International Symposium on Graph Drawing and Network Visualization

Graph Drawing and Network Visualization pp 99-110

• Michael Pelsmajer
• Marcus Schaefer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

## Abstract

A drawing of a graph G is radial if the vertices of G are placed on concentric circles $$C_1, \ldots , C_k$$ with common center c, and edges are drawn radially: every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling.

We show that a graph G is radial planar if G has a radial drawing in which every two edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Tóth.

### References

1. 1.
Bachmaier, C., Brandenburg, F.J., Forster, M.: Radial level planarity testing and embedding in linear time. J. Graph Algorithms Appl. 9, 2005 (2005)
2. 2.
Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)
3. 3.
Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23(2), 191–206 (2000)
4. 4.
Chimani, M., Zeranski, R.: Upward planarity testing: a computational study. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 13–24. Springer, Heidelberg (2013)
5. 5.
Chojnacki, C., Hanani, H.: Über wesentlich unplättbare Kurven im dreidimensionalen Raume. Fundamenta Mathematicae 23, 135–142 (1934)Google Scholar
6. 6.
Di Battista, G., Nardelli, E.: Hierarchies and planarity theory. IEEE Trans. Syst. Man Cybern. 18(6), 1035–1046 (1989)
7. 7.
Di Giacomo, E., Didimo, W., Liotta, G.: Spine and radial drawings, chapter 8. In: Roberto, T. (ed.) Handbook of Graph Drawing and Visualization. Discrete Mathematics and Its Applications. Chapman and Hall/CRC, Boca Raton (2013)Google Scholar
8. 8.
Fulek, R., Kynčl, J., Malinović, I., Pálvölgyi, D.: Clustered planarity testing revisited. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 428–439. Springer, Heidelberg (2014) Google Scholar
9. 9.
Fulek, R., Pelsmajer, M., Schaefer, M., Štefankovič, D.: Hanani-Tutte, monotone drawings, and level-planarity. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 263–287. Springer, New York (2013)
10. 10.
Gross, J.L., Tucker, T.W.: Topological Graph Theory. Dover Publications Inc., Mineola (2001). Reprint of the 1987 original
11. 11.
Gutwenger, C., Mutzel, P., Schaefer, M.: Practical experience with Hanani-Tutte for testing $$c$$-planarity. In: McGeoch, C.C., Meyer, U. (eds.) 2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 86–97. SIAM, Portland (2014)
12. 12.
Jünger, M., Leipert, S.: Level planar embedding in linear time. J. Graph Algorithms Appl. 6(1), 72–81 (2002)
13. 13.
Northway, M.L.: A method for depicting social relationships obtained by sociometric testing. Sociometry 3(2), 144–150 (1940)
14. 14.
Pach, J., Tóth, G.: Monotone drawings of planar graphs. J. Graph Theory 46(1), 39–47 (2004). Updated version: arXiv:1101.0967
15. 15.
Pelsmajer, M.J., Schaefer, M., Stasi, D.: Strong Hanani-Tutte on the projective plane. SIAM J. Discrete Math. 23(3), 1317–1323 (2009)
16. 16.
Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings. J. Combin. Theor. Ser. B 97(4), 489–500 (2007)
17. 17.
Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings on surfaces. Eur. J. Comb. 30(7), 1704–1717 (2009)
18. 18.
Schaefer, M.: Toward a theory of planarity: Hanani-Tutte and planarity variants. J. Graph Algortihms Appl. 17(4), 367–440 (2013)
19. 19.
Schaefer, M.: Hanani-Tutte and related results. In: Bárány, I., Böröczky, K.J., Tóth, G.F., Pach, J. (eds.) A Tribute to László Fejes Tóth. Bolyai Society Mathematical Studies, vol. 24, pp. 259–299. Springer, Berlin (2014)Google Scholar
20. 20.
Tutte, W.T.: Toward a theory of crossing numbers. J. Comb. Theor. 8, 45–53 (1970)
21. 21.
Wiedemann, D.H.: Solving sparse linear equations over finite fields. IEEE Trans. Inf. Theor. 32(1), 54–62 (1986)

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