GD 2014: Graph Drawing pp 476-487

# On Self-Approaching and Increasing-Chord Drawings of 3-Connected Planar Graphs

• Martin Nöllenburg
• Roman Prutkin
• Ignaz Rutter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8871)

## Abstract

An st-path in a drawing of a graph is self-approaching if during a traversal of the corresponding curve from s to any point t′ on the curve the distance to t′ is non-increasing. A path has increasing chords if it is self-approaching in both directions. A drawing is self-approaching (increasing-chord) if any pair of vertices is connected by a self-approaching (increasing-chord) path.

We study self-approaching and increasing-chord drawings of triangulations and 3-connected planar graphs. We show that in the Euclidean plane, triangulations admit increasing-chord drawings, and for planar 3-trees we can ensure planarity. Moreover, we give a binary cactus that does not admit a self-approaching drawing. Finally, we show that 3-connected planar graphs admit increasing-chord drawings in the hyperbolic plane and characterize the trees that admit such drawings.

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### References

1. 1.
Alamdari, S., Chan, T.M., Grant, E., Lubiw, A., Pathak, V.: Self-approaching Graphs. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 260–271. Springer, Heidelberg (2013)
2. 2.
Angelini, P., Colasante, E., Di Battista, G., Frati, F., Patrignani, M.: Monotone drawings of graphs. J. Graph Algorithms Appl. 16(1), 5–35 (2012)
3. 3.
Angelini, P., Di Battista, G., Frati, F.: Succinct greedy drawings do not always exist. Networks 59(3), 267–274 (2012)
4. 4.
Angelini, P., Didimo, W., Kobourov, S., Mchedlidze, T., Roselli, V., Symvonis, A., Wismath, S.: Monotone drawings of graphs with fixed embedding. In: Speckmann, B. (ed.) GD 2011. LNCS, vol. 7034, pp. 379–390. Springer, Heidelberg (2012)
5. 5.
Angelini, P., Frati, F., Grilli, L.: An algorithm to construct greedy drawings of triangulations. J. Graph Algorithms Appl. 14(1), 19–51 (2010)
6. 6.
Dhandapani, R.: Greedy drawings of triangulations. Discrete Comput. Geom. 43, 375–392 (2010)
7. 7.
Eppstein, D., Goodrich, M.T.: Succinct greedy geometric routing using hyperbolic geometry. IEEE Trans. Computers 60(11), 1571–1580 (2011)
8. 8.
Felsner, S.: Geometric Graphs and Arrangements. Vieweg+Teubner Verlag (2004)Google Scholar
9. 9.
Goodrich, M.T., Strash, D.: Succinct greedy geometric routing in the Euclidean plane. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 781–791. Springer, Heidelberg (2009)
10. 10.
Huang, W., Eades, P., Hong, S.-H.: A graph reading behavior: Geodesic-path tendency. In: IEEE Pacific Visualization Symposium (PacificVis 2009), pp. 137–144 (2009)Google Scholar
11. 11.
Icking, C., Klein, R., Langetepe, E.: Self-approaching curves. Math. Proc. Camb. Phil. Soc. 125, 441–453 (1999)
12. 12.
Kleinberg, R.: Geographic routing using hyperbolic space. In: Computer Communications (INFOCOM 2007), pp. 1902–1909. IEEE (2007)Google Scholar
13. 13.
Moitra, A., Leighton, T.: Some Results on Greedy Embeddings in Metric Spaces. Discrete Comput. Geom. 44, 686–705 (2010)
14. 14.
Nöllenburg, M., Prutkin, R.: Euclidean greedy drawings of trees. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 767–778. Springer, Heidelberg (2013)
15. 15.
Nöllenburg, M., Prutkin, R., Rutter, I.: On Self-Approaching and Increasing-Chord Drawings of 3-Connected Planar Graphs. CoRR arXiv:1409.0315 (2014)Google Scholar
16. 16.
Papadimitriou, C.H., Ratajczak, D.: On a conjecture related to geometric routing. Theor. Comput. Sci. 344(1), 3–14 (2005)
17. 17.
Purchase, H.C., Hamer, J., Nöllenburg, M., Kobourov, S.G.: On the usability of Lombardi graph drawings. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 451–462. Springer, Heidelberg (2013)
18. 18.
Rao, A., Ratnasamy, S., Papadimitriou, C., Shenker, S., Stoica, I.: Geographic routing without location information. In: Mobile Computing and Networking (MobiCom 2003), pp. 96–108. ACM (2003)Google Scholar
19. 19.
Rote, G.: Curves with increasing chords. Math. Proc. Camb. Phil. Soc. 115, 1–12 (1994)
20. 20.
Schnyder, W.: Embedding planar graphs on the grid. In: Discrete Algorithms (SODA 1990), pp. 138–148. SIAM (1990)Google Scholar
21. 21.
Wang, J.-J., He, X.: Succinct strictly convex greedy drawing of 3-connected plane graphs. Theor. Comput. Sci. 532, 80–90 (2014)