Linear Kernels on Graphs Excluding Topological Minors

We show that problems which have finite integer index and satisfy a requirement we call treewidth-bounding admit linear kernels on the class of $H$-topological-minor free graphs, for an arbitrary fixed graph $H$. This builds on earlier results by Fomin et al.\ on linear kernels for $H$-minor-free graphs and by Bodlaender et al.\ on graphs of bounded genus. Our framework encompasses several problems, the prominent ones being Chordal Vertex Deletion, Feedback Vertex Set and Edge Dominating Set.Comment: 19 pages. A simpler proof of the results of this paper appears in http://arxiv.org/abs/1207.0835. This new paper contains additional result

Planar F-Deletion: Approximation and Optimal FPT Algorithms

Let F be a finite set of graphs. In the F-Deletion problem, we are given an n-vertex, m-edge graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. F-Deletion is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as Vertex Cover, Feedback Vertex Set or Treewidth t-Deletion. In this paper we obtain a number of generic algorithmic results about Planar F-Deletion, when F contains at least one planar graph. The highlights of our work are - A randomized O(nm) time constant factor approximation algorithm for the optimization version of Planar F-Deletion. - A randomized O(2^{O(k)} n) time parameterized algorithm for Planar F-Deletion when F is connected. Here a family F is called connected if every graph in F is connected. The algorithm can be made deterministic at the cost of making the polynomial factor in the running time n\log^2 n rather than linear. These algorithms unify, generalize, and improve over a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results -- constant factor approximation and FPT algorithms -- are stringed together by a common theme of polynomial time preprocessing