Fulek, RadoslavISTA ; Kynčl, Jan; Malinovič, Igor; Pálvölgyi, Dömötör
The Hanani-Tutte theorem is a classical result proved for the first time in the 1930s that characterizes planar graphs as graphs that admit a drawing in the plane in which every pair of edges not sharing a vertex cross an even number of times. We generalize this result to clustered graphs with two disjoint clusters, and show that a straightforward extension to flat clustered graphs with three or more disjoint clusters is not possible. For general clustered graphs we show a variant of the Hanani-Tutte theorem in the case when each cluster induces a connected subgraph. Di Battista and Frati proved that clustered planarity of embedded clustered graphs whose every face is incident to at most five vertices can be tested in polynomial time. We give a new and short proof of this result, using the matroid intersection algorithm.
Electronic Journal of Combinatorics
e research leading to these results has received funding fromthe People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme(FP7/2007-2013) under REA grant agreement no , and ESF Eurogiga project GraDR as GAˇCRGIG/11/E023.
Fulek R, Kynčl J, Malinovič I, Pálvölgyi D. Clustered planarity testing revisited. Electronic Journal of Combinatorics. 2015;22(4). doi:10.37236/5002
Fulek, R., Kynčl, J., Malinovič, I., & Pálvölgyi, D. (2015). Clustered planarity testing revisited. Electronic Journal of Combinatorics. Electronic Journal of Combinatorics. https://doi.org/10.37236/5002
Fulek, Radoslav, Jan Kynčl, Igor Malinovič, and Dömötör Pálvölgyi. “Clustered Planarity Testing Revisited.” Electronic Journal of Combinatorics. Electronic Journal of Combinatorics, 2015. https://doi.org/10.37236/5002.
R. Fulek, J. Kynčl, I. Malinovič, and D. Pálvölgyi, “Clustered planarity testing revisited,” Electronic Journal of Combinatorics, vol. 22, no. 4. Electronic Journal of Combinatorics, 2015.
Fulek R, Kynčl J, Malinovič I, Pálvölgyi D. 2015. Clustered planarity testing revisited. Electronic Journal of Combinatorics. 22(4), P4.24.
Fulek, Radoslav, et al. “Clustered Planarity Testing Revisited.” Electronic Journal of Combinatorics, vol. 22, no. 4, P4.24, Electronic Journal of Combinatorics, 2015, doi:10.37236/5002.
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