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 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(k²·5.1926^k+|G|) time and the 1-LAYER PLANARIZATION problem in O(k³·2.5616^k+|G|²) time. Moreover, we derive a small problem kernel for 1-LAYER PLANARIZATION.