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# 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.

## 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Farb, B., Thurston, B.: Homeomorphisms and simple closed curves, unpublished manuscriptGoogle Scholar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