The Erdős–Pósa Property For Long Circuits

Birmelé, Étienne; Bondy, J. A.; Reed, Bruce

doi:10.1007/s00493-007-0047-0

Cited by 48 publications

(134 citation statements)

References 8 publications

(11 reference statements)

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“…As in [2], a path between two vertices x and y of C that is internally disjoint from C is Proof of Claim A. Let P be a long path between two vertices x and y of C. Let L P denote the length of P. We may assume that C[x, y] is at least as long as C [y, x].…”

Section: Lemma 4 Let G Be a Graph Containing No Two Disjoint Long CImentioning

confidence: 96%

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The Erdős–Pósa Property for Long Circuits

Meierling

Rautenbach

Sasse

2013

Journal of Graph Theory

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Section: Lemma 4 Let G Be a Graph Containing No Two Disjoint Long CImentioning

confidence: 96%

“…Let F denote the family of circuits of length at least . In [2], Birmelé, Bondy, and Reed show that F has the Erdős-Pósa property with…”

Section: Introductionmentioning

confidence: 99%

The Erdős–Pósa Property for Long Circuits

Meierling

Rautenbach

Sasse

2013

Journal of Graph Theory

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“…Finally, asking whether |E(G)| − e-cover C θ 3 (G) ≤ k corresponds to the Maximum Cactus Subgraph. 2 All parameters keep being NP-complete to compute because the aforementioned base cases can be reduced to the general one by replacing each edge by one of multiplicity r − 1.…”

Section: Approximation Algorithmsmentioning

confidence: 99%

An $$O(\log \mathrm {OPT})$$ O ( log OPT ) -Approximation for Covering and Packing Minor Models of $$\theta _r$$ θ r

Chatzidimitriou

Raymond

et al. 2017

Algorithmica

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Given two graphs G and H , we define v-cover H (G) (resp. e-cover H (G)) as the minimum number of vertices (resp. edges) whose removal from G produces a graph without any minor isomorphic to H . Also v-pack H (G) (resp. e-pack H (G)) is the maximum number of vertex-(resp. edge-) disjoint subgraphs of G that contain a minor isomorphic to H . We denote by θ r the graph with two vertices and r parallel edges between them. When H = θ r , the parameters v-cover H , e-cover H , v-pack H , and e-pack H are NP-hard to compute (for sufficiently big values of r ). Drawing upon combinatorial results in Chatzidimitriou et al. (Minors in graphs of large θ r -girth, 2015, arXiv:1510.03041), we give an algorithmic proof that if v-pack θ r (G) ≤ k, then v-cover θ r (G) = O(k log k), and similarly for e-pack θ r and e-cover θ r . In other words, the class of graphs containing θ r as a minor has the vertex/edge Erdős-Pósa property, for every positive integer r . Using the algorithmic machinery of our proofs we introduce a unified approach for the design of an O(log OPT)-approximation algorithm for v-pack θ r , v-cover θ r , e-pack θ r , and e-cover θ r that runs in O(n · log(n) · m) steps. Also, we derive several new Erdős-Pósa-type results from the techniques that we introduce.

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“…Erdős and Pósa also showed that the function cannot be improved to o(k log k), and later Simonovits [25] provided a construction achieving the lower bound. The result of Erdős and Pósa has been strengthened for cycles with additional constraints; for example, long cycles [24,4,10,18,5], directed cycles [23,14], cycles with modularity constraints [26,12] or cycles intersecting a prescribed vertex set [15,19,5,12]. Reed [22] showed that the class of odd cycles does not satisfy the Erdős-Pósa property.…”

Section: Introductionmentioning

confidence: 99%

Erdős-Pósa property of chordless cycles and its applications

Kim¹,

Kwon²

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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A chordless cycle in a graph G is an induced subgraph of G which is a cycle of length at least four. We prove that the Erdős-Pósa property holds for chordless cycles, which resolves the major open question concerning the Erdős-Pósa property. Our proof for chordless cycles is constructive: in polynomial time, one can find either k + 1 vertex-disjoint chordless cycles, or ck 2 log k vertices hitting every chordless cycle for some constant c. It immediately implies an approximation algorithm of factor O(opt log opt) for Chordal Vertex Deletion. We complement our main result by showing that chordless cycles of length at least for any fixed ≥ 5 do not have the Erdős-Pósa property.As a corollary, for a non-negative integral function w defined on the vertex set of a graph G, the minimum value x∈S w(x) over all vertex sets S hitting all cycles of G is at most O(k 2 log k) where k is the maximum number of cycles (not necessarily vertex-disjoint) in G such that each vertex v is used at most w(v) times.

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The Erdős–Pósa Property For Long Circuits

Cited by 48 publications

References 8 publications

The Erdős–Pósa Property for Long Circuits

The Erdős–Pósa Property for Long Circuits

An $$O(\log \mathrm {OPT})$$ O ( log OPT ) -Approximation for Covering and Packing Minor Models of $$\theta _r$$ θ r

Erdős-Pósa property of chordless cycles and its applications

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