Geodesic
Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces
Journal of Graph Algorithms and Applications, Special Issue on Selected Papers from the Fifteenth International Symposium on Graph Drawing, GD 2007 , Tome 13 (2009) no. 3, pp. 349-378.

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Let C be a clustered graph and suppose that the planar embedding of its underlying graph is fixed. Is testing the c-planarity of C easier than in the variable embedding setting? In this paper we give a first contribution towards answering the above question. Namely, we characterize c-planar embedded flat clustered graphs with at most five vertices per face and give an efficient testing algorithm for such graphs. The results are based on a more general methodology that sheds new light on the c-planarity testing problem. 
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     author = {Giuseppe Di Battista and Fabrizio Frati},
     title = {Efficient {C-Planarity} {Testing} for {Embedded} {Flat} {Clustered} {Graphs} with {Small} {Faces}},
     journal = {Journal of Graph Algorithms and Applications},
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Giuseppe Di Battista; Fabrizio Frati. Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces. Journal of Graph Algorithms and Applications, 
							Special Issue on Selected Papers from the Fifteenth International Symposium on Graph Drawing, GD 2007
					, Tome 13 (2009) no. 3, pp. 349-378. doi : 10.7155/jgaa.00191. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00191/

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