# Variants of the Segment Number of a Graph

• Yoshio Okamoto
• Alexander Ravsky
• Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)

## Abstract

The segment number of a planar graph is the smallest number of line segments whose union represents a crossing-free straight-line drawing of the given graph in the plane. The segment number is a measure for the visual complexity of a drawing; it has been studied extensively.

In this paper, we study three variants of the segment number: for planar graphs, we consider crossing-free polyline drawings in 2D; for arbitrary graphs, we consider crossing-free straight-line drawings in 3D and straight-line drawings with crossings in 2D. We first construct an infinite family of planar graphs where the classical segment number is asymptotically twice as large as each of the new variants of the segment number. Then we establish the $$\exists \mathbb {R}$$-completeness (which implies the NP-hardness) of all variants. Finally, for cubic graphs, we prove lower and upper bounds on the new variants of the segment number, depending on the connectivity of the given graph.

## Notes

### Acknowledgments

We thank the organizers and participants of the 2019 Dagstuhl seminar “Beyond-planar graphs: Combinatorics, Models and Algorithms”. In particular, we thank Günter Rote and Martin Gronemann for suggestions that led to some of this research. We also thank Carlos Alegría. We thank our reviewers for an idea that improved the bound in Proposition 5, for suggesting the statement of Lemma 1, and for many other helpful comments.

## References

1. 1.
Bose, P., Everett, H., Wismath, S.K.: Properties of arrangement graphs. Int. J. Comput. Geom. Appl. 13(6), 447–462 (2003).
2. 2.
Chaplick, S., Fleszar, K., Lipp, F., Ravsky, A., Verbitsky, O., Wolff, A.: Drawing graphs on few lines and few planes. In: Hu, Y., Nöllenburg, M. (eds.) GD 2016. LNCS, vol. 9801, pp. 166–180. Springer, Cham (2016).
3. 3.
Chaplick, S., Fleszar, K., Lipp, F., Ravsky, A., Verbitsky, O., Wolff, A.: The complexity of drawing graphs on few lines and few planes. In: Ellen, F., Kolokolova, A., Sack, J.R. (eds.) Algorithms and Data Structures. LNCS, vol. 10389, pp. 265–276. Springer, Cham (2017). . arxiv.org/1607.06444
4. 4.
Chartrand, G., Zhang, P.: Chromatic Graph Theory, 1st edn. Chapman & Hall/CRC, Boca Raton (2008)
5. 5.
Dujmović, V., Eppstein, D., Suderman, M., Wood, D.R.: Drawings of planar graphs with few slopes and segments. Comput. Geom. Theory Appl. 38(3), 194–212 (2007).
6. 6.
Durocher, S., Mondal, D.: Drawing plane triangulations with few segments. In: Proceedings Canadian Conference on Computational Geometry (CCCG 2014), pp. 40–45 (2014). http://cccg.ca/proceedings/2014/papers/paper06.pdf
7. 7.
Durocher, S., Mondal, D., Nishat, R., Whitesides, S.: A note on minimum-segment drawings of planar graphs. J. Graph Algorithms Appl. 17(3), 301–328 (2013).
8. 8.
Eppstein, D.: Drawing arrangement graphs in small grids, or how to play planarity. J. Graph Algorithms Appl. 18(2), 211–231 (2014).
9. 9.
Even, S., Tarjan, R.E.: Computing an $$st$$-numbering. Theoret. Comput. Sci. 2(3), 339–344 (1976).
10. 10.
Hoffmann, U.: On the complexity of the planar slope number problem. J. Graph Algorithms Appl. 21(2), 183–193 (2017).
11. 11.
Hültenschmidt, G., Kindermann, P., Meulemans, W., Schulz, A.: Drawing planar graphs with few geometric primitives. In: Bodlaender, H.L., Woeginger, G.J. (eds.) WG 2017. LNCS, vol. 10520, pp. 316–329. Springer, Cham (2017).
12. 12.
Igamberdiev, A., Meulemans, W., Schulz, A.: Drawing planar cubic 3-connected graphs with few segments: algorithms and experiments. J. Graph Algorithms Appl. 21(4), 561–588 (2017).
13. 13.
Kindermann, P., Mchedlidze, T., Schneck, T., Symvonis, A.: Drawing planar graphs with few segments on a polynomial grid. In: Archambault, D., Tóth, C.D. (eds.) GD 2019. LNCS, vol. 11904, pp. 416–429. Springer. Cham (2019). https://arxiv.org/abs/1903.08496
14. 14.
Liu, Y., Marchioro, P., Petreschi, R.: At most single-bend embeddings of cubic graphs. Appl. Math. 9(2), 127–142 (1994).
15. 15.
Matoušek, J.: Intersection graphs of segments and $$\exists \mathbb{R}$$. ArXiv report (2014). http://arxiv.org/abs/1406.2636
16. 16.
Mondal, D., Nishat, R.I., Biswas, S., Rahman, M.S.: Minimum-segment convex drawings of 3-connected cubic plane graphs. J. Comb. Optim. 25(3), 460–480 (2013).
17. 17.
Mukkamala, P., Szegedy, M.: Geometric representation of cubic graphs with four directions. Comput. Geom. Theory Appl. 42(9), 842–851 (2009).
18. 18.
Schaefer, M.: Complexity of some geometric and topological problems. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 334–344. Springer, Heidelberg (2010).
19. 19.
Schulz, A.: Drawing graphs with few arcs. J. Graph Algorithms Appl. 19(1), 393–412 (2015).
20. 20.
Shor, P.W.: Stretchability of pseudolines is NP-hard. In: Gritzmann, P., Sturmfels, B. (eds.) Applied Geometry and Discrete Mathematics-The Victor Klee Festschrift. DIMACS Series in Mathematics and Theoretical Computer Science, vol. 4, pp. 531–554. American Mathematical Society (1991)Google Scholar