Two-Layer Planarization parameterized by feedback edge set

Abstract

Given an undirected graph G and an integer k≥0, the NP-hard 2-Layer Planarization problem asks whether G can be transformed into a forest of caterpillar trees by removing at most k edges. 2-Layer Planarization was known to be fixed-parameter tractable with respect to the parameter k. The state of the art is an O(3.562k⋅k+|G|)-time search tree algorithm and an O(k)-size problem kernel. Since transforming G into a forest of caterpillar trees requires breaking every cycle, the size f of a minimum feedback edge set is a natural parameter with f≤k. We improve on previous fixed-parameter tractability results with respect to k by presenting new polynomial-time data reduction rules leading to a problem kernel with O(f) vertices and edges and a new search-tree based algorithm. We expect f to be significantly smaller than k for a wide range of input instances.