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.
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References
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)
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)
Chrobak, M., Eppstein, D.: Planar Orientations with Low Out-degree and Compaction of Adjacency Matrices. Theor. Comput. Sci. 86(2), 243–266 (1991)
Fabrici, I., Madaras, T.: The structure of 1-planar graphs. Discrete Mathematics 307(7-8), 854–865 (2007)
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall (1999)
Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM J. on Comput. 25(5), 956–997 (1996)
Fáry, I.: On straight line representations of planar graphs. Acta Sci. Math. Szeged 11, 229–233 (1948)
Hong, S., Nagamochi, H.: An algorithm for constructing star-shaped drawings of plane graphs. Comput. Geom. 43(2), 191–206 (2010)
Hopcroft, J.E., Tarjan, R.E.: Dividing a graph into triconnected components. SIAM J. on Comput. 2, 135–158 (1973)
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)
Kuratowski, K.: Sur le problme des courbes gauches en topologie. Fund. Math. 15, 271–283 (1930)
Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. World Scientific (2004)
Pach, J., Toth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997)
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)
Suzuki, Y.: Optimal 1-planar graphs which triangulate other surfaces. Discrete Mathematics 310(1), 6–11 (2010)
Tutte, W.T.: How to draw a graph. Proc. of the London Mathematical Society 13, 743–767 (1963)
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Hong, SH., Eades, P., Liotta, G., Poon, SH. (2012). Fáry’s Theorem for 1-Planar Graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_29
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DOI: https://doi.org/10.1007/978-3-642-32241-9_29
Publisher Name: Springer, Berlin, Heidelberg
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