Geodesic
Two-Layer Planarization: Improving on Parameterized Algorithmics
Journal of Graph Algorithms and Applications, Tome 9 (2005) no. 2, pp. 205-238.

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A bipartite graph is biplanar if the vertices can be placed on two parallel lines in the plane such that there are no edge crossings when edges are drawn as straight-line segments connecting vertices on one line to vertices on the other line. We study two problems: 2-LAYER PLANARIZATION: can k edges be deleted from a given graph G so that the remaining graph is biplanar? 1-LAYER PLANARIZATION: same question, but the order of the vertices on one layer is fixed. Improving on earlier works of Dujmovi\'c et al. (Proc. Graph Drawing GD 2001, pp. 1-15, 2002), we solve the 2-LAYER PLANARIZATION problem in O(k2·5.1926k+|G|) time and the 1-LAYER PLANARIZATION problem in O(k3·2.5616k+|G|2) time. Moreover, we derive a small problem kernel for 1-LAYER PLANARIZATION. 
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     author = {Henning Fernau},
     title = {Two-Layer {Planarization:} {Improving} on {Parameterized} {Algorithmics}},
     journal = {Journal of Graph Algorithms and Applications},
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     doi = {10.7155/jgaa.00106},
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     url = {http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00106/}
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Henning Fernau. Two-Layer Planarization: Improving on Parameterized Algorithmics. Journal of Graph Algorithms and Applications, Tome 9 (2005) no. 2, pp. 205-238. doi : 10.7155/jgaa.00106. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00106/

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