On the excluded minor structure theorem for graphs of large tree-width

The width of a delta-matroid is the difference in size between a maximal and minimal feasible set. We give a Rough Structure Theorem for delta-matroids that admit a twist of width one. We apply this theorem to give an excluded minor characterisation of delta-matroids that admit a twist of width at most one.

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The excluded minor structure theorem with planarly embedded wall

Ars Mathematica Contemporanea ◽

10.26493/1855-3974.118.3dd ◽

2012 ◽

Vol 6 (2) ◽

pp. 187-196

Author(s):

Bojan Mohar

Keyword(s):

Structure Theorem ◽

Excluded Minor

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Algebraically Grid-Like Graphs have Large Tree-Width

The Electronic Journal of Combinatorics ◽

10.37236/7691 ◽

2019 ◽

Vol 26 (1) ◽

Author(s):

Daniel Weißauer

Keyword(s):

Real Number ◽

Long Cycle ◽

Large Tree ◽

Tree Width ◽

A Minor ◽

Grid Minor ◽

Large Grid

By the Grid Minor Theorem of Robertson and Seymour, every graph of sufficiently large tree-width contains a large grid as a minor. Tree-width may therefore be regarded as a measure of 'grid-likeness' of a graph. The grid contains a long cycle on the perimeter, which is the $\mathbb{F}_2$-sum of the rectangles inside. Moreover, the grid distorts the metric of the cycle only by a factor of two. We prove that every graph that resembles the grid in this algebraic sense has large tree-width: Let $k, p$ be integers, $\gamma$ a real number and $G$ a graph. Suppose that $G$ contains a cycle of length at least $2 \gamma p k$ which is the $\mathbb{F}_2$-sum of cycles of length at most $p$ and whose metric is distorted by a factor of at most $\gamma$. Then $G$ has tree-width at least $k$.

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On planar graphs with large tree-width and small grid minors

Electronic Notes in Discrete Mathematics ◽

10.1016/j.endm.2009.02.006 ◽

2009 ◽

Vol 32 ◽

pp. 35-42 ◽

Cited By ~ 5

Author(s):

Alexander Grigoriev ◽

Bert Marchal ◽

Natalya Usotskaya

Keyword(s):

Planar Graphs ◽

Large Tree ◽

Tree Width

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In absence of long chordless cycles, large tree-width becomes a local phenomenon

Journal of Combinatorial Theory Series B ◽

10.1016/j.jctb.2019.04.004 ◽

2019 ◽

Vol 139 ◽

pp. 342-352

Author(s):

Daniel Weißauer

Keyword(s):

Large Tree ◽

Local Phenomenon ◽

Tree Width

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Existence

10.23943/princeton/9780691166902.003.0016 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Structure Theorem ◽

Quadratic Extension ◽

Quadratic Space ◽

Division Algebras ◽

Separable Extension ◽

Valued Fields ◽

Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

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On Lattice-Ordered Rees Matrix Γ-Semigroups

Annals of the Alexandru Ioan Cuza University - Mathematics ◽

10.2478/v10157-012-0033-8 ◽

2013 ◽

Vol 59 (1) ◽

pp. 209-218 ◽

Cited By ~ 2

Author(s):

Kostaq Hila ◽

Edmond Pisha

Keyword(s):

Structure Theorem ◽

Lattice Of Congruences

Abstract The purpose of this paper is to introduce and give some properties of l-Rees matrix Γ-semigroups. Generalizing the results given by Guowei and Ping, concerning the congruences and lattice of congruences on regular Rees matrix Γ-semigroups, the structure theorem of l-congruences lattice of l - Γ-semigroup M = μº(G : I; L; Γe) is given, from which it follows that this l-congruences lattice is distributive.

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A structure theorem for polyharmonic maps between Riemannian manifolds

Journal of Differential Equations ◽

10.1016/j.jde.2020.11.046 ◽

2021 ◽

Vol 273 ◽

pp. 14-39

Author(s):

Volker Branding

Keyword(s):

Riemannian Manifolds ◽

Structure Theorem

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Quasi-ideal Ehresmann transversals: The spined product structure

Open Mathematics ◽

10.1515/math-2020-0071 ◽

2021 ◽

Vol 19 (1) ◽

pp. 77-86

Author(s):

Xiangjun Kong ◽

Pei Wang ◽

Jian Tang

Keyword(s):

Structure Theorem ◽

Product Structure ◽

Abundant Semigroup ◽

Spined Product ◽

Equivalent Conditions

Abstract In any U-abundant semigroup with an Ehresmann transversal, two significant components R and L are introduced in this paper and described by Green’s ∼ \sim -relations. Some interesting properties associated with R and L are explored and some equivalent conditions for the Ehresmann transversal to be a quasi-ideal are acquired. Finally, a spined product structure theorem is established for a U-abundant semigroup with a quasi-ideal Ehresmann transversal by means of R and L.

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