Brambles, Prisms and Grids

For every r ∈ N, we denote by θ r the multigraph with two vertices and r parallel edges. Given a graph G, we say that a subgraph H of G is a model of θ r in G if H contains θ r as a contraction. We prove that the following edge variant of the Erdős-Pósa property holds for every r 2: if G is a graph and k is a positive integer, then either G contains a packing of k mutually edge-disjoint models of θ r , or it contains a set S of f r (k) edges such that G \ S has no θ r -model, for both f r (k) = O(k 2 r 3 polylog kr) and f r (k) = O(k 4 r 2 polylog kr). Erdős-Pósa property with gap O(k · log k). Given a graph J, we denote by M(J) the set of all graphs containing J as a contraction. Robertson and Seymour proved the following proposition, which in particular can be seen as an extension of the Erdős-Pósa Theorem. Proposition 1. Let J be a graph. The class M(J) satisfies the Erdős-Pósa property if and only if J is planar. A proof of Proposition 1 appeared for the first time in [18]. Another proof can be found in Diestel's monograph [4, Corollary 12.4.10 and Exercise 40 of Chapter 12]. In view of Proposition 1, it is natural to try to derive good estimations of the gap function f M(J) in the case where J is a planar graph. In this direction, the recent breakthrough results of Chekuri and Chuzhoy imply that f M(J) (k) = k · polylog k [2] when J is a planar graph and, even more, that f M(J) = (k + |V (J)|) O(1) [3]. Before this, the best known estimation of the gap for planar graphs was exponential, namely f M(J) (k) = 2 O(k log k) , and could be deduced from [14] using the proof of [18]. Moreover, some improved polynomial gaps have been proven for particular instantiations of the graph J (see [9,15,16,7,8,6]). Another direction is to add restrictions on the graphs G that we consider, which usually allows to optimize the gap f M(J) . In this direction, it is known that f M(J) = O(k) in the case where graphs are restricted to some non-trivial minor-closed class [10].We consider the edge counterpart of the Erdős-Pósa property, where packings are edgedisjoint (instead of vertex-disjoint) and coverings contain edges instead of vertices. We say that a graph class G satisfies the edge variant of the Erdős-Pósa property if there exists a function f G such that, for every graph G and every positive integer k, either G contains k mutually edge-disjoint subgraphs, each isomorphic to a graph in G, or it contains a set X of f G (k) edges meeting every subgraph of G that is isomorphic to a graph in G. Recently, the edge variant of the Erdős-Pósa property was proved in [12] for 4-edge-connected graphs in the case where G contains all odd cycles. J (G) for the minimum size of a subset S ⊆ E(G) (called J-edge-hitting set) that meets the edge sets of all models of J in G. Obviously, for every two graphs G and J, the following inequality holds:A graph J is said to satisfy the (vertex-)Erdős-Pósa property for minors (vertex-Erdős-Pósa property for short) if there is a function f J : N → N, called vertex-Erdős-Pósa gap of J, such that for e...

show abstract

“…These propositions as stated thereafter. Proposition 4 ([9, Lemma 1], see also [1]). Let k and r be two positive integers.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

An edge variant of the Erdős–Pósa property

Raymond

Thilikos

2016

Discrete Mathematics

View full text Add to dashboard Cite

show abstract

“…From Lemmas 2.1 and 2.3, G does not contain F 2n 2 −2n as a minor; otherwise we are done. A graph with no K 2 × C k minor, where C k is a cycle on k vertices, has treewidth at most 60k 2 − 120k + 63 [1]. As F k is a minor of K 2 × C k , the same bound holds for graphs with no F k minor.…”

Section: The Adhesion If a Tree Decomposition ({Xmentioning

confidence: 93%

Lift Contractions

Golovach

Kamiński

Paulusma

et al. 2011

Electronic Notes in Discrete Mathematics

View full text Add to dashboard Cite

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe introduce and study a partial order on graphs -lift-contractions. A graph H is a lift-contraction of a graph G if H can be obtained from G by a sequence of edge lifts and edge contractions. We give sufficient conditions for a connected graph to contain every n-vertex graph as a lift-contraction and describe the structure of graphs with an excluded lift-contraction.

show abstract

“…Notice that none of the above classes is minor comparable with the classes

P_{k}

and

F_{k}

treated in and . Moreover,

{scriptH}_{k}^{1} \subset {scriptH}_{k}^{2} \subset {scriptH}_{k}^{3}

, while

H_{k}^{0}

is not minor comparable with the other three.…”

Section: Introductionmentioning

confidence: 92%

Low Polynomial Exclusion of Planar Graph Patterns

Raymond

Thilikos

2015

J. Graph Theory

View full text Add to dashboard Cite

The celebrated grid exclusion theorem states that for every h‐vertex planar graph H, there is a constant ch such that if a graph G does not contain H as a minor then G has treewidth at most ch. We are looking for patterns of H where this bound can become a low degree polynomial. We provide such bounds for the following parameterized graphs: the wheel (ch=O(h)), the double wheel (ch=O(h2·log2h)), any graph of pathwidth at most 2 (ch=O(h2)), and the yurt graph (ch=O(h4)).

show abstract

Brambles, Prisms and Grids

Cited by 14 publications

References 3 publications

An edge variant of the Erdős–Pósa property

An edge variant of the Erdős–Pósa property

Lift Contractions

Low Polynomial Exclusion of Planar Graph Patterns

Contact Info

Product

Resources

About