Corpus ID: 220768974

Half-integral Erd\H{o}s-P\'osa property of directed odd cycles

@article{Kawarabayashi2020HalfintegralEP,
  title={Half-integral Erd\H\{o\}s-P\'osa property of directed odd cycles},
  author={Ken-ichi Kawarabayashi and Stephan Kreutzer and O-joung Kwon and Qiqin Xie},
  journal={arXiv: Combinatorics},
  year={2020}
}
We prove that there exists a function $f:\mathbb{N}\rightarrow \mathbb{R}$ such that every digraph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a vertex set $X$ of size at most $f(k)$ hitting all directed odd cycles. This extends the half-integral Erdős-Posa property of undirected odd cycles, proved by Reed [Mangoes and blueberries. Combinatorica 1999], to digraphs. 

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References

SHOWING 1-10 OF 41 REFERENCES
Packing Topological Minors Half-Integrally
TLDR
It is proved that this half-integral version of Erd\H{o}s-P\'osa property holds with respect to the topological minor containment, which easily implies Thomas' conjecture. Expand
The Erdős–Pósa Property For Long Circuits
TLDR
It is obtained as a corollary that graphs without k disjoint circuits of length l or more have tree-width O(lk2), thereby sharpening a result of C. Thomassen. Expand
Packing Directed Circuits through Prescribed Vertices Bounded Fractionally
TLDR
It is shown that the fractionality can be bounded by at most one-fifth, and a fixed-parameter tractable approximation algorithm is given for finding a 1/5-integral packing of S-circuits through a prescribed set of vertices. Expand
Long cycles through prescribed vertices have the Erdős-Pósa property
We prove that for every graph, any vertex subset S, and given integers k, `: there are k disjoint cycles of length at least ` that each contain at least one vertex from S, or a vertex set of size O(`Expand
A Tighter Erdős-Pósa Function for Long Cycles
We prove that there exists a bivariate function f with f(k,l) = O(l k log k) such that for every naturals k and l, every graph G has at least k vertex-disjoint cycles of length at least l or a set ofExpand
Packing cycles through prescribed vertices
TLDR
It is proved that either G has k vertex-disjoint cycles, each of which contains at least one vertex of S, or G has a vertex set X of order at most f(k)=40k^2log"2k such that [email protected]?X has no cycle that intersects S. Expand
Parameterized Complexity and Approximability of Directed Odd Cycle Transversal
TLDR
The parameterized complexity of DOCT when parameterized by the solution size $k$ is settled by showing that DOCT does not admit an algorithm with running time $f(k)n^{O(1)}$ unless FPT = W[1]. Expand
Disjoint cycles intersecting a set of vertices
TLDR
This work considers the following generalization of the problem: fix a subset S of vertices of G, and shows that again there exists a constant c such that G either contains k disjoint S-cycles, or there is a set of at most cklogk vertices intersecting every S-cycle. Expand
On the presence of disjoint subgraphs of a specified type
  • C. Thomassen
  • Mathematics, Computer Science
  • J. Graph Theory
  • 1988
TLDR
A general sufficient condition for a family of graphs to have the Erdos-Posa property is derived and a polynomially bounded algorithm for finding a cycle of length divisible by m is obtained. Expand
Erdős-Pósa property of chordless cycles and its applications
TLDR
It is shown that chordless cycles of length at least e for any fixed e ≥ 5 do not have the Erdős-Posa property, and for a non-negative integral function w defined on the vertex set of a graph G, the minimum value of w hitting all cycles of G is at most O(k2 log k) where k is the maximum number of cycles in G such that each vertex v is used at most w(v) times. Expand
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