Parity Linkage and the Erdős–Pósa Property of Odd Cycles through Prescribed Vertices in Highly Connected Graphs

Joos, Felix

doi:10.1002/jgt.22103

Cited by 5 publications

(5 citation statements)

References 18 publications

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“…As discussed in Section 3.2, an analogue of the Erdős-Pósa theorem does not hold for the cycles described in Theorem 1.1. It was shown that in graphs of sufficiently high connectivity [11,13,18,30], an analogue of the Erdős-Pósa theorem holds for odd cycles. We ask whether a similar phenomenon happens for the cycles described in Theorem 1.1.…”

Section: Discussionmentioning

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: Discussionmentioning

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-Pósa Property of Obstructions to Interval Graphs graphs for which the family of odd cycles (subgraphs and induced subgraphs) admits the Erdős-Pósa property. This includes planar graphs [15], or graphs with certain connectivity constraints [48,36,28,24]. Not only the family of odd cycles does not admit the Erdős-Pósa property, but also subfamilies such as the family of chordless cycles of length at least 5 [25].…”

Section: :4mentioning

confidence: 99%

“…Since the emergence of the result of Erdős and Pósa [14], a multitude of studies on the Erdős-Pósa property have appeared in the literature for several combinatorial objects beyond graphs. This includes extensions to digraphs [32,44,40,22,20], rooted graphs [9,26,35,24], labeled graphs [29], signed graphs [23,3], hypergraphs [1,6,7], matroids [16], helly-type theorems [21], H-minors [41], H-immersions [17,31], and H-butterfly directed minors [2] (also see [38]). This list is not comprehensive but rather illustrative.…”

Section: Introductionmentioning

confidence: 99%

Erdős–Pósa property of obstructions to interval graphs

Agrawal

Lokshtanov

Misra

et al. 2022

Journal of Graph Theory

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The duality between packing and covering problems lies at the heart of fundamental combinatorial proofs, as well as well-known algorithmic methods such as the primal-dual method for approximation and win/win-approach for parameterized analysis. The very essence of this duality is encompassed by a well-known property called the Erdős-Pósa property, which has been extensively studied for over five decades. Informally, we say that a class of graphs F admits the Erdős-Pósa property if there exists f such that for any graph G, either G has k vertex-disjoint "copies" of the graphs in F, or there is a set S ⊆ V (G) of f (k) vertices that intersects all copies of the graphs in F. In the context of any graph class G, the most natural question that arises in this regard is as follows -do obstructions to G have the Erdős-Pósa property? Having this view in mind, we focus on the class of interval graphs. Structural properties of interval graphs are intensively studied, also as they lead to the design of polynomial-time algorithms for classic problems that are NP-hard on general graphs. Nevertheless, about one of the most basic properties of such graphs, namely, the Erdős-Pósa property, nothing is known. In this paper, we settle this anomaly: we prove that the family of obstructions to interval graphs -namely, the family of chordless cycles and ATs -admits the Erdős-Pósa property. Our main theorem immediately results in an algorithm to decide whether an input graph G has k vertex-disjoint ATs and chordless cycles, or there exists a set of O(k 2 log k) vertices in G that hits all ATs and chordless cycles.

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“…Just as odd cycles, odd -cycles do not have the Erdős-Pósa property in general, but gain it in highly connected graphs; see [12]. For cycles in digraphs the situation is slightly different as demonstrated by an example of Wollan (see Kakimura and Kawarabayashi [13]): while (directed) cycles in digraphs have the Erdős-Pósa property, the property is lost when cycles are replaced by -cycles.…”

Section: Theorem 4 Let and Be Integers For Any Graph And Any Subsetmentioning

confidence: 99%

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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Parity Linkage and the Erdős–Pósa Property of Odd Cycles through Prescribed Vertices in Highly Connected Graphs

Cited by 5 publications

References 18 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

Erdős–Pósa property of obstructions to interval graphs

Long cycles through prescribed vertices have the Erdős‐Pósa property

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