Improved FPT Algorithms for Weighted Independent Set in Bull-Free Graphs

Very recently, Thomassé, Trotignon and Vuskovic [WG 2014] have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time 2 ) · n. In this article we improve this running time to 2 ) ·n. As a byproduct, we also improve the previous Turing-kernel for this problem from O(k) to O(k). Furthermore, for the subclass of bull-free graphs without holes of length at most 2p − 1 for p > 3, we speed up the running time to 2O(k·k 1 p−1 ) · n. As p grows, this running time is asymptotically tight in terms of k, since we prove that for each integer p > 3, Weighted Independent Set cannot be solved in time 2 · n in the class of {bull, C4, . . . , C2p−1}-free graphs unless the ETH fails.