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### Hitting minors on bounded treewidth graphs. I. General upper bounds [article]

Julien Baste, Ignasi Sau, Dimitrios M. Thilikos
2021 arXiv   pre-print
the graphs in F as a minor.  ...  We also consider the version of the problem where the graphs in F are forbidden as topological minors, called F-TM-DELETION.  ...  versions containing some of the results of this article for helpful remarks that improved the presentation of the manuscript, and Édouard Bonnet, Eun Jung Kim, and Juanjo Rué for insightful discussions on  ...

### On the treewidth of toroidal grids

Masashi Kiyomi, Yoshio Okamoto, Yota Otachi
2016 Discrete Applied Mathematics
Treewidth of graphs is a graph parameter for measuring how close a graph is to a tree.  ...  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.  ...  Furthermore, it is known that a graph has bounded treewidth if and only if the size of a maximum grid minor in the graph is bounded  .  ...

### Quadratic Upper Bounds on the Erdős-Pósa Property for a Generalization of Packing and Covering Cycles

Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, Saket Saurabh
2013 Journal of Graph Theory
Robertson and Seymour [Graph minors. V. Excluding a planar graph. J. Comb. Theory Series B, 41:92-114, 1986] generalized this result in the best possible way.  ...  According to the classical Erdős-Pósa theorem, given a positive integer k, every graph G either contains k vertex disjoint cycles or a set of at most O(k log k) vertices that hits all its cycles.  ...  We believe that even for θ c , the correct upper bound on the size of a minimum hitting set when a graph G does not have k vertex disjoint θ c -minor models is O(k log k).  ...

### Coalition Games on Interaction Graphs: A Horticultural Perspective [article]

Nicolas Bousquet, Zhentao Li, Adrian Vetta
2015 arXiv   pre-print
This gap is upper bounded by the packing-covering ratio which, for graphical coalition games, is known to be at most the treewidth of the interaction graph plus one (Meir et al. 2013).  ...  The thicket number provides an upper bound of both integrality gaps.  ...  Robertson and Seymour  proved that every graph of treewidth k admits a grid minor of size f (k), that is, the f (k) × f (k) grid is a minor of every graph of treewidth at least k.  ...

### Coalition Games on Interaction Graphs

Nicolas Bousquet, Zhentao Li, Adrian Vetta
2015 Proceedings of the Sixteenth ACM Conference on Economics and Computation - EC '15
This gap is upper bounded by the packing-covering ratio which, for graphical coalition games, is known to be at most the treewidth of the interaction graph plus one  .  ...  The thicket number provides an upper bound of both integrality gaps.  ...  Robertson and Seymour  proved that every graph of treewidth k admits a grid minor of size f (k), that is, the f (k) × f (k) grid is a minor of every graph of treewidth at least k.  ...

### Hitting forbidden subgraphs in graphs of bounded treewidth

Marek Cygan, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk
2017 Information and Computation
We study the complexity of a generic hitting problem H-Subgraph Hitting, where given a fixed pattern graph H and an input graph G, we seek for the minimum size of a set X ⊆ V (G) that hits all subgraphs  ...  For the colorful variant, we demonstrate matching upper and lower bounds showing that the dependence of the running time on treewidth of G is tightly governed by µ(H), the maximum size of a minimal vertex  ...  Classic results on model checking monadic second-order logic on graphs of bounded treewidth, such as Courcelle's Theorem, provide a unified and generic way of proving fixed-parameter tractability of most  ...

### Bounds on vertex colorings with restrictions on the union of color classes

N. R. Aravind, C. R. Subramanian
2011 Journal of Graph Theory
of graphs as well as a general upper bound of O(d 3/2 ) where d denotes the maximum degree of G.  ...  In this paper, we find upper bounds for these general chromatic numbers in terms of the maximum degree of a graph.  ...

### Heuristic computation of exact treewidth [article]

Hisao Tamaki
2022 arXiv   pre-print
We are interested in computing the treewidth (G) of a given graph G.  ...  The upper bound algorithm extends and simplifies Tamaki's unpublished work on a heuristic use of the dynamic programming algorithm for deciding treewidth due to Bouchitté and Todinca.  ...  Acknowledgment I thank Holger Dell for posing the challenging bonus instances, which have kept defying my "great ideas", showing how they fail, and pointing to yet greater ideas.  ...

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

Michael Harp, Elijah Jackson, David Jensen, Noah Speeter
2021 arXiv   pre-print
We define a new graph invariant called the scramble number. 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.  ...  We show that the scramble number of a graph is a lower bound for the gonality and an upper bound for the treewidth.  ...

### Treewidth Lower Bounds with Brambles [chapter]

Hans L. Bodlaender, Alexander Grigoriev, Arie M. C. A. Koster
2005 Lecture Notes in Computer Science
Our technique is based on the fact that the treewidth of a graph G is the maximum order of a bramble of G minus one. We give two algorithms: one for general graphs, and one for planar graphs.  ...  In this paper we present a new technique for computing lower bounds for graph treewidth.  ...  Acknowledgements We thank Illya Hicks for providing us with the planar graphs for our experiments.  ...

### Treewidth Lower Bounds with Brambles

Hans L. Bodlaender, Alexander Grigoriev, Arie M. C. A. Koster
2007 Algorithmica
Our technique is based on the fact that the treewidth of a graph G is the maximum order of a bramble of G minus one. We give two algorithms: one for general graphs, and one for planar graphs.  ...  In this paper we present a new technique for computing lower bounds for graph treewidth.  ...  Acknowledgements We thank Illya Hicks for providing us with the planar graphs for our experiments.  ...

### Treewidth and gonality of glued grid graphs [article]

Ivan Aidun, Frances Dean, Ralph Morrison, Teresa Yu, Julie Yuan
2019 arXiv   pre-print
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 discuss connections to divisorial graph theory coming from tropical geometry, and use our results to compute the divisorial gonality of these graphs.  ...  We combine these with upper bounds on treewidth to achieve our desired results from Theorem 1.1.  ...

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

Stephan Kreutzer, Siamak Tazari
2009 arXiv   pre-print
can be decided by a linear time algorithm on classes of graphs of bounded treewidth.  ...  Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour.  ...  They use the grid-minor theorem for general graphs, together with ideas from the bidimensionality theory [DFHT05] , to obtain this bound.  ...

### Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs [chapter]

Erik D. Demaine, MohammadTaghi Hajiaghayi, Ken-ichi Kawarabayashi
2010 Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms
an algorithmic decomposition into two bounded-treewidth graphs, generalizing a similar result for minors.  ...  Our decomposition results provide new structural insights into odd-H-minor-free graphs, on the one hand generalizing the central structural result from Graph Minor Theory, and on the other hand providing  ...  We also thank Paul Seymour for general insights into Graph Minor Theory.  ...

### 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.  ...
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