Linear time algorithms for NP-hard problems restricted to partial k-trees
Discrete Applied Mathematics
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem
SIAM Journal on Discrete Mathematics
A Polyhedral Approach to the Feedback Vertex Set Problem
Proceedings of the 5th International IPCO Conference on Integer Programming and Combinatorial Optimization
A Min-Max Theorem on Feedback Vertex Sets
Mathematics of Operations Research
An 8-approximation algorithm for the subset feedback vertex set problem
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Packing cycles in undirected graphs
Journal of Algorithms - Special issue: Twelfth annual ACM-SIAM symposium on discrete algorithms
On the connectivity of minimum and minimal counterexamples to Hadwiger's Conjecture
Journal of Combinatorial Theory Series B
Approximation algorithms and hardness results for cycle packing problems
ACM Transactions on Algorithms (TALG)
On the odd-minor variant of Hadwiger's conjecture
Journal of Combinatorial Theory Series B
Proceedings of the forty-second ACM symposium on Theory of computing
An (almost) linear time algorithm for odd cycles transversal
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Notes: Packing cycles through prescribed vertices
Journal of Combinatorial Theory Series B
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
The Graph Minor Algorithm with Parity Conditions
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Packing cycles with modularity constraints
Combinatorica
The disjoint paths problem in quadratic time
Journal of Combinatorial Theory Series B
Graph Minors. XXII. Irrelevant vertices in linkage problems
Journal of Combinatorial Theory Series B
Parameterized complexity of generalized vertex cover problems
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
Finding a minimum feedback vertex set in time O(1.7548n)
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
Finding odd cycle transversals
Operations Research Letters
Parameterized Complexity
The Multivariate Algorithmic Revolution and Beyond
Directed subset feedback vertex set is fixed-parameter tractable
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Parameterized tractability of multiway cut with parity constraints
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Parameterized algorithms for even cycle transversal
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
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The well-known Erdös-Pósa theorem says that for any integer k and any graph G, either G contains k vertex-disjoint cycles or a vertex set X of order at most c·k log k (for some constant c) such that G - X is a forest. Thomassen [39] extended this result to the even cycles, but on the other hand, it is well-known that this theorem is no longer true for the odd cycles. However, Reed [31] proved that this theorem still holds if we relax k vertex-disjoint odd cycles to k odd cycles with each vertex in at most two of them. These theorems initiate many researches in both graph theory and theoretical computer science. In the graph theory side, our problem setting is that we are given a graph and a vertex set S, and we want to extend all the above results to cycles that are required to go through a subset of S, i.e., each cycle contains at least one vertex in S (such a cycle is called an S-cycle). It was shown in [20] that the above Erdős-Pósa theorem still holds for this subset version. In this paper, we extend both Thomassen's result and Reed's result in this way. In the theoretical computer science side, we investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our purpose here is to consider the following problems: the feedback set problem with respect to the S-cycles, and the S-cycle packing problem. We give the first fixed parameter algorithms for the two problems. Namely; 1. For fixed k, we can either find a vertex set X of size k such that G -- X has no S-cycle, or conclude that such a vertex set does not exist in O(n2m) time (independently obtained in [7]). 2. For fixed k, we can either find k vertex-disjoint S-cycles, or conclude that such k disjoint cycles do not exist in O(n2m) time. We also extend the above results to those with the parity constraints as follows; 1. For a parameter k, there exists a fixed parameter algorithm that either finds a vertex set X of size k such that G -- X has no even S-cycle, or concludes that such a vertex set does not exist. 2. For a parameter k, there exists a fixed parameter algorithm that either finds a vertex set X of size k such that G -- X has no odd S-cycle, or concludes that such a vertex set does not exist. 3. For a parameter k, there exists a fixed parameter algorithm that either finds k vertex-disjoint even S-cycles, or concludes that such k disjoint cycles do not exist. 4. For a parameter k, there exists a fixed parameter algorithm that either finds k odd S-cycles with each vertex in at most two of them, or concludes that such k cycles do not exist.