A Tighter Erdős-Pósa Function for Long Cycles

Fiorini, Samuel; Herinckx, Audrey

doi:10.1002/jgt.21776

Cited by 26 publications

(34 citation statements)

References 7 publications

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“…Fiorini and Herinckx recently improved the above‐mentioned result of Birmele et al. by showing that cycles of length at least t satisfy the Erdős‐Pósa property with

f (k) = O (t k log k)

(which is optimal for fixed k or fixed t ).…”

Section: Notes Added In Proofmentioning

confidence: 91%

Circumference and Pathwidth of Highly Connected Graphs

Marshall

Wood

2014

Journal of Graph Theory

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Birmele [J. Graph Theory, 2003] proved that every graph with circumference t has treewidth at most t − 1. Under the additional assumption of 2-connectivity, such graphs have bounded pathwidth, which is a qualitatively stronger conclusion. Birmele's theorem was extended by Birmele, Bondy and Reed [Combinatorica, 2007] who showed that every graph without k disjoint cycles of length at least t has treewidth O(tk 2 ). Our main result states that, under the additional assumption of (k + 1)-connectivity, such graphs have bounded pathwidth. In fact, they have pathwidth O(t 3 +tk 2 ). Moreover, examples show that (k+1)-connectivity is required for bounded pathwidth to hold. These results suggest the following general question: for which values of k and graphs H does every k-connected H-minor-free graph have bounded pathwidth? We discuss this question and provide a few observations.

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“…Fiorini and Herinckx recently improved the above‐mentioned result of Birmele et al. by showing that cycles of length at least t satisfy the Erdős‐Pósa property with

f (k) = O (t k log k)

(which is optimal for fixed k or fixed t ).…”

Section: Notes Added In Proofmentioning

confidence: 91%

Circumference and Pathwidth of Highly Connected Graphs

Marshall

Wood

2014

Journal of Graph Theory

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“…That there is always such a hitting set, a vertex set meeting all cycles of length at least , of a size depending only on and is a consequence of a more general result by Robertson and Seymour [21]. The bound on the hitting set was subsequently improved by Thomassen [23], followed by Birmelé et al [1], and Fiorini and Herinckx [9], until Mousset et al established the currently best bound stated above.…”

Section: Theorem 1 (Erdős and Pósamentioning

confidence: 99%

“…We note that the proof of the theorem can be turned into an algorithm that runs in ( )-time. For the proof of our main result we naturally borrow some arguments from Fiorini and Herinckx [9], and Pontecorvi and Wollan [18]. In particular, both pairs of authors adapt Simonovits' graph so that it only contains cycles of the desired kind, that is, either long cycles or -cycles.…”

Section: Theorem 6 (Simonovits [22]) Every Cubic Multigraph With At mentioning

confidence: 99%

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Long cycles through prescribed vertices have the Erdős‐Pósa property

Bruhn

Joos

Schaudt

2017

Journal of Graph Theory

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We prove that for every graph, any vertex subset S, and given integers k,ℓ: there are k disjoint cycles of length at least ℓ that each contain at least one vertex from S, or a vertex set of size O(ℓ·klogk) that meets all such cycles. This generalizes previous results of Fiorini and Herinckx and of Pontecorvi and Wollan. In addition, we describe an algorithm for our main result that runs in O(klogk·s2·false(f(ℓ)·n+mfalse)) time, where s denotes the cardinality of S.

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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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A Tighter Erdős-Pósa Function for Long Cycles

Cited by 26 publications

References 7 publications

Circumference and Pathwidth of Highly Connected Graphs

Circumference and Pathwidth of Highly Connected Graphs

Long cycles through prescribed vertices have the Erdős‐Pósa property

Erdős-Pósa property of chordless cycles and its applications

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