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Treewidth Lower Bounds with Brambles

Hans L. Bodlaender, Alexander Grigoriev, Arie M. C. A. Koster
2007 Algorithmica  
In this paper we present a new technique for computing 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.  ... 
doi:10.1007/s00453-007-9056-z fatcat:lamgyxjesrggze6wppny5xk7he

Treewidth Lower Bounds with Brambles [chapter]

Hans L. Bodlaender, Alexander Grigoriev, Arie M. C. A. Koster
2005 Lecture Notes in Computer Science  
In this paper we present a new technique for computing 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.  ... 
doi:10.1007/11561071_36 fatcat:hvhke4uh3nghdod5qd724jdyuq

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

Stephan Kreutzer, Siamak Tazari
2010 Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms  
Perfect 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  ... 
doi:10.1137/1.9781611973075.30 dblp:conf/soda/KreutzerT10 fatcat:udgkaapsynhqlpbm5pg3vfqdxy

On the treewidth of toroidal grids

Masashi Kiyomi, Yoshio Okamoto, Yota Otachi
2016 Discrete Applied Mathematics  
To show the 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.  ... 
doi:10.1016/j.dam.2015.06.027 fatcat:k33laemlunhwpcl4hnrt6kzsdu

Treewidth and gonality of glued grid graphs [article]

Ivan Aidun, Frances Dean, Ralph Morrison, Teresa Yu, Julie Yuan
2019 arXiv   pre-print
Our main technique is constructing strict 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.  ... 
arXiv:1808.09475v2 fatcat:2imvhh4iavcp3pqdyuaykpyoci

Treewidth of Cartesian Products of Highly Connected Graphs

David R. Wood
2012 Journal of Graph Theory  
For n≫ k this 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.  ... 
doi:10.1002/jgt.21677 fatcat:gl2fucw7orcf3hnpvjovnawsle

Treewidth computations II. Lower bounds

Hans L. Bodlaender, Arie M.C.A. Koster
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.  ... 
doi:10.1016/j.ic.2011.04.003 fatcat:tndpvxgrv5d6par5f7mib3ft24

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

Stephan Kreutzer, Siamak Tazari
2009 arXiv   pre-print
Perfect 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  ... 
arXiv:0907.3076v1 fatcat:ajfgzk6oqregpiwfrvvkec3viu

Constant Congestion Brambles

Meike Hatzel, Pawel Komosa, Marcin Pilipczuk, Manuel Sorge
2022 Discrete Mathematics & Theoretical Computer Science  
Second, we provide a tight upper 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.  ... 
doi:10.46298/dmtcs.6699 fatcat:37wgzyuulveffb4nx5ubujphri

Treewidth of the Line Graph of a Complete Graph

Daniel J. Harvey, David R. Wood
2014 Journal of Graph Theory  
In recent articles by Grohe and Marx, the 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.  ... 
doi:10.1002/jgt.21813 fatcat:vw6pvekhofaozfgcvvbeoziowi

Constant Congestion Brambles [article]

Meike Hatzel, Pawel Komosa, Marcin Pilipczuk, Manuel Sorge
2022 arXiv   pre-print
Second, we provide a tight upper 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.  ... 
arXiv:2008.02133v3 fatcat:hvipwqd77jhpvacmpfazb2mkae

Treewidth Bounds for Planar Graphs Using Three-Sided Brambles [article]

Karen L. Collins, Brett C. Smith
2017 arXiv   pre-print
We use nets in an O(n^3) time algorithm that computes both upper and 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.  ... 
arXiv:1706.08581v1 fatcat:c7ion3x4vbawvppzbzj3z2coeq

Polynomial treewidth forces a large grid-like-minor

Bruce A. Reed, David R. Wood
2012 European journal of combinatorics (Print)  
Robertson and Seymour proved that every graph 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.  ... 
doi:10.1016/j.ejc.2011.09.004 fatcat:3jcng4dsxrce3hmhpuzo2fmvla

Discovering Treewidth [chapter]

Hans L. Bodlaender
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  ... 
doi:10.1007/978-3-540-30577-4_1 fatcat:p6qw6ze5jndr7mr7arggak5oo4

A New Lower Bound on Graph Gonality [article]

Michael Harp, Elijah Jackson, David Jensen, Noah Speeter
2021 arXiv   pre-print
We show that the scramble number of a graph is a 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.  ... 
arXiv:2006.01020v2 fatcat:bolsuykbdngwxoeha3ox6pwhfm
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