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Treewidth Lower Bounds with Brambles

2007
*
Algorithmica
*

In this paper we present a new technique for computing

doi:10.1007/s00453-007-9056-z
fatcat:lamgyxjesrggze6wppny5xk7he
*lower**bounds*for graph*treewidth*. ... The algorithm for planar graphs is shown to give a*lower**bound*for both the*treewidth*and branchwidth that is at most a constant factor away from the optimum. ... Acknowledgements We thank Illya Hicks for providing us*with*the planar graphs for our experiments. ...##
###
Treewidth Lower Bounds with Brambles
[chapter]

2005
*
Lecture Notes in Computer Science
*

In this paper we present a new technique for computing

doi:10.1007/11561071_36
fatcat:hvhke4uh3nghdod5qd724jdyuq
*lower**bounds*for graph*treewidth*. ... The algorithm for planar graphs is shown to give a*lower**bound*for both the*treewidth*and branchwidth that is at most a constant factor away from the optimum. ... Acknowledgements We thank Illya Hicks for providing us*with*the planar graphs for our experiments. ...##
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On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic
[chapter]

2010
*
Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms
*

Perfect

doi:10.1137/1.9781611973075.30
dblp:conf/soda/KreutzerT10
fatcat:udgkaapsynhqlpbm5pg3vfqdxy
*brambles*are*brambles**with*a particularly simple structure and they also provide us*with*a subgraph that has*bounded*degree and still large*treewidth*; we use them to obtain a meta-theorem on deciding ... Whereas much work has been done on designing, improving, and applying algorithms on graphs of*bounded**treewidth*, not much is known on the side of*lower**bounds*: what*bound*on the*treewidth*of a class of ... So, there is a huge gap between the known*lower*and upper*bounds*of this theorem; Robertson and Seymour conjecture that the true value should be closer to the*lower**bound*, i.e. that every graph should ...##
###
On the treewidth of toroidal grids

2016
*
Discrete Applied Mathematics
*

To show the

doi:10.1016/j.dam.2015.06.027
fatcat:k33laemlunhwpcl4hnrt6kzsdu
*lower**bounds*, we construct*brambles*of high orders. ... In this paper, we study the*treewidth*of toroidal grids and show that the*treewidth*of the n × n toroidal grid is either 2n − 2 or 2n − 1. We then show that these*bounds*are tight in some cases. ... By this fact, we can show a*lower**bound*of*treewidth*by constructing a*bramble*of high order. ...##
###
Treewidth and gonality of glued grid graphs
[article]

2019
*
arXiv
*
pre-print

Our main technique is constructing strict

arXiv:1808.09475v2
fatcat:2imvhh4iavcp3pqdyuaykpyoci
*brambles*of large orders. ... We compute the*treewidth*of a family of graphs we refer to as the glued grids, consisting of the stacked prism graphs and the toroidal grids. ... We first present strict*brambles*of order min{m, 2n} in the cases when 2n = m to achieve a*lower**bound*on*treewidth*. ...##
###
Treewidth of Cartesian Products of Highly Connected Graphs

2012
*
Journal of Graph Theory
*

For n≫ k this

doi:10.1002/jgt.21677
fatcat:gl2fucw7orcf3hnpvjovnawsle
*lower**bound*is asymptotically tight for particular graphs G and H. This theorem generalises a well known result about the*treewidth*of planar grid graphs. ... The following theorem is proved: For all k-connected graphs G and H each*with*at least n vertices, the*treewidth*of the cartesian product of G and H is at least k(n -2k+2)-1. ... This paper proves the following general*lower**bound*on the*treewidth*of cartesian products of highly connected graphs. Theorem 2. ...##
###
Treewidth computations II. Lower bounds

2011
*
Information and Computation
*

*Lower*

*bounds*give indications of the quality of

*treewidth*upper

*bounds*and upper

*bound*algorithms. • Better

*lower*

*bounds*can make preprocessing more effective. ... There are preprocessing rules that apply only when a large enough

*lower*

*bound*on the

*treewidth*of the graph is known. ... In Section 9, a different characterisation of the notion of

*treewidth*in terms of

*brambles*is used to obtain

*treewidth*

*lower*

*bounds*. ...

##
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On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic
[article]

2009
*
arXiv
*
pre-print

Perfect

arXiv:0907.3076v1
fatcat:ajfgzk6oqregpiwfrvvkec3viu
*brambles*are*brambles**with*a particularly simple structure and they also provide us*with*a subgraph that has*bounded*degree and still large*treewidth*; we use them to obtain a meta-theorem on deciding ... The second part of our work deals*with*providing a*lower**bound*to Courcelle's famous theorem, stating that every graph property that can be expressed by a sentence in monadic second-order logic (MSO), ... So, there is a huge gap between the known*lower*and upper*bounds*of this theorem; Robertson and Seymour conjecture that the true value should be closer to the*lower**bound*, i.e. that every graph should ...##
###
Constant Congestion Brambles

2022
*
Discrete Mathematics & Theoretical Computer Science
*

Second, we provide a tight upper

doi:10.46298/dmtcs.6699
fatcat:37wgzyuulveffb4nx5ubujphri
*bound*for the*lower**bound*of Grohe and Marx: For every $\delta \in (0,\frac{1}{2}]$, every graph $G$ of*treewidth*at least $k$ contains a*bramble*of order $\widetilde{\ ... In this note, we first sharpen the second*bound*by proving that every graph $G$ of*treewidth*at least $k$ contains a*bramble*of order $\widetilde{\Omega}(k^{1/2})$ and congestion $2$, i.e., every vertex ... Second, we provide a tight matching*bound*(up to polylogarithmic factors) to the Grohe-Marx*lower**bound*on the size of a*bramble*. Theorem 1.2. ...##
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Treewidth of the Line Graph of a Complete Graph

2014
*
Journal of Graph Theory
*

In recent articles by Grohe and Marx, the

doi:10.1002/jgt.21813
fatcat:vw6pvekhofaozfgcvvbeoziowi
*treewidth*of the line graph of a complete graph is a critical example-in a certain sense, every graph*with*large*treewidth*"contains" L(K n ). ... However, the*treewidth*of L(K n ) was not determined exactly. We determine the exact*treewidth*of the line graph of a complete graph. ... First construct a*bramble*of large order, thus proving a*lower**bound*on tw(G). Then to prove an upper*bound*, construct a path decomposition of small width. ...##
###
Constant Congestion Brambles
[article]

2022
*
arXiv
*
pre-print

Second, we provide a tight upper

arXiv:2008.02133v3
fatcat:hvipwqd77jhpvacmpfazb2mkae
*bound*for the*lower**bound*of Grohe and Marx: For every δ∈ (0,1/2], every graph G of*treewidth*at least k contains a*bramble*of order Ω(k^1/2+δ) and size 2^𝒪(k^2δ). ... A*bramble*in an undirected graph G is a family of connected subgraphs of G such that for every two subgraphs H_1 and H_2 in the*bramble*either V(H_1) ∩ V(H_2) ≠∅ or there is an edge of G*with*one endpoint ... Second, we provide a tight matching*bound*(up to polylogarithmic factors) to the Grohe-Marx*lower**bound*on the size of a*bramble*. Theorem 1.2. ...##
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Treewidth Bounds for Planar Graphs Using Three-Sided Brambles
[article]

2017
*
arXiv
*
pre-print

We use nets in an O(n^3) time algorithm that computes both upper and

arXiv:1706.08581v1
fatcat:c7ion3x4vbawvppzbzj3z2coeq
*lower**bounds*on the*bramble*number (hence*treewidth*) of any planar graph. ... We correct a*lower**bound*of Bodlaender, Grigoriev and Koster (2008) to s(G)/5 (instead of s(G)/4) and thus the*lower**bound*of λ(G)/4 on our approximation is an improvement. ... Our method for*bounding**treewidth*will be to define a special class of*brambles*called nets. ...##
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Polynomial treewidth forces a large grid-like-minor

2012
*
European journal of combinatorics (Print)
*

Robertson and Seymour proved that every graph

doi:10.1016/j.ejc.2011.09.004
fatcat:3jcng4dsxrce3hmhpuzo2fmvla
*with*sufficiently large*treewidth*contains a large grid minor. ... However, the best known*bound*on the*treewidth*that forces an ℓ×ℓ grid minor is exponential in ℓ. It is unknown whether polynomial*treewidth*suffices. We prove a result in this direction. ... [18] also proved that certain random graphs have*treewidth*proportional to ℓ 2 log ℓ, yet do not contain an ℓ × ℓ grid minor. This is the best known*lower**bound*on the function in Theorem 1.1. ...##
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Discovering Treewidth
[chapter]

2005
*
Lecture Notes in Computer Science
*

*Treewidth*is a graph parameter

*with*several interesting theoretical and practical applications. ... Both theoretical results, establishing the asymptotic computational complexity of the problem, as experimental work on heuristics (both for upper

*bounds*as for

*lower*

*bounds*), preprocessing, exact algorithms ... Acknowledgement I want to express my gratitude to the many colleagues who collaborated

*with*me on the research on

*treewidth*and other topics, helped me

*with*so many things, and from who I learned so much ...

##
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A New Lower Bound on Graph Gonality
[article]

2021
*
arXiv
*
pre-print

We show that the scramble number of a graph is a

arXiv:2006.01020v2
fatcat:bolsuykbdngwxoeha3ox6pwhfm
*lower**bound*for the gonality and an upper*bound*for the*treewidth*. ... Unlike the*treewidth*, the scramble number is not minor monotone, but it is subgraph monotone and invariant under refinement. ...*Treewidth*is a*lower**bound*on graph gonality. Algebr. Comb., 3(4):941–953, 2020. ...
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