COCOON 2012: Computing and Combinatorics pp 335-346

# Fáry’s Theorem for 1-Planar Graphs

• Seok-Hee Hong
• Giuseppe Liotta
• Sheung-Hung Poon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

## Abstract

A plane graph is a graph embedded in a plane without edge crossings. Fáry’s theorem states that every plane graph can be drawn as a straight-line drawing, preserving the embedding of the plane graph. In this paper, we extend Fáry’s theorem to a class of non-planar graphs. More specifically, we study the problem of drawing 1-plane graphs with straight-line edges. A 1-plane graph is a graph embedded in a plane with at most one crossing per edge. We give a characterisation of those 1-plane graphs that admit a straight-line drawing. The proof of the characterisation consists of a linear time testing algorithm and a drawing algorithm. Further, we show that there are 1-plane graphs for which every straight-line drawing has exponential area. To the best of our knowledge, this is the first result to extend Fáry’s theorem to non-planar graphs.

## Keywords

Convex Polygon Topological Graph Linear Time Algorithm Outer Face Virtual Edge
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Borodin, O.V., Kostochka, A.V., Raspaud, A., Sopena, E.: Acyclic colouring of 1-planar graphs. Discrete Applied Mathematics 114(1-3), 29–41 (2001)
2. 2.
Chiba, N., Yamanouchi, T., Nishizeki, T.: Linear time algorithms for convex drawings of planar graphs. In: Progress in Graph Theory, pp. 153–173. Academic Press (1984)Google Scholar
3. 3.
Chrobak, M., Eppstein, D.: Planar Orientations with Low Out-degree and Compaction of Adjacency Matrices. Theor. Comput. Sci. 86(2), 243–266 (1991)
4. 4.
Fabrici, I., Madaras, T.: The structure of 1-planar graphs. Discrete Mathematics 307(7-8), 854–865 (2007)
5. 5.
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)
6. 6.
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall (1999)Google Scholar
7. 7.
Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM J. on Comput. 25(5), 956–997 (1996)
8. 8.
Fáry, I.: On straight line representations of planar graphs. Acta Sci. Math. Szeged 11, 229–233 (1948)
9. 9.
Hong, S., Nagamochi, H.: An algorithm for constructing star-shaped drawings of plane graphs. Comput. Geom. 43(2), 191–206 (2010)
10. 10.
Hopcroft, J.E., Tarjan, R.E.: Dividing a graph into triconnected components. SIAM J. on Comput. 2, 135–158 (1973)
11. 11.
Korzhik, V.P., Mohar, B.: Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 302–312. Springer, Heidelberg (2009)
12. 12.
Kuratowski, K.: Sur le problme des courbes gauches en topologie. Fund. Math. 15, 271–283 (1930)
13. 13.
Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. World Scientific (2004)Google Scholar
14. 14.
Pach, J., Toth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997)
15. 15.
Read, R.C.: A new method for drawing a planar graph given the cyclic order of the edges at each vertex. Congr. Numer. 56, 31–44 (1987)
16. 16.
Suzuki, Y.: Optimal 1-planar graphs which triangulate other surfaces. Discrete Mathematics 310(1), 6–11 (2010)
17. 17.
Tutte, W.T.: How to draw a graph. Proc. of the London Mathematical Society 13, 743–767 (1963)

## Authors and Affiliations

• Seok-Hee Hong
• 1
• 1
• Giuseppe Liotta
• 2
• Sheung-Hung Poon
• 3
1. 1.University of SydneyAustralia
2. 2.University of PerugiaItaly
3. 3.National Tsing Hua UniversityTaiwan