Abstract
We investigate the number of plane geometric, i.e., straight-line, graphs, a set S of n points in the plane admits. We show that the number of plane geometric graphs and connected plane geometric graphs as well as the number of cycle-free plane geometric graphs is minimized when S is in convex position. Moreover, these results hold for all these graphs with an arbitrary but fixed number of edges. Consequently, we provide a unified proof that the cardinality of any family of acyclic graphs (for example spanning trees, forests, perfect matchings, spanning paths, and more) is minimized for point sets in convex position.
In addition we construct a new maximizing configuration, the so-called double zig-zag chain. Most noteworthy this example bears Θ* \({{(\sqrt{72}\,}^n)}\) = Θ*(8.4853n) triangulations (omitting polynomial factors), improving the previously known best maximizing examples.
Similar content being viewed by others
References
Aichholzer, O., Aurenhammer, F., Krasser, H.: Enumerating order types for small point sets with applications. Order 19, 265–281 (2002)
Aichholzer, O., Aurenhammer, F., Krasser, H.: On the crossing number of complete graphs. Computing 76(1–2), 165–176 (2006)
Aichholzer, O., Aurenhammer, F., Krasser, H.,Speckmann, B.: Convexity minimizes pseudo-triangulations. Comput. Geom. Theory Appl. 28, 3–10 (2004)
Aichholzer, O., Hackl, T., Huemer, C., Hurtado, F., Krasser, H., Vogtenhuber, B.: On the number of plane graphs. FSP S92 Industrial Geometry, Technical Report No.8, available online at http://www.ig.jku.at under FSP-Reports (2006)
Aichholzer, O., Hurtado, F., Noy, M.: A lower bound on the number of triangulations of planar point sets. Comput. Geom. Theory Appl. 29(2), 135–145 (2004)
Aichholzer, O., Krasser, H.: The point set order type data base: a collection of applications and results. In: Proceedings of 13th Canadian Conference on Computational Geometry, Waterloo, Ontario, Canada, 17–20 (2001)
Aichholzer, O., Krasser, H.: Abstract order type extension and new results on the rectilinear crossing number. Comput. Geom. Theory Appl. 36(1), 2–15 (2006)
Aichholzer, O., Orden, D., Santos, F., Speckmann, B.: On the number of pseudo- triangulations of certain point sets. In: Proceedings of 20th European Workshop on Computational Geometry, Sevilla, Spain, pp. 119–122 (2004)
Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs. Ann. Discrete Math. 12, 9–12 (1982)
Brass, P., Moser W., Pach, J.: Research problems in discrete geometry. Springer, New York (2005)
Dumitrescu, A.: On two lower bound constructions. In: Proceedings of 11th Canadian Conference on Computational Geometry, Vancouver, British Columbia, Canada, pp. 111–114 (1999)
Flajolet, P., Noy, M.: Analytic combinatorics of non-crossing configurations. Discrete Math. 204, 203–229 (1999)
García, A., Noy, M., Tejel, J.: Lower bounds on the number of crossing-free subgraphs of K n . Comput. Geom. Theory Appl. 16, 211–221 (2000)
Goodman, J. E., O’Rourke, J. (eds): Handbook of Discrete and Computational Geometry, 2nd edn. CRC Press LLC, Boca Raton (2004)
Leighton, T.: Complexity issues in VLSI. MIT Press, Cambridge (1983)
Orden, D., Santos, F.: The polytope of non-crossing graphs on a planar point set. Discrete Comput. Geom. 33(2), 275–305 (2005)
Randall, D., Rote, G., Santos, F., Snoeyink, J.: Counting triangulations and pseudo- triangulations of wheels. In: Proceedings of 13th Canadian Conference on Computational Geometry, Waterloo, Ontario, Canada, pp. 149–152 (2001)
Ribo Mor, A., Rote, G.: Locked and unlocked self-touching linkages. Ph.D. thesis of Ares Ribo Mor personal communication (2005)
Santos, F., Seidel, R.: A better upper bound on the number of triangulations of a planar point set. J. Combin. Theory Ser. A 102, 186–193 (2003)
Sharir, M., Welzl, E.: On the number of crossing-free matchings, (cycles, and partitions). In: Proceedings of 17th Annual ACM-SIAM Symposium on Discrete Algorithms, Miami, Florida, pp. 860–869 (2006) (to appear in SIAM Journal of Computing 2007)
Sharir, M., Welzl, E.: Random triangulations of point sets to appear. In: Proceedings of 22nd Annual ACM-SIAM Symposium on Computational Geometry, Sedona, Arizona, pp. 273–281 (2006)
Sharir, M., Welzl, E.: On the number of crossing-free cycles and spanning trees (in preparation)
Sloane, N.J.A.: The on-line encyclopedia of integer sequences.http://www.research.att. com/~njas/sequences/
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aichholzer, O., Hackl, T., Huemer, C. et al. On the Number of Plane Geometric Graphs. Graphs and Combinatorics 23 (Suppl 1), 67–84 (2007). https://doi.org/10.1007/s00373-007-0704-5
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s00373-007-0704-5