International Symposium on Graph Drawing and Network Visualization

Graph Drawing and Network Visualization pp 309-320

Simple Realizability of Complete Abstract Topological Graphs Simplified

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

Abstract

An abstract topological graph (briefly an AT-graph) is a pair $$A=(G,\mathcal {X})$$ where $$G=(V,E)$$ is a graph and $$\mathcal {X}\subseteq \left( {\begin{array}{c}E\\ 2\end{array}}\right)$$ is a set of pairs of its edges. The AT-graph A is simply realizable if G can be drawn in the plane so that each pair of edges from $$\mathcal {X}$$ crosses exactly once and no other pair crosses. We characterize simply realizable complete AT-graphs by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author.

Notes

Acknowledgements

I thank Martin Balko for his comments on an earlier version of the manuscript. I also thank all the reviewers for their suggestions for improving the presentation.

References

1. 1.
Ábrego, B.M., Aichholzer, O., Fernández-Merchant, S., Hackl, T., Pammer, J., Pilz, A., Ramos, P., Salazar, G., Vogtenhuber, B.: All good drawings of small complete graphs, EuroCG 2015, Book of abstracts, pp. 57–60 (2015)Google Scholar
2. 2.
Aichholzer, O.: Personal communication. 2014Google Scholar
3. 3.
Armas-Sanabria, L., González-Acuña, F., Rodríguez-Viorato, J.: Self-intersection numbers of paths in compact surfaces. J. Knot Theor. Ramif. 20(3), 403–410 (2011)
4. 4.
Balko, M., Fulek, R., Kynčl, J.: Crossing numbers and combinatorial characterization of monotone drawings of $$K_n$$. Discrete Comput. Geom. 53(1), 107–143 (2015)
5. 5.
Chimani, M.: Facets in the crossing number polytope. SIAM J. Discrete Math. 25(1), 95–111 (2011)
6. 6.
Farb, B., Thurston, B.: Homeomorphisms and simple closed curves, unpublished manuscriptGoogle Scholar
7. 7.
Gioan, E.: Complete graph drawings up to triangle mutations. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 139–150. Springer, Heidelberg (2005)
8. 8.
Harborth, H., Mengersen, I.: Drawings of the complete graph with maximum number of crossings, In: Proceedings of the Twenty-third Southeastern International Conference on Combinatorics, Graph Theory, and Computing, (Boca Raton, FL, 1992), Congressus Numerantium, 88 pp. 225–228 (1992)Google Scholar
9. 9.
Hass, J., Scott, P.: Intersections of curves on surfaces. Isr. J. Math. 51(1–2), 90–120 (1985)
10. 10.
Kratochvíl, J., Lubiw, A., Nešetřil, J.: Noncrossing subgraphs in topological layouts. SIAM J. Discrete Math. 4(2), 223–244 (1991)
11. 11.
Kratochvíl, J., Matoušek, J.: NP-hardness results for intersection graphs. Commentationes Math. Univ. Carol. 30, 761–773 (1989)
12. 12.
Kynčl, J.: Enumeration of simple complete topological graphs. Eur. J. Comb. 30(7), 1676–1685 (2009)
13. 13.
Kynčl, J.: Simple realizability of complete abstract topological graphs in P. Discrete Comput. Geom. 45(3), 383–399 (2011)
14. 14.
Kynčl, J.: Improved enumeration of simple topological graphs. Discrete Comput. Geom. 50(3), 727–770 (2013)
15. 15.
Mutzel, P.: Recent advances in exact crossing minimization (extended abstract). Electron. Notes Discrete Math. 31, 33–36 (2008)
16. 16.
Pach, J., Tóth, G.: How many ways can one draw a graph? Combinatorica 26(5), 559–576 (2006)
17. 17.
Schaefer, M., Sedgwick, E., Štefankovič, D.: Computing Dehn twists and geometric intersection numbers in polynomial time, In: Proceedings of the 20th Canadian Conference on Computational Geometry, CCCG 2008, pp. 111–114 2008Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

1. 1.Department of Applied Mathematics and Institute for Theoretical Computer Science, Faculty of Mathematics and PhysicsCharles UniversityPraha 1Czech Republic
2. 2.Chair of Combinatorial Geometry, EPFL-SB-MATHGEOM-DCGÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland