Item

Abstract

We study a general class of problems called p-\textsc\mathcal{F}-Deletion} problems. In an p-\textsc{\mathcal{F}-Deletion} problem, we are asked whether a subset of at most k vertices can be deleted from a graph G such that the resulting graph does not contain as a minor any graph from the family {\cal F} of forbidden minors. We obtain a number of algorithmic results on the p-\textsc{\mathcal{F}-Deletion} problem when \mathcal{F} contains a planar graph. We give \begin{itemize} \item a linear vertex kernel on graphs excluding t-claw K_{1,t}, the star with t leves, as an induced subgraph, where t is a fixed integer. \item an approximation algorithm achieving an approximation ratio of O(\log^{3/2} OPT), where OPT is the size of an optimal solution on general undirected graphs. \end{itemize} Finally, we obtain polynomial kernels for the case when \cal F only contains graph θ_c as a minor for a fixed integer c. The graph θ_c consists of two vertices connected by c parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback Vertex Set} and \textsc{Diamond Hitting Set . The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes.