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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  ...  Let W_t denote the wheel on t+1 vertices.  ...  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

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

Wouter Cames van Batenburg, Tony Huynh, Gwenaël Joret, Jean-Florent Raymond
2019 arXiv   pre-print
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  ...  The proof is constructive and yields a polynomial-time O(OPT)-approximation algorithm for packing subgraphs containing an H-minor.  ...  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

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

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  
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  ...  The proof is constructive and yields a polynomial-time O(log OPT)-approximation algorithm for packing subgraphs containing an H-minor.  ...  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 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

Packing and covering immersion-expansions of planar sub-cubic graphs

Archontia C. Giannopoulou, O-joung Kwon, Jean-Florent Raymond, Dimitrios M. Thilikos
2017 European journal of combinatorics (Print)  
We prove that there is a polynomial function f : N × N → N, such that if H is a connected planar sub-cubic graph on h > 0 edges, G is a graph, and k is a non-negative integer, then either G contains k  ...  A graph H is an immersion of a graph G if H can be obtained by some subgraph G after lifting incident edges.  ...  We say that a graph class C has the vertex/edge Erdős-Pósa property (shortly v/e-E&P property) for some graph class G if there is a function f : N → N, called a gap function, such that, for every graph  ... 
doi:10.1016/j.ejc.2017.05.009 fatcat:ibvjk3xpwrdgralwm6v6zshqwi

Packing and Covering Immersion Models of Planar subcubic Graphs [article]

Archontia Giannopoulou, O-joung Kwon, Jean-Florent Raymond, Dimitrios M. Thilikos
2016 arXiv   pre-print
We prove that there is a polynomial function f:N×N→N, such that if H is a connected planar subcubic graph on h>0 edges, G is a graph, and k is a non-negative integer, then either G contains k vertex/edge-disjoint  ...  A graph H is an immersion of a graph G if H can be obtained by some sugraph G after lifting incident edges.  ...  Erdős-Pósa for immersions of subcubic planar graphs Grids and Walls. Let k and r be positive integers where k, r ≥ 2.  ... 
arXiv:1602.04042v2 fatcat:ovicanxmkzahtmdaubdfurlufy

Packing and Covering Immersion Models of Planar Subcubic Graphs [chapter]

Archontia C. Giannopoulou, O-joung Kwon, Jean-Florent Raymond, Dimitrios M. Thilikos
2016 Lecture Notes in Computer Science  
We prove that there is a polynomial function f : N × N → N, such that if H is a connected planar subcubic graph on h > 0 edges, G is a graph, and k is a non-negative integer, then either G contains k vertex  ...  A graph H is an immersion of a graph G if H can be obtained by some subgraph G after lifting incident edges.  ...  Erdős-Pósa for immersions of subcubic planar graphs Grids and Walls. Let k and r be positive integers where k, r ≥ 2.  ... 
doi:10.1007/978-3-662-53536-3_7 fatcat:4vqkbwhkmbcirkroy7kj4ywphm

Recent developments in graph Ramsey theory [article]

David Conlon, Jacob Fox, Benny Sudakov
2015 arXiv   pre-print
Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H.  ...  Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics.  ...  The authors would like to thank the anonymous referee for a number of useful comments.  ... 
arXiv:1501.02474v3 fatcat:q3qcowfhgjbp5j36oetad6qdnq

Open problems of Paul Erd�s in graph theory

F. R. K. Chung
1997 Journal of Graph Theory  
To honestly follow the unique style of Paul Erdős, we will mention the fact that Erdős often offered monetary awards for solutions to a number of his favorite problems.  ...  Solutions or partial solutions to Erdős problems usually lead to further questions, often in new directions. These problems provide inspiration and serve as a common focus for all graph theorists.  ...  Acknowledgement The author wishes to thank Paul Seymour for suggesting writing this paper.  ... 
doi:10.1002/(sici)1097-0118(199705)25:1<3::aid-jgt1>3.0.co;2-r fatcat:ja46si6w75gvzh2ksrkvblnznm

Cyclewidth and the Grid Theorem for Perfect Matching Width of Bipartite Graphs [article]

Meike Hatzel, Roman Rabinovich, Sebastian Wiederrecht
2019 arXiv   pre-print
Norine conjectured that graphs of high perfect matching width would contain a large grid as a matching minor, similar to the result on treewidth by Robertson and Seymour.  ...  Perfect matching width is a width parameter for matching covered graphs based on a branch decomposition.  ...  ] , or in the proofs of the (general) Erdős-Pósa property for undirected graphs [RS86] .  ... 
arXiv:1902.01322v3 fatcat:f3oauyjo4zg2vagxbtvysouxia

Jones' Conjecture in subcubic graphs [article]

Marthe Bonamy, François Dross, Tomáš Masařík, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk
2019 arXiv   pre-print
We confirm Jones' Conjecture for subcubic graphs. Namely, if a subcubic planar graph does not contain k+1 vertex-disjoint cycles, then it suffices to delete 2k vertices to obtain a forest.  ...  Erdős and Pósa [EP65] showed that there is a constant c such that for any graph G, fvs(G) c · cp(G) log cp(G), and that this upper-bound is tight for some graphs.  ...  Every planar graph G satisfies fvs(G) 2 · cp(G). 1 Note that Conjecture 1 is tight for wheels or for the dodecahedron.  ... 
arXiv:1912.01570v1 fatcat:tg2tfy7pzzfqhetloigzl34grq

On obstructions to small face covers in planar graphs

D Bienstock, N Dean
1992 Journal of combinatorial theory. Series B (Print)  
If the embedding of the graph is fixed, this problem leads to variants of the ErdGs-Posa theorem on independent cycles in a graph.  ...  If the embedding of the graph is not fixed, the analysis leads to generalizations of outerplanar graphs, and we obtain an explicit upper bound on the size of the minimal excluded minors for such classes  ...  a classical theorem of Erdos and Posa [4] , which we sketch next.  ... 
doi:10.1016/0095-8956(92)90040-5 fatcat:6qnv2u7klzgs5n5ldt326gqdke

Jones' Conjecture in Subcubic Graphs

Marthe Bonamy, François Dross, Tomáš Masařík, Andrea Munaro, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk
2021 Electronic Journal of Combinatorics  
We confirm Jones' Conjecture for subcubic graphs. Namely, if a subcubic planar graph does not contain $k+1$ vertex-disjoint cycles, then it suffices to delete $2k$ vertices to obtain a forest.  ...  Erdős and Pósa [4] showed that there is a constant c such that for any graph G, fvs(G) c • cp(G) log cp(G), and that this upper-bound is tight for some graphs.  ...  Note that Conjecture 1 is tight for wheels or for the dodecahedron. Currently, the best known bound is that every planar graph G satisfies fvs(G) 3 • cp(G), as proved independently by Chappel et al.  ... 
doi:10.37236/9192 fatcat:g4lyyhzrrjcprka2oyydnoct3m

Structural graph theory meets algorithms: covering and connectivity problems in graphs [article]

Saeed Akhoondian Amiri, Technische Universität Berlin, Technische Universität Berlin, Stephan Kreutzer
2017
This question first raised by Erdos and Posa.  ...  Structural graph theory proved itself a valuable tool for designing efficient algorithms for hard problems over recent decades.  ...  We prove the cases for butterfly minors, the cases of topological minors are analogous. Towards a contradiction, suppose that H has the Erdős-Pósa property witnessed by a function f : N → N.  ... 
doi:10.14279/depositonce-6538 fatcat:yc6w5ouqcjby5nt4fs64kemo4q
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