Two-layer planarization: Improving on parameterized algorithmics

Abstract

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. 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 Dujmović et al., we solve the 2-LAYER PLANARIZATION problem in script O sign(k2 · 5.1926k + |G|) time and the 1-LAYER PLANARIZATION problem in script O sign(k3 · 2.5616 k + |G|2) time. Moreover, we derive a small problem kernel for 1-LAYER PLANARIZATION. © Springer-Verlag Berlin Heidelberg 2005.