Abstract
We prove that every planar graph is an intersection graph of strings in the plane such that any two strings intersect at most once.
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de Castro, N., Cobos, F., Dana, J.C., Márquez, A., Noy, M.: Triangle-free planar graphs as segment intersection graphs. J. Graph Algorithms Appl. 6(1), 7–26 (2002)
Chalopin, J., Gonçalves, D.: Every planar graph is the intersection graph of segments in the plane. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing (2009)
Ehrlich, G., Even, S., Tarjan, R.E.: Intersection graphs of curves in the plane. J. Combin. Theory., Ser. B 21, 8–20 (1976)
de Fraysseix, H., Ossona de Mendez, P., Pach, J.: Representation of planar graphs by segments. Intuit. Geom. (Szeged, 1991), Colloq. Math. Soc. János Bolyai 63, 109–117 (1994)
de Fraysseix, H., Ossona de Mendez, P.: Intersection graphs of Jordan arcs. DIMACS Ser. Discrete Math. Theor. Comput. Sci. 49, 11–28 (1999)
de Fraysseix, H., Ossona de Mendez, P.: Representations by contact and intersection of segments. Algorithmica 47(4), 453–463 (2007)
Gonçalves, D.: Edge-partition of planar graphs into two outerplanar graphs. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pp. 504–512 (2005)
Hartman, I.B.-A., Newman, I., Ziv, R.: On grid intersection graphs. Discrete Math. 87(1), 41–52 (1991)
Koebe, P.: Kontaktprobleme der Konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.—Phys. Kl. 88, 141–164 (1936)
Scheinerman, E.R.: Intersection classes and multiple intersection parameters of graphs. PhD Thesis, Princeton University (1984)
West, D.: Open problems. SIAM J. Discrete Math. Newslett. 2(1), 10–12 (1991)
Whitney, H.: A theorem on graphs. Ann. Math. (2) 32(2), 378–390 (1931)
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An abstract of this paper appeared in the Proceedings of the eighteenth annual ACM–SIAM Symposium on Discrete algorithms (SODA 2007).
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Chalopin, J., Gonçalves, D. & Ochem, P. Planar Graphs Have 1-string Representations. Discrete Comput Geom 43, 626–647 (2010). https://doi.org/10.1007/s00454-009-9196-9
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DOI: https://doi.org/10.1007/s00454-009-9196-9