scholarly journals A Note on Forbidding Clique Immersions

10.37236/2533 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Matt DeVos ◽  
Jessica McDonald ◽  
Bojan Mohar ◽  
Diego Scheide
Keyword(s):  
Short Proof ◽  
Structure Theorem ◽  
Graph Minors ◽  
Finite Graphs

Robertson and Seymour proved that the relation of graph immersion is well-quasi-ordered for finite graphs. Their proof uses the results of graph minors theory. Surprisingly, there is a very short proof of the corresponding rough structure theorem for graphs without $K_t$-immersions; it is based on the Gomory-Hu theorem. The same proof also works to establish a rough structure theorem for Eulerian digraphs without $\vec{K}_t$-immersions, where $\vec{K}_t$ denotes the bidirected complete digraph of order $t$.

scholarly journals Optimizing the Graph Minors Weak Structure Theorem

2013 ◽  
Vol 27 (3) ◽  
pp. 1209-1227 ◽  
Author(s):  
Archontia C. Giannopoulou ◽  
Dimitrios M. Thilikos
Keyword(s):  
Structure Theorem ◽  
Graph Minors ◽  
Weak Structure

Graph Minors I: A Short Proof of the Path-width Theorem

1995 ◽  
Vol 4 (1) ◽  
pp. 27-30 ◽  
Author(s):  
Reinhard Diestel
Keyword(s):  
Short Proof ◽  
Graph Minors

Robertson and Seymour proved that excluding any fixed forest F as a minorimposes a bound on the path-width of a graph. We give a short proof of this, reobtaining the best possible bound of |F| – 2.

A Short Proof of the Factor Theorem for Finite Graphs

1954 ◽  
Vol 6 ◽  
pp. 347-352 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Short Proof ◽  
Common Element ◽  
Finite Graphs ◽  
Factor Theorem

We define a graph as a set V of objects called vertices together with a set E of objects called edges, the two sets having no common element. With each edge there are associated just two vertices, called its ends. We say that an edge joins its ends. Two vertices may be joined by more than one edge.

Existence

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.

On Lattice-Ordered Rees Matrix Γ-Semigroups

2013 ◽  
Vol 59 (1) ◽  
pp. 209-218 ◽  
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.

A note on generalized quasi-contraction

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3091-3093
Author(s):  
Dejan Ilic ◽  
Darko Kocev
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Short Proof

In this paper we give a short proof of the main results of Kumam, Dung and Sitthithakerngkiet (P. Kumam, N.V. Dung, K. Sitthithakerngkiet, A Generalization of Ciric Fixed Point Theorems, FILOMAT 29:7 (2015), 1549-1556).

scholarly journals Mass-conserving diffusion-based dynamics on graphs

2021 ◽  
pp. 1-49
Author(s):  
J.M BUDD ◽  
Y. VAN GENNIP
Keyword(s):  
Multiscale Modeling ◽  
Classification Problems ◽  
Theoretical Justification ◽  
Finite Graphs ◽  
Discrete Scheme ◽  
Cahn Equation ◽  
Separating Flows ◽  
Special Case ◽  
Appl Math ◽  
Convergence Of The Scheme

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

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