A Polynomial Turing-Kernel for Weighted Independent Set in Bull-Free Graphs

Abstract

The maximum stable set problem is NP-hard, even when restricted to triangle-free graphs. In particular, one cannot expect a polynomial time algorithm deciding if a bull-free graph has a stable set of size \(k\), when \(k\) is part of the instance. Our main result in this paper is to show the existence of an FPT algorithm when we parameterize the problem by the solution size \(k\). A polynomial kernel is unlikely to exist for this problem. We show however that our problem has a polynomial size Turing-kernel. More precisely, the hard cases are instances of size \({O}(k^5)\). All our results rely on a decomposition theorem of bull-free graphs due to Chudnovsky which is modified here, allowing us to provide extreme decompositions, adapted to our computational purpose.