Abstract
This paper shows that any planar graph with n vertices can be point-set embedded with at most one bend per edge on a universal set of n points in the plane. An implication of this result is that any number of planar graphs admit a simultaneous embedding without mapping with at most one bend per edge.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bernhart, F., Kainen, P.C.: The book thickness of a graph. J. Comb. Theory, Ser. B 27, 320–331 (1979)
Braß, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous planar graph embeddings. Comput. Geom. Theory Appl. 36(2), 117–130 (2007)
Chrobak, M., Karloff, H.: A lower bound on the size of universal sets for planar graphs. SIGACT News 20(4), 83–86 (1989)
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)
Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Curve-constrained drawings of planar graphs. Comput. Geom. Theory Appl. 30, 1–23 (2005)
Enomoto, H., Miyauchi, M.S.: Embedding graphs into a three page book with o(mlog n) crossings of edges over the spine. SIAM J. Discrete Math. 12(3), 337–341 (1999)
Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. Am. Math. Mon. 98(2), 165–166 (1991)
Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. J. Graph Algorithms Appl. 6(1), 115–129 (2002)
Kurowski, M.: A 1.235 lower bound on the number of points needed to draw all n-vertex planar graphs. Inf. Process. Lett. 92(2), 95–98 (2004)
Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs Comb. 17, 717–728 (2001)
Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM–SIAM Sympos. Discrete Algorithms (SODA’90), pp. 138–148 (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by NSERC and the MIUR Project “MAINSTREAM: Algorithms for Massive Information Structures and Data Streams.” Work initiated during the “Workshop on Graph Drawing and Computational Geometry,” Bertinoro, Italy, March 2007. We are grateful to the other participants and, in particular, to W. Didimo and E. Di Giacomo for useful discussions. An earlier version of this paper was presented at the 15th International Symposium on Graph Drawing, GD 2007.
Rights and permissions
About this article
Cite this article
Everett, H., Lazard, S., Liotta, G. et al. Universal Sets of n Points for One-bend Drawings of Planar Graphs with n Vertices. Discrete Comput Geom 43, 272–288 (2010). https://doi.org/10.1007/s00454-009-9149-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-009-9149-3