scholar16. G. Laman , On graphs and rigidity of plane skeletal structures , J. Engrg. Math. , 4 ( 1970 ), pp. 331 -- 340 . CrossrefISIGoogle Scholar17. D. Orden, F. Santos, B. Servatius and H. Servatius , Combinatorial pseudo-triangulations , Discrete Math. , 307 ( 2007 ), pp. 554 -- 566 . CrossrefISIGoogle Scholar18. G. Rote, F. Santos and I. Streinu , Pseudo-triangulations---a survey, in Surveys on Discrete and Computational Geometry---Twenty Years Later, Contemp. Math., 453, R. Pollack, E. Goodman, and J. Pach, eds., American Mathematical Society, Providence , RI , 2008 , pp. 343 -- 410 . Google Scholar19. C. Thomassen , Plane representations of graphs, in Progress in Graph Theory, Academic Press , Toronto , 1984 , pp. 43 -- 69 . Google Scholar20. W. T. Tutte , Convex representations of graphs , Proc. London Math. Soc. (3) , 10 ( 1960 ), pp. 304 -- 320 . CrossrefGoogle Scholar21. K. Wagner and Bemerkungen zum Vierfarbenproblem . Deutsch . Math.-Verein , 2 ( 1936 ), pp. 26 -- 32 . Google ScholarKeywordsgraph drawingconvex drawingplanar graph Previous article Next article FiguresRelatedReferencesCited ByDetails Volume 26, Issue 4| 2012SIAM Journal on Discrete Mathematics1471-1819 History Submitted:04 February 2009Accepted:10 September 2012Published online:27 November 2012 Information© 2012, Society for Industrial and Applied MathematicsKeywordsgraph drawingconvex drawingplanar graphMSC codes05C1068R10PDF Download Article & Publication DataArticle DOI:10.1137/090748640Article page range:pp. 1670-1681ISSN (print):0895-4801ISSN (online):1095-7146Publisher:Society for Industrial and Applied Mathematics" @default.
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