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GraphMaps: Browsing Large Graphs as Interactive Maps Read chapter Read first chapter

2015 | OriginalPaper | Chapter

Recognizing and Drawing IC-Planar Graphs

Authors : Franz J. Brandenburg, Walter Didimo, William S. Evans, Philipp Kindermann, Giuseppe Liotta, Fabrizio Montecchiani

Published in: Graph Drawing and Network Visualization

Publisher: Springer International Publishing

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Abstract

IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph G with n vertices, we present an O(n)-time algorithm that computes a straight-line drawing of G in quadratic area, and an \(O(n^3)\)-time algorithm that computes a straight-line drawing of G with right-angle crossings in exponential area. Both these area requirements are worst-case optimal. We also show that it is \(\mathrm {NP}\)-complete to test IC-planarity both in the general case and in the case in which a rotation system is fixed for the input graph. Furthermore, we describe a polynomial-time algorithm to test whether a set of matching edges can be added to a triangulated planar graph such that the resulting graph is IC-planar.

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previous chapter 2-Layer Fan-Planarity: From Caterpillar to Stegosaurus
next chapter Simple Realizability of Complete Abstract Topological Graphs Simplified
Literature
1.
Ackerman, E.: On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete Comput. Geom. 41(3), 365–375 (2009)MathSciNetCrossRefMATH
2.
Alam, M.J., Brandenburg, F.J., Kobourov, S.G.: Straight-line grid drawings of 3-connected 1-planar graphs. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 83–94. Springer, Heidelberg (2013) CrossRef
3.
Argyriou, E.N., Bekos, M.A., Symvonis, A.: The straight-line RAC drawing problem is NP-hard. J. Graph Algorithms Appl. 16(2), 569–597 (2012)MathSciNetCrossRefMATH
4.
Auer, C., Bachmaier, C., Brandenburg, F.J., Gleißner, A., Hanauer, K., Neuwirth, D., Reislhuber, J.: Recognizing outer 1-planar graphs in linear time. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 107–118. Springer, Heidelberg (2013) CrossRef
5.
Auer, C., Brandenburg, F.J., Gleiner, A., Reislhuber, J.: 1-planarity of graphs with a rotation system. J. Graph Algorithms Appl. 19(1), 67–86 (2015)MathSciNetCrossRefMATH
6.
Bekos, M.A., Cornelsen, S., Grilli, L., Hong, S.-H., Kaufmann, M.: On the recognition of fan-planar and maximal outer-fan-planar graphs. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 198–209. Springer, Heidelberg (2014)
7.
Binucci, C., Di Giacomo, E., Didimo, W., Montecchiani, F., Patrignani, M., Symvonis, A., Tollis, I.G.: Fan-planarity: properties and complexity. Theor. Comput. Sci. 589, 76–85 (2015)CrossRef
8.
Borodin, O.V.: Solution of the Ringel problem on vertex-face coloring of planar graphs and coloring of \(1\)-planar graphs. Metody Diskret Analiz. 41, 12–26 (1984)MATH
9.
Brandenburg, F.J., Eppstein, D., Gleißner, A., Goodrich, M.T., Hanauer, K., Reislhuber, J.: On the density of maximal 1-planar graphs. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 327–338. Springer, Heidelberg (2013) CrossRef
10.
11.
Cheong, O., Har-Peled, S., Kim, H., Kim, H.-S.: On the number of edges of fan-crossing free graphs. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) Algorithms and Computation. LNCS, vol. 8283, pp. 163–173. Springer, Heidelberg (2013) CrossRef
12.
Chrobak, M., Payne, T.H.: A linear-time algorithm for drawing a planar graph on a grid. Inf. Process. Lett. 54(4), 241–246 (1995)MathSciNetCrossRefMATH
13.
14.
15.
Di Giacomo, E., Didimo, W., Liotta, G., Montecchiani, F.: h-Quasi planar drawings of bounded treewidth graphs in linear area. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds.) WG 2012. LNCS, vol. 7551, pp. 91–102. Springer, Heidelberg (2012) CrossRef
16.
Di Giacomo, E., Didimo, W., Liotta, G., Montecchiani, F.: Area requirement of graph drawings with few crossings per edge. Comput. Geom. 46(8), 909–916 (2013)MathSciNetCrossRefMATH
17.
18.
Didimo, W., Liotta, G.: The crossing angle resolution in graph drawing. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 167–184. Springer, New York (2012)
19.
20.
21.
22.
Hong, S., Eades, P., Katoh, N., Liotta, G., Schweitzer, P., Suzuki, Y.: A linear-time algorithm for testing outer-1-planarity. Algorithmica 46, 1–22 (2014)
23.
Hong, S.-H., Eades, P., Liotta, G., Poon, S.-H.: Fáry’s theorem for 1-planar graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 335–346. Springer, Heidelberg (2012) CrossRef
24.
25.
Korzhik, V.P., Mohar, B.: Minimal obstructions for 1-immersions and hardness of 1-planarity testing. J. Graph Theor. 72(1), 30–71 (2013)MathSciNetCrossRefMATH
26.
Král, D., Stacho, L.: Coloring plane graphs with independent crossings. J. Graph Theor. 64(3), 184–205 (2010)MATH
27.
Liotta, G.: Graph drawing beyond planarity: some results and open problems. In: Theoretical Computer Science (ICTCS 2014), pp. 3–8 (2014)
28.
29.
30.
Zhang, X.: Drawing complete multipartite graphs on the plane with restrictions on crossings. Acta Math. Sinica 30(12), 2045–2053 (2014)CrossRefMATH
31.
Zhang, X., Liu, G.: The structure of plane graphs with independent crossings and its applications to coloring problems. Central Eur. J. Math. 11(2), 308–321 (2013)MATH
Metadata
Title
Recognizing and Drawing IC-Planar Graphs
Authors
Franz J. Brandenburg
Walter Didimo
William S. Evans
Philipp Kindermann
Giuseppe Liotta
Fabrizio Montecchiani
Copyright Year
2015
Book
Graph Drawing and Network Visualization

Print ISBN: 978-3-319-27260-3

Electronic ISBN: 978-3-319-27261-0

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-27261-0_25

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