We show that c-planarity is solvable in quadratic time for flat clustered graphs with three clusters if the combinatorial embedding of the underlying graph is fixed. In simpler graph-theoretical terms our result can be viewed as follows. Given a graph G with the vertex set partitioned into three parts embedded on a 2-sphere, our algorithm decides if we can augment G by adding edges without creating an edge-crossing so that in the resulting spherical graph the vertices of each part induce a connected sub-graph. We proceed by a reduction to the problem of testing the existence of a perfect matching in planar bipartite graphs. We formulate our result in a slightly more general setting of cyclic clustered graphs, i.e., the simple graph obtained by contracting each cluster, where we disregard loops and multi-edges, is a cycle.
R. Fulek—The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no . I would like to thank Jan Kynčl and Dömötör Pálvölgyi for many comments and suggestions that helped to improve the presentation of the result.
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GD: Graph Drawing and Network Visualization
2016-09-19 – 2016-09-21
Fulek R. C-planarity of embedded cyclic c-graphs. In: Vol 9801. Springer; 2016:94-106. doi:10.1007/978-3-319-50106-2_8
Fulek, R. (2016). C-planarity of embedded cyclic c-graphs (Vol. 9801, pp. 94–106). Presented at the GD: Graph Drawing and Network Visualization, Athens, Greece: Springer. https://doi.org/10.1007/978-3-319-50106-2_8
Fulek, Radoslav. “C-Planarity of Embedded Cyclic c-Graphs,” 9801:94–106. Springer, 2016. https://doi.org/10.1007/978-3-319-50106-2_8.
R. Fulek, “C-planarity of embedded cyclic c-graphs,” presented at the GD: Graph Drawing and Network Visualization, Athens, Greece, 2016, vol. 9801, pp. 94–106.
Fulek R. 2016. C-planarity of embedded cyclic c-graphs. GD: Graph Drawing and Network Visualization, LNCS, vol. 9801. 94–106.
Fulek, Radoslav. C-Planarity of Embedded Cyclic c-Graphs. Vol. 9801, Springer, 2016, pp. 94–106, doi:10.1007/978-3-319-50106-2_8.
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