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

2019
*
arXiv
*
pre-print

By

arXiv:1807.04969v4
fatcat:j3v6shfh35hqznj2cjg2i6broa
*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. ...##
###
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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A tight Erdős-Pósa function for planar minors
[chapter]

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

By

doi:10.1137/1.9781611975482.90
dblp:conf/soda/BatenburgHJR19
fatcat:tqnlqnmtx5bppnjial7huwcryi
*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. ...##
###
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. ...

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

2017
*
European journal of combinatorics (Print)
*

We prove that there is

doi:10.1016/j.ejc.2017.05.009
fatcat:ibvjk3xpwrdgralwm6v6zshqwi
*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 ...##
###
Packing and Covering Immersion Models of Planar subcubic Graphs
[article]

2016
*
arXiv
*
pre-print

We prove that there is

arXiv:1602.04042v2
fatcat:ovicanxmkzahtmdaubdfurlufy
*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. ...##
###
Packing and Covering Immersion Models of Planar Subcubic Graphs
[chapter]

2016
*
Lecture Notes in Computer Science
*

We prove that there is

doi:10.1007/978-3-662-53536-3_7
fatcat:4vqkbwhkmbcirkroy7kj4ywphm
*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. ...##
###
Recent developments in graph Ramsey theory
[article]

2015
*
arXiv
*
pre-print

Given

arXiv:1501.02474v3
fatcat:q3qcowfhgjbp5j36oetad6qdnq
*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. ...##
###
Open problems of Paul Erd�s in graph theory

1997
*
Journal of Graph Theory
*

To honestly follow the unique style of Paul

doi:10.1002/(sici)1097-0118(199705)25:1<3::aid-jgt1>3.0.co;2-r
fatcat:ja46si6w75gvzh2ksrkvblnznm
*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. ...##
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Cyclewidth and the Grid Theorem for Perfect Matching Width of Bipartite Graphs
[article]

2019
*
arXiv
*
pre-print

Norine conjectured that graphs of high perfect matching width would contain

arXiv:1902.01322v3
fatcat:f3oauyjo4zg2vagxbtvysouxia
*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] . ...##
###
Jones' Conjecture in subcubic graphs
[article]

2019
*
arXiv
*
pre-print

We confirm Jones' Conjecture

arXiv:1912.01570v1
fatcat:tg2tfy7pzzfqhetloigzl34grq
*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. ...##
###
On obstructions to small face covers in planar graphs

1992
*
Journal of combinatorial theory. Series B (Print)
*

If the embedding of the graph is fixed, this problem leads to variants of the ErdGs-

doi:10.1016/0095-8956(92)90040-5
fatcat:6qnv2u7klzgs5n5ldt326gqdke
*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. ...##
###
Jones' Conjecture in Subcubic Graphs

2021
*
Electronic Journal of Combinatorics
*

We confirm Jones' Conjecture

doi:10.37236/9192
fatcat:g4lyyhzrrjcprka2oyydnoct3m
*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. ...##
###
Structural graph theory meets algorithms: covering and connectivity problems in graphs
[article]

2017

This question first raised by

doi:10.14279/depositonce-6538
fatcat:yc6w5ouqcjby5nt4fs64kemo4q
*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. ...
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