A Fixed-Parameter Approach To Two-Layer Planarization

A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2-LAYER PLANARIZATION problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar. This problem is NP-complete, as is the 1-LAYER PLANARIZATION problem in which the permutation of the vertices in one layer is fixed. We give the following fixed parameter tractability results: an O(k (.) 6(k) + \G\) algorithm for 2-LAYER PLANARIZATION and an O(3(k) (.) \G\) algorithm for 1-LAYER PLANARIZATION, thus achieving linear time for fixed k.