article published yes Radoslav Fulek author 39F3FFE4-F248-11E8-B48F-1D18A9856A870000-0001-8485-1774 Jan Kynčl author Igor Malinovič author Dömötör Pálvölgyi author UlWa department International IST Postdoc Fellowship Programme project 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. https://research-explorer.ista.ac.at/download/1642/5120/IST-2016-714-v1+1_5002-15499-3-PB.pdf application/pdfno Electronic Journal of Combinatorics2015 eng 1077-8926 1305.451910.37236/5002 224 https://research-explorer.ista.ac.at/record/10793 Fulek R, Kynčl J, Malinovič I, Pálvölgyi D. Clustered planarity testing revisited. <i>Electronic Journal of Combinatorics</i>. 2015;22(4). doi:<a href="https://doi.org/10.37236/5002">10.37236/5002</a> R. Fulek, J. Kynčl, I. Malinovič, D. Pálvölgyi, Electronic Journal of Combinatorics 22 (2015). R. Fulek, J. Kynčl, I. Malinovič, and D. Pálvölgyi, “Clustered planarity testing revisited,” <i>Electronic Journal of Combinatorics</i>, vol. 22, no. 4. Electronic Journal of Combinatorics, 2015. Fulek, Radoslav, Jan Kynčl, Igor Malinovič, and Dömötör Pálvölgyi. “Clustered Planarity Testing Revisited.” <i>Electronic Journal of Combinatorics</i>. Electronic Journal of Combinatorics, 2015. <a href="https://doi.org/10.37236/5002">https://doi.org/10.37236/5002</a>. 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.” <i>Electronic Journal of Combinatorics</i>, vol. 22, no. 4, P4.24, Electronic Journal of Combinatorics, 2015, doi:<a href="https://doi.org/10.37236/5002">10.37236/5002</a>. Fulek, R., Kynčl, J., Malinovič, I., &#38; Pálvölgyi, D. (2015). Clustered planarity testing revisited. <i>Electronic Journal of Combinatorics</i>. Electronic Journal of Combinatorics. <a href="https://doi.org/10.37236/5002">https://doi.org/10.37236/5002</a> 16422018-12-11T11:53:12Z2025-06-26T07:57:45Z