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

Wouter Cames van Batenburg, Tony Huynh, Gwenaël Joret, Jean-Florent Raymond
2019 arXiv   pre-print
Let H be 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.  ... 
arXiv:1807.04969v4 fatcat:j3v6shfh35hqznj2cjg2i6broa

A tight Erdős-Pósa function for planar minors

Wouter Cames van Batenburg, Tony Huyn, Gwenaël Joret, Jean-Florent Raymond
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.  ... 
doi:10.19086/aic.10807 fatcat:yv2ppc32cbdavjy3gmzg4byfd4

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

Wouter Cames van Batenburg, Tony Huynh, Gwenaël Joret, Jean-Florent Raymond
2019 Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms  
Let H be 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.  ... 
doi:10.1137/1.9781611975482.90 dblp:conf/soda/BatenburgHJR19 fatcat:tqnlqnmtx5bppnjial7huwcryi

A Tight Erdös--Pósa Function for Wheel Minors

Pierre Aboulker, Samuel Fiorini, Tony Huynh, Gwenaël Joret, Jean-Florent Raymond, Ignasi Sau
2018 SIAM Journal on Discrete Mathematics  
We prove that 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.  ... 
doi:10.1137/17m1153169 fatcat:5h5ht5zgc5fihksxfraa6vg6z4

Quadratic Upper Bounds on the Erdős-Pósa Property for a Generalization of Packing and Covering Cycles

Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, Saket Saurabh
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.  ... 
doi:10.1002/jgt.21720 fatcat:tl2srfx6hjfqvp4iptphb3xueq

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

Samuel Fiorini, Audrey Herinckx
2012 arXiv   pre-print
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 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] .  ... 
arXiv:1205.0940v1 fatcat:7pbxivuqqvbsxh6zo6hmmi4ype

Long A-B-paths have the edge-Erdős-Pósa property [article]

Matthias Heinlein, Arthur Ulmer
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  ... 
arXiv:1903.07989v1 fatcat:xw4wztkk6ndqfhwrue7i4ga2oy

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

Jean-Florent Raymond, Dimitrios M. Thilikos
2013 The Seventh European Conference on Combinatorics, Graph Theory and Applications  
The following extension of the 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.  ... 
doi:10.1007/978-88-7642-475-5_3 fatcat:nwvygyvpzjbndhby3rl6fxtm6i

Bidimensional Structures: Algorithms, Combinatorics and Logic (Dagstuhl Seminar 13121)

Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi, Dimitrios M. Thilikos, Marc Herbstritt
2013 Dagstuhl Reports  
We provide 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.  ... 
doi:10.4230/dagrep.3.3.51 dblp:journals/dagstuhl-reports/DemaineFHT13 fatcat:vbtdm45hovds5eyorghjbz4kr4

Polynomial Gap Extensions of the Erdős-Pósa Theorem [article]

Jean-Florent Raymond, Dimitrios M. Thilikos
2013 arXiv   pre-print
The following extension of the 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.  ... 
arXiv:1305.7376v2 fatcat:odrhhxeabzebpkdarru4wpvv24

On the Erdős-Pósa property for immersions and topological minors in tournaments [article]

Łukasz Bożyk, Michał Pilipczuk
2022 arXiv   pre-print
We consider the 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  ... 
arXiv:2101.06732v4 fatcat:bx3kgpdhc5bi3daicmgzip5kvi

Packing cycles through prescribed vertices

Naonori Kakimura, Ken-ichi Kawarabayashi, Dániel Marx
2011 Journal of combinatorial theory. Series B (Print)  
The well-known theorem of 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.  ... 
doi:10.1016/j.jctb.2011.03.004 fatcat:jag442sbbjfrlppziac3maskbm

On the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments

Łukasz Bożyk, Michał Pilipczuk
2022 Discrete Mathematics & Theoretical Computer Science  
We consider the Erd\H{o}s-P\'osa 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 $\mathcal{O}_H(k\log k)$ 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  ... 
doi:10.46298/dmtcs.7099 fatcat:gyvhc7t3ifdadazm2uzhif7zry

Uniquely restricted (g, f )-factors

Miklós Krész
2019 Zenodo  
Conference abstract 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  ... 
doi:10.5281/zenodo.3376246 fatcat:2mzhwbmddfeendtik76rzrrswy

On the edge-Erdős-Pósa property of Ladders [article]

Raphael Steck, Arthur Ulmer
2022 arXiv   pre-print
We prove that ladders with 3 rungs and 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.  ... 
arXiv:2003.03236v4 fatcat:53t4oi2ckzg3vnssx47rz3v63a
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