Geodesic
Matched Drawings of Planar Graphs
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. 423-445.

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A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which is a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3-connected planar graphs or a 3-connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a sufficiently restricted planar graph-such as an unlabeled level planar (ULP) graph or a graph of the family of "carousel graphs"-are always matched drawable. 
@article{JGAA_2009_13_3_a5,
     author = {Emilio Di Giacomo and Walter Didimo and Marc van Kreveld and Giuseppe Liotta and Bettina Speckmann},
     title = {Matched {Drawings} of {Planar} {Graphs}},
     journal = {Journal of Graph Algorithms and Applications},
     pages = {423--445},
     publisher = {mathdoc},
     volume = {13},
     number = {3},
     year = {2009},
     doi = {10.7155/jgaa.00193},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00193/}
}
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Emilio Di Giacomo; Walter Didimo; Marc van Kreveld; Giuseppe Liotta; Bettina Speckmann. Matched Drawings of Planar Graphs. 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. 423-445. doi : 10.7155/jgaa.00193. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00193/

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