GD 2008: Graph Drawing pp 50-60

# Cubic Graphs Have Bounded Slope Parameter

• Balázs Keszegh
• János Pach
• Dömötör Pálvölgyi
• Géza Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)

## Abstract

We show that every finite connected graph G with maximum degree three and with at least one vertex of degree smaller than three has a straight-line drawing in the plane satisfying the following conditions. No three vertices are collinear, and a pair of vertices form an edge in G if and only if the segment connecting them is parallel to one of the sides of a previously fixed regular pentagon. It is also proved that every finite graph with maximum degree three permits a straight-line drawing with the above properties using only at most seven different edge slopes.

### References

1. 1.
Ambrus, G., Barát, J., Hajnal, P.: The slope parameter of graphs. Acta Sci. Math.(Szeged) 72(3–4), 875–889 (2006)
2. 2.
Barát, J., Matoušek, J., Wood, D.R.: Bounded-degree graphs have arbitrarily large geometric thickness. Electr. J. Combin. 13(1), R3, 14pp. (2006)Google Scholar
3. 3.
Dillencourt, M.B., Eppstein, D., Hirschberg, D.S.: Geometric thickness of complete graphs. J. Graph Algorithms Appl. 4(3), 5–17 (2000)
4. 4.
Dujmović, V., Suderman, M., Wood, D.R.: Graph drawings with few slopes. Comput. Geom. 38, 181–193 (2007)
5. 5.
Dujmović, V., Wood, D.R.: Graph treewidth and geometric thickness parameters. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 129–140. Springer, Heidelberg (2006)
6. 6.
Duncan, C.A., Eppstein, D., Kobourov, S.G.: The geometric thickness of low degree graphs. In: SoCG 2004, pp. 340–346. ACM Press, New York (2004)Google Scholar
7. 7.
Eppstein, D.: Separating thickness from geometric thickness. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Math, vol. 342, pp. 75–86. AMS, Providence (2004)
8. 8.
Fáry, I.: On straight line representation of planar graphs. Acta Univ. Szeged. Sect. Sci. Math. 11, 229–233 (1948)
9. 9.
Hutchinson, J.P., Shermer, T.C., Vince, A.: On representations of some thickness-two graphs. Comput. Geom. 13, 161–171 (1999)
10. 10.
Jamison, R.E.: Few slopes without collinearity. Discrete Math. 60, 199–206 (1986)
11. 11.
Kainen, P.C.: Thickness and coarseness of graphs. Abh. Math. Sem. Univ. Hamburg 39, 88–95 (1973)
12. 12.
Keszegh, B., Pach, J., Pálvölgyi, D., Tóth, G.: Drawing cubic graphs with at most five slopes. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 114–125. Springer, Heidelberg (2007)
13. 13.
Mukkamala, P., Szegedy, M.: Geometric representation of cubic graphs with four directions (manuscript, 2007)Google Scholar
14. 14.
Mutzel, P., Odenthal, T., Scharbrodt, M.: The thickness of graphs: a survey. Graphs Combin. 14, 59–73 (1998)
15. 15.
Pach, J., Pálvölgyi, D.: Bounded-degree graphs can have arbitrarily large slope numbers. Electr. J. Combin. 13(1), Note 1, 4pp. (2006)Google Scholar
16. 16.
Wade, G.A., Chu, J.-H.: Drawability of complete graphs using a minimal slope set. The Computer J. 37, 139–142 (1994)

© Springer-Verlag Berlin Heidelberg 2009

## Authors and Affiliations

• Balázs Keszegh
• 1
• 5
• János Pach
• 2
• 4
• 5
• Dömötör Pálvölgyi
• 3
• 4
• Géza Tóth
• 5
1. 1.Central European UniversityBudapestHungary
2. 2.City College, CUNYNew YorkUSA
3. 3.Eötvös UniversityBudapestHungary
4. 4.Ecole Polytechnique Fédérale de LausanneSwitzerland
5. 5.A. Rényi Institute of MathematicsBudapestHungary