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A tight Erdős-Pósa function for planar minors
[article]

2019
*
arXiv
*
pre-print

Let H be

arXiv:1807.04969v4
fatcat:j3v6shfh35hqznj2cjg2i6broa
*a**planar*graph. ... By*a*classical result of Robertson and Seymour, there is*a**function*f:N→R such that*for*all k ∈N and all graphs G, either G contains k vertex-disjoint subgraphs each containing H as*a**minor*, or there is ... Acknowledgements We are much grateful to the three anonymous referees*for*their careful reading of the paper and their very helpful comments. ...##
###
A tight Erdős-Pósa function for planar minors

2019
*
Advances in Combinatorics
*

*A*very general theorem of Robertson and Seymour says that

*for*every

*planar*graph H the family F(H) of all graphs with

*a*

*minor*isomorphic to H has the

*Erdős*-

*Pósa*property. ... As the next best weakening we say that F has the

*Erdős*-

*Pósa*property if there exists

*a*

*function*f such that

*for*every graph G and integer k>0 the graph G has either k disjoint subgraphs each isomorphic ... Acknowledgements We are much grateful to the three anonymous referees

*for*their careful reading of the paper and their very helpful comments. ...

##
###
A tight Erdős-Pósa function for planar minors
[chapter]

2019
*
Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms
*

Let H be

doi:10.1137/1.9781611975482.90
dblp:conf/soda/BatenburgHJR19
fatcat:tqnlqnmtx5bppnjial7huwcryi
*a**planar*graph. ... By*a*classical result of Robertson and Seymour, there is*a**function*f : N → R such that*for*all k ∈ N and all graphs G, either G contains k vertex-disjoint subgraphs each containing H as*a**minor*, or there ... We prove the result*for*f (r) := (c r + f 4.1 (|Γ r |)). Let us consider*a*graph G of treewidth at least f (r) · k log(k + 1). If ν Γr (G) ≥ k, then we are done. ...##
###
A Tight Erdös--Pósa Function for Wheel Minors

2018
*
SIAM Journal on Discrete Mathematics
*

We prove that

doi:10.1137/17m1153169
fatcat:5h5ht5zgc5fihksxfraa6vg6z4
*for*every integer t ≥ 3 there is*a*constant c=c(t) such that*for*every integer k≥ 1 and every graph G, either G has k vertex-disjoint subgraphs each containing W_t as*minor*, or there is*a*... We conjecture that the result remains true more generally if we replace W_t with any fixed*planar*graph H. ... We also thank Hong Liu*for*spotting*a*slight inaccuracy in the proof of Theorem 1.5 in an earlier version of this paper. ...##
###
Quadratic Upper Bounds on the Erdős-Pósa Property for a Generalization of Packing and Covering Cycles

2013
*
Journal of Graph Theory
*

*For*

*a*fixed c, θ c is the graph with two vertices and c ≥ 1 parallel edges. Observe that

*for*c = 2 this corresponds to the classical

*Erdős*-

*Pósa*theorem. ... However the

*function*f is exponential. In this note, we prove that this

*function*becomes quadratic when H consists all graphs that can be contracted to

*a*fixed

*planar*graph θ c . ... An interesting question will be to classify those

*planar*graphs H, such that MH has

*Erdős*-

*Pósa*property with

*a*polynomial

*function*on all graphs. ...

##
###
A tighter Erdös-Pósa function for long cycles
[article]

2012
*
arXiv
*
pre-print

We prove that there exists

arXiv:1205.0940v1
fatcat:7pbxivuqqvbsxh6zo6hmmi4ype
*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 of at ... This improves*a*result by Birmel\'e, Bondy and Reed (Combinatorica, 2007), who proved the same result with f(k,l) = \Theta(l k^2). ... Robertson and Seymour have shown that H has the*Erdős*-*Pósa*property if and only if H is*planar*[5] . ...##
###
Long A-B-paths have the edge-Erdős-Pósa property
[article]

2019
*
arXiv
*
pre-print

*For*

*a*fixed integer ℓ

*a*path is long if its length is at least ℓ. ... We prove that

*for*all integers k and ℓ there is

*a*number f(k,ℓ) such that

*for*every graph G and vertex sets

*A*,B the graph G either contains k edge-disjoint long

*A*-B-paths or it contains an edge set F of ... Raymond,

*A*

*tight*

*Erdős*-

*Pósa*

*function*

*for*

*planar*

*minors*, 2018. 12 ...

##
###
Polynomial gap extensions of the Erdős-Pósa theorem
[chapter]

2013
*
The Seventh European Conference on Combinatorics, Graph Theory and Applications
*

The following extension of the

doi:10.1007/978-88-7642-475-5_3
fatcat:nwvygyvpzjbndhby3rl6fxtm6i
*Erdős*-*Pósa*theorem holds:*for*every h-vertex*planar*graph H, there exists*a**function*f H such that every graph G, either contains k disjoint copies of graphs in M(H), or ... As*a*main ingredient of the proof of our result, we show that*for*every graph H on h vertices and pathwidth at most 2, either G contains k disjoint copies of H as*a**minor*or the treewidth of G is upper-bounded ... does not contain*a**planar*graph as*minor*to*a*gap*for*the*Erdős*-*Pósa*Property. ...##
###
Bidimensional Structures: Algorithms, Combinatorics and Logic (Dagstuhl Seminar 13121)

2013
*
Dagstuhl Reports
*

We provide

doi:10.4230/dagrep.3.3.51
dblp:journals/dagstuhl-reports/DemaineFHT13
fatcat:vbtdm45hovds5eyorghjbz4kr4
*a*report on the Dagstuhl Seminar 13121: Bidimensional Structures: Algorithms, Combinatorics and Logic held at Schloss Dagstuhl in Wadern, Germany between Monday 18 and Friday 22 of March 2013 ...*A*possible approach is to use the fact that, if (G, k) is*a*yes instance, then G has treewidth bounded by some*function*of k. ... Then*a*dynamic programming algorithm*for**Planar*Completion to Bounded Diameter on graphs of bound treewidth would result in the construction of the desired algorithm. ...##
###
Polynomial Gap Extensions of the Erdős-Pósa Theorem
[article]

2013
*
arXiv
*
pre-print

The following extension of the

arXiv:1305.7376v2
fatcat:odrhhxeabzebpkdarru4wpvv24
*Erdős*-*Pósa*Theorem holds:*for*every h-vertex*planar*graph H, there exists*a**function*f_H such that every graph G, either contains k disjoint copies of graphs in M(H), or ... As*a*main ingredient of the proof of our result, we show that*for*every graph H on h vertices and pathwidth at most 2, either G contains k disjoint copies of H as*a**minor*or the treewidth of G is upper-bounded ... does not contain*a**planar*graph as*minor*to*a*gap*for*the*Erdős*-*Pósa*Property. ...##
###
On the Erdős-Pósa property for immersions and topological minors in tournaments
[article]

2022
*
arXiv
*
pre-print

We consider the

arXiv:2101.06732v4
fatcat:bx3kgpdhc5bi3daicmgzip5kvi
*Erdős*-*Pósa*property*for*immersions and topological*minors*in tournaments. ... (ii) If in T one cannot find k vertex-disjoint topological*minor*copies of H, then there exists*a*set of 𝒪_H(klog k) vertices that intersects all topological*minor*copies of H in T. ... from O H (k 3 ) to O H (k 2 )*for*the*Erdős*-*Pósa*property*for*immersions of*a*strongly connected H, and from O H (k 2 ) to O H (k log k)*for*the*Erdős*-*Pósa*property*for*topological*minors*, (ii) drawing ...##
###
Packing cycles through prescribed vertices

2011
*
Journal of combinatorial theory. Series B (Print)
*

The well-known theorem of

doi:10.1016/j.jctb.2011.03.004
fatcat:jag442sbbjfrlppziac3maskbm
*Erdős*and*Pósa*says that*a*graph G has either k vertex-disjoint cycles or*a*vertex set X of order at most f (k) such that G\X is*a*forest. ... In this paper, we generalize*Erdős*-Pósa's result to cycles that are required to go through*a*set S of vertices. ... Acknowledgements The authors would like to thank Jim Geelen*for*suggesting*a*simpler proof of the main theorem, which allows us to shorten our proof and to improve the bound. ...##
###
On the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments

2022
*
Discrete Mathematics & Theoretical Computer Science
*

We consider the Erd\H{o}s-P\'osa property

doi:10.46298/dmtcs.7099
fatcat:gyvhc7t3ifdadazm2uzhif7zry
*for*immersions and topological*minors*in tournaments. ... (ii) If in $T$ one cannot find $k$ vertex-disjoint topological*minor*copies of $H$, then there exists*a*set of $\mathcal{O}_H(k\log k)$ vertices that intersects all topological*minor*copies of $H$ in $ ... from O H (k 3 ) to O H (k 2 )*for*the*Erdős*-*Pósa*property*for*immersions of*a*strongly connected H, and from O H (k 2 ) to O H (k log k)*for*the*Erdős*-*Pósa*property*for*topological*minors*, (ii) drawing ...##
###
Uniquely restricted (g, f )-factors

2019
*
Zenodo
*

Conference abstract

doi:10.5281/zenodo.3376246
fatcat:2mzhwbmddfeendtik76rzrrswy
*for*9th Slovenian International Conference on Graph Theory ... Becerra López: Integer invariants of*a*graph manifold using Hirzenbruch-Jung continued fractions on the linking matrix. . . . . . . . . . 97 Simona Bonvicini:*A*variant of orthogonality*for*symmetric ... schurian coherent configurations . . . . . . . . . . . . . . . . . . . 31 Janoš Vidali: On*tight*4-designs in Hamming association schemes . . . . . . . . 32 Paul-Hermann Zieschang: Residually Thin ...##
###
On the edge-Erdős-Pósa property of Ladders
[article]

2022
*
arXiv
*
pre-print

We prove that ladders with 3 rungs and

arXiv:2003.03236v4
fatcat:53t4oi2ckzg3vnssx47rz3v63a
*a**minor*of it (the house graph) have the edge-*Erdős*-*Pósa*property, while ladders with 14 rungs or more have not. ... Additionally, we prove that the latter bound is optimal in the sense that the only known counterexample graph does not permit*a*better result. ... We call f an edge-*Erdős*-*Pósa**function**for*H. ...
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