Half-integral packing of odd cycles through prescribed vertices

Kakimura, Naonori; Kawarabayashi, Ken-ichi

doi:10.1007/s00493-013-2865-6

Cited by 8 publications

(6 citation statements)

References 19 publications

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“…Very recently, Thomas and Yoo [28] extended this result to arbitrary abelian groups: they showed that there is a function 𝑓 such that given a graph whose edges are labelled by an abelian group, there is a half-integral packing of at least 𝑘 cycles each of non-zero total weight, or a vertex set of size at most 𝑓(𝑘) hitting all such cycles. Kakimura and Kawarabayashi [12] proved a different kind of strengthening of the theorem of Reed. They showed that there is a function 𝑓 such that every graph contains a half-integral packing of 𝑘 odd cycles each of which intersects a prescribed set 𝑆 of vertices or a vertex set of size at most 𝑓(𝑘) hitting all such cycles.…”

Section: Introductionmentioning

confidence: 99%

A unified half‐integral Erdős–Pósa theorem for cycles in graphs labelled by multiple abelian groups

Gollin,

Hendrey,

Kawarabayashi

et al. 2024

Journal of London Math Soc

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Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half‐integral packing. We prove a far‐reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half‐integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles. A multitude of natural properties of cycles can be encoded in this setting, for example, cycles of length at least , cycles of length modulo , cycles intersecting a prescribed set of vertices at least times and cycles contained in given ‐homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties.

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Section: Introductionmentioning

confidence: 99%

A unified half‐integral Erdős–Pósa theorem for cycles in graphs labelled by multiple abelian groups

Gollin,

Hendrey,

Kawarabayashi

et al. 2024

Journal of London Math Soc

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“…Very recently, Thomas and Yoo [25] extended this result to arbitrary abelian groups: they showed that there is a function f such that given a graph whose edges are labelled by an abelian group, there is a half-integral packing of at least k cycles each of non-zero total weight, or a vertex set of size at most f (k) hitting all such cycles. Kakimura and Kawarabayashi [12] proved a different kind of strengthening of the theorem of Reed. They showed that there is a function f such that every graph contains a half-integral packing of k odd cycles each of which intersects a prescribed set S of vertices or a vertex set of size at most f (k) hitting all such cycles.…”

Section: Introductionmentioning

confidence: 99%

A unified half-integral Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups

Gollin¹,

Hendrey²,

Kawarabayashi³

et al. 2021

Preprint

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Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half-integral packing. We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half-integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles.A multitude of natural properties of cycles can be encoded in this setting, for example cycles of length at least ℓ, cycles of length p modulo q, cycles intersecting a prescribed set of vertices at least t times, and cycles contained in given Z2-homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties.

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“…For example, the variant of Cycle Packing where one seeks k odd vertex-disjoint cycles has been widely studied [37,41,36,30,28,29]. Another well-known variant, where the cycles need to contain a prescribed set of vertices, has also been extensively investigated [25,34,26,24,27]. Furthermore, a combination of these two variants has been considered in [24,23].…”

Section: Introductionmentioning

confidence: 99%

“…Another well-known variant, where the cycles need to contain a prescribed set of vertices, has also been extensively investigated [25,34,26,24,27]. Furthermore, a combination of these two variants has been considered in [24,23].…”

Section: Introductionmentioning

confidence: 99%

Packing Cycles Faster Than Erdos--Posa

Lokshtanov¹,

Mouawad²,

Saurabh³

et al. 2019

SIAM J. Discrete Math.

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The Cycle Packing problem asks whether a given undirected graph G = (V, E) contains k vertex-disjoint cycles. Since the publication of the classic Erdős-Pósa theorem in 1965, this problem received significant scientific attention in the fields of Graph Theory and Algorithm Design. In particular, this problem is one of the first problems studied in the framework of Parameterized Complexity. The non-uniform fixed-parameter tractability of Cycle Packing follows from the Robertson-Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle Packing can be solved in time 2 O(k 2 ) · |V | using exponential space. In case a solution exists, Bodlaender's algorithm also outputs a solution (in the same time). It has later become common knowledge that Cycle Packing admits a 2 O(k log 2 k) · |V |-time (deterministic) algorithm using exponential space, which is a consequence of the Erdős-Pósa theorem. Nowadays, the design of this algorithm is given as an exercise in textbooks on Parameterized Complexity. Yet, no algorithm that runs in time 2 o(k log 2 k) · |V | O(1) , beating the bound 2 O(k log 2 k) · |V | O(1) , has been found. In light of this, it seems natural to ask whether the 2 O(k log 2 k) · |V | O(1) bound is essentially optimal. In this paper, we answer this question negatively by developing a 2 O( k log 2 k log log k ) · |V |-time (deterministic) algorithm for Cycle Packing. In case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the bound 2 O(k log 2 k) · |V | O(1) , our algorithm runs in time linear in |V |, and its space complexity is polynomial in the input size.

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Half-integral packing of odd cycles through prescribed vertices

Cited by 8 publications

References 19 publications

A unified half‐integral Erdős–Pósa theorem for cycles in graphs labelled by multiple abelian groups

A unified half‐integral Erdős–Pósa theorem for cycles in graphs labelled by multiple abelian groups

A unified half-integral Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups

Packing Cycles Faster Than Erdos--Posa

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