{"abstract":[{"lang":"eng","text":"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."}],"acknowledgement":"I would like to thank Jan Kynčl, Dömötör Pálvölgyi and anonymous referees for many comments and suggestions that helped to improve the presentation of the result.","_id":"794","year":"2017","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","date_published":"2017-12-01T00:00:00Z","date_updated":"2021-01-12T08:16:07Z","main_file_link":[{"url":"https://arxiv.org/abs/1602.01346","open_access":"1"}],"publication":"Computational Geometry: Theory and Applications","doi":"10.1016/j.comgeo.2017.06.016","author":[{"first_name":"Radoslav","last_name":"Fulek","full_name":"Fulek, Radoslav","orcid":"0000-0001-8485-1774","id":"39F3FFE4-F248-11E8-B48F-1D18A9856A87"}],"day":"01","oa_version":"Preprint","related_material":{"record":[{"id":"1165","relation":"earlier_version","status":"public"}]},"month":"12","oa":1,"publisher":"Elsevier","page":"1 - 13","publist_id":"6860","scopus_import":1,"quality_controlled":"1","type":"journal_article","intvolume":" 66","department":[{"_id":"UlWa"}],"publication_status":"published","date_created":"2018-12-11T11:48:32Z","title":"C-planarity of embedded cyclic c-graphs","citation":{"ama":"Fulek R. C-planarity of embedded cyclic c-graphs. Computational Geometry: Theory and Applications. 2017;66:1-13. doi:10.1016/j.comgeo.2017.06.016","short":"R. Fulek, Computational Geometry: Theory and Applications 66 (2017) 1–13.","apa":"Fulek, R. (2017). C-planarity of embedded cyclic c-graphs. Computational Geometry: Theory and Applications. Elsevier. https://doi.org/10.1016/j.comgeo.2017.06.016","chicago":"Fulek, Radoslav. “C-Planarity of Embedded Cyclic c-Graphs.” Computational Geometry: Theory and Applications. Elsevier, 2017. https://doi.org/10.1016/j.comgeo.2017.06.016.","mla":"Fulek, Radoslav. “C-Planarity of Embedded Cyclic c-Graphs.” Computational Geometry: Theory and Applications, vol. 66, Elsevier, 2017, pp. 1–13, doi:10.1016/j.comgeo.2017.06.016.","ista":"Fulek R. 2017. C-planarity of embedded cyclic c-graphs. Computational Geometry: Theory and Applications. 66, 1–13.","ieee":"R. Fulek, “C-planarity of embedded cyclic c-graphs,” Computational Geometry: Theory and Applications, vol. 66. Elsevier, pp. 1–13, 2017."},"status":"public","language":[{"iso":"eng"}],"volume":66}