Two-Layer Planarization: Improving on Parameterized Algorithmics - Fernau, Henning (Journal of Graph Algorithms and Applications, 2005)
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. <!—->