A quartic kernel for pathwidth-one vertex deletion
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Kernel bounds for disjoint cycles and disjoint paths
Theoretical Computer Science
Subset feedback vertex set is fixed-parameter tractable
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Parameterized complexity of vertex deletion into perfect graph classes
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
Parameterized complexity of vertex deletion into perfect graph classes
Theoretical Computer Science
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The Feedback Vertex Set problem on unweighted, undirected graphs is considered. Improving upon a result by Burrage et al. (Proceedings 2nd International Workshop on Parameterized and Exact Computation, pp. 192–202, 2006), we show that this problem has a kernel with O(k 3) vertices, i.e., there is a polynomial time algorithm, that given a graph G and an integer k, finds a graph G′ with O(k 3) vertices and integer k′≤k, such that G has a feedback vertex set of size at most k, if and only if G′ has a feedback vertex set of size at most k′. Moreover, the algorithm can be made constructive: if the reduced instance G′ has a feedback vertex set of size k′, then we can easily transform a minimum size feedback vertex set of G′ into a minimum size feedback vertex set of G. This kernelization algorithm can be used as the first step of an FPT algorithm for Feedback Vertex Set, but also as a preprocessing heuristic for Feedback Vertex Set. We also show that the related Loop Cutset problem also has a kernel of cubic size. The kernelization algorithms are experimentally evaluated, and we report on these experiments.