scholarly journals Fan-Extensions in Fragile Matroids

10.37236/3994 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Carolyn Chun ◽  
Deborah Chun ◽  
Dillon Mayhew ◽  
Stefan H. M. Van Zwam
Keyword(s):  
Case Analysis ◽  
Finite Case ◽  
Closed Class ◽  
Excluded Minor ◽  
A Minor

If $\mathcal{S}$ is a set of matroids, then the matroid $M$ is $\mathcal{S}$-fragile if, for every element $e\in E(M)$, either $M\backslash e$ or $M/e$ has no minor isomorphic to a member of $\mathcal{S}$. Excluded-minor characterizations often depend, implicitly or explicitly, on understanding classes of fragile matroids. In certain cases, when $\mathcal{M}$ is a minor-closed class of $\mathcal{S}$-fragile matroids, and $N\in \mathcal{M}$, the only members of $\mathcal{M}$ that contain $N$ as a minor are obtained from $N$ by increasing the length of fans. We prove that if this is the case, then we can certify it with a finite case-analysis. The analysis  involves examining matroids that are at most two elements larger than $N$.

Random Graphs from a Minor-Closed Class

2009 ◽  
Vol 18 (4) ◽  
pp. 583-599 ◽  
Author(s):  
COLIN McDIARMID
Keyword(s):  
Poisson Distribution ◽  
Random Graph ◽  
Generating Function ◽  
Random Graphs ◽  
Giant Component ◽  
Closed Class ◽  
Exponential Generating Function ◽  
Excluded Minor ◽  
A Minor ◽  
Labelled Graphs

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a classof labelled graphs has smooth growth; and, for the random graphRnsampled uniformly from then-vertex graphs in, the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, asn→ ∞ the probability thatRnis connected tends to 1/A(ρ), whereA(x) is the exponential generating function forand ρ is its radius of convergence.

scholarly journals Asymptotic Properties of Some Minor-Closed Classes of Graphs

2014 ◽  
Vol 23 (5) ◽  
pp. 749-795 ◽  
Author(s):  
MIREILLE BOUSQUET-MÉLOU ◽  
KERSTIN WELLER
Keyword(s):  
Asymptotic Properties ◽  
Systematic Investigation ◽  
Exact Enumeration ◽  
Limit Laws ◽  
Closed Class ◽  
Connected Case ◽  
Non Gaussian ◽  
A Minor ◽  
Excluded Minors

Let${\cal A}$be a minor-closed class of labelled graphs, and let${\cal G}_{n}$be a random graph sampled uniformly from the set ofn-vertex graphs of${\cal A}$. Whennis large, what is the probability that${\cal G}_{n}$is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected.Using exact enumeration, we study a collection of classes${\cal A}$excluding non-2-connected minors, and show that their asymptotic behaviour may be rather different from the 2-connected case. This behaviour largely depends on the nature of the dominant singularity of the generating functionC(z) that counts connected graphs of${\cal A}$. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. It follows non-Gaussian limit laws (Beta and Gamma), and clearly merits a systematic investigation.

scholarly journals An Extension of Matroid Rank Submodularity and the $Z$-Rayleigh Property

10.37236/600 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Arun P. Mani
Keyword(s):  
Half Plane ◽  
Closed Class ◽  
Tutte Polynomials ◽  
A Minor

We define an extension of matroid rank submodularity called $R$-submodularity, and introduce a minor-closed class of matroids called extended submodular matroids that are well-behaved with respect to $R$-submodularity. We apply $R$-submodularity to study a class of matroids with negatively correlated multivariate Tutte polynomials called the $Z$-Rayleigh matroids. First, we show that the class of extended submodular matroids are $Z$-Rayleigh. Second, we characterize a minor-minimal non-$Z$-Rayleigh matroid using its $R$-submodular properties. Lastly, we use $R$-submodularity to show that the Fano and non-Fano matroids (neither of which is extended submodular) are $Z$-Rayleigh, thus giving the first known examples of $Z$-Rayleigh matroids without the half-plane property.

On random graphs from a minor-closed class

2016 ◽  
pp. 102-120
Author(s):  
Colin McDiarmid ◽  
Michael Krivelevich ◽  
Konstantinos Panagiotou ◽  
Mathew Penrose ◽  
Colin McDiarmid
Keyword(s):  
Random Graphs ◽  
Closed Class ◽  
A Minor

scholarly journals The Minor Crossing Number of Graphs with an Excluded Minor

10.37236/728 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Drago Bokal ◽  
Gašper Fijavž ◽  
David R. Wood
Keyword(s):  
Crossing Number ◽  
Excluded Minor ◽  
A Minor

The minor crossing number of a graph $G$ is the minimum crossing number of a graph that contains $G$ as a minor. It is proved that for every graph $H$ there is a constant $c$, such that every graph $G$ with no $H$-minor has minor crossing number at most $c|V(G)|$.

Serial Verbs

2018 ◽  
Author(s):  
Alexandra Y. Aikhenvald
Keyword(s):  
Minor Component ◽  
Serial Verb Constructions ◽  
Closed Class ◽  
Verb Constructions ◽  
Powerful Means ◽  
Serial Verbs ◽  
Serial Verb ◽  
A Minor ◽  
Open Class ◽  
Syntactic Dependency

In many languages of the world, a sequence of several verbs act together as one unit. These sequences—known as serial verbs—form one predicate and contain no overt marker of coordination, subordination, or syntactic dependency of any sort. Serial verbs describe what can be conceptualized as one single event. They are often pronounced as if they were one word, and tend to share subjects and objects. The whole serial verb will have one value for tense, aspect, mood, modality, and evidentiality. Their components cannot be negated or questioned separately without negating or questioning the whole construction. Asymmetrical serial verbs consist of a ‘minor’ verb from a closed class and a major verb from an open class. The minor component tends to grammaticalize giving rise to markers of aspect, directionality, valency increase, prepositions, and coordinators. Symmetrical serial verbs consist of several components each from an open class. They may undergo lexicalization and become non-compositional idioms. Various grammatical categories—including person, tense, aspect, and negation—can be marked on each component, or just once per construction. Serial verb constructions are a powerful means for a detailed portrayal of various facets of one event. They have numerous grammatical and discourse functions. Serial verbs have to be distinguished from verb sequences of other kinds, including constructions with converbs and auxiliaries, and from verbal compounds. The book sets out cross-linguistic parameters of variation for serial verbs based on an inductive approach and discusses their synchronic and diachronic properties, functions, and histories.

A counter-example to ‘Wagner's conjecture’ for infinite graphs

1988 ◽  
Vol 103 (1) ◽  
pp. 55-57 ◽  
Author(s):  
Robin Thomas
Keyword(s):  
Graph Theory ◽  
Planar Graphs ◽  
Infinite Sequence ◽  
Infinite Graphs ◽  
Famous Theorem ◽  
Finite Graphs ◽  
Counter Example ◽  
Excluded Minor ◽  
A Minor ◽  
Forbidden Minors

Wagner made the conjecture that given an infinite sequence G1, G2, … of finite graphs there are indices i < j such that Gi is a minor of Gj. (A graph is a minor of another if the first can be obtained by contraction from a subgraph of the second.) The importance of this conjecture is that it yields excluded minor theorems in graph theory, where by an excluded minor theorem we mean a result asserting that a graph possesses a specified property if and only if none of its minors belongs to a finite list of ‘forbidden minors’. A widely known example of an excluded minor theorem is Kuratowski's famous theorem on planar graphs; one of its formulations says that a graph is planar if and only if it has neither K5 nor K3, 3 as a minor. But several other excluded minor theorems have been discovered by now (see e.g. [7–9]).

scholarly journals On the Excluded Minors for Matroids of Branch-Width Three

10.37236/1648 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Petr Hliněný
Keyword(s):  
Computer Assisted ◽  
Binary Matroids ◽  
Alternative Characterization ◽  
Excluded Minor ◽  
Practical Algorithm ◽  
Branch Width ◽  
A Minor ◽  
Excluded Minors

Knowing the excluded minors for a minor-closed matroid property provides a useful alternative characterization of that property. It has been shown in [R. Hall, J. Oxley, C. Semple, G. Whittle, On Matroids of Branch-Width Three, submitted 2001] that if $M$ is an excluded minor for matroids of branch-width $3$, then $ M$ has at most $14$ elements. We show that there are exactly $10$ such binary matroids $M$ (7 of which are regular), proving a conjecture formulated by Dharmatilake in 1994. We also construct numbers of such ternary and quaternary matroids $ M$, and provide a simple practical algorithm for finding a width-$3$ branch-decomposition of a matroid. The arguments in our paper are computer-assisted — we use a program $MACEK$ [P. Hliněný, The MACEK Program, http://www.mcs.vuw.ac.nz/research/macek, 2002] for structural computations with represented matroids. Unfortunately, it seems to be infeasible to search through all matroids on at most $14$ elements.

scholarly journals Asymptotic properties of some minor-closed classes of graphs (conference version)

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
Kerstin Weller
Keyword(s):  
Asymptotic Properties ◽  
Systematic Investigation ◽  
Limit Laws ◽  
Closed Class ◽  
Connected Case ◽  
International Audience ◽  
Non Gaussian ◽  
A Minor ◽  
Excluded Minors

International audience Let $\mathcal{A}$ be a minor-closed class of labelled graphs, and let $G_n$ be a random graph sampled uniformly from the set of n-vertex graphs of $\mathcal{A}$. When $n$ is large, what is the probability that $G_n$ is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected. Using exact enumeration, we study a collection of classes $\mathcal{A}$ excluding non-2-connected minors, and show that their asymptotic behaviour is sometimes rather different from the 2-connected case. This behaviour largely depends on the nature of the dominant singularity of the generating function $C(z)$ that counts connected graphs of $\mathcal{A}$. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. This follows non-gaussian limit laws (beta and gamma), and clearly deserves a systematic investigation.

Vertex-Bipartition Method for Colouring Minor-Closed Classes of Graphs

2010 ◽  
Vol 19 (4) ◽  
pp. 579-591 ◽  
Author(s):  
GUOLI DING ◽  
STAN DZIOBIAK
Keyword(s):  
Approximation Algorithms ◽  
Absolute Constant ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Fixed Number ◽  
Time Approximation ◽  
Closed Class ◽  
Tree Width ◽  
A Minor ◽  
Minor Free Graphs

Thomas conjectured that there is an absolute constant c such that for every proper minor-closed class of graphs, there is a polynomial-time algorithm that can colour every G ∈ with at most χ(G) + c colours. We introduce a parameter ρ(), called the degenerate value of , which is defined to be the smallest r such that every G ∈ can be vertex-bipartitioned into a part of bounded tree-width (the bound depending only on ), and a part that is r-degenerate. Although the existence of one global bound for the degenerate values of all proper minor-closed classes would imply Thomas's conjecture, we prove that the values ρ() can be made arbitrarily large. The problem lies in the clique sum operation. As our main result, we show that excluding a planar graph with a fixed number of apex vertices gives rise to a minor-closed class with small degenerate value. As corollaries, we obtain that (i) the degenerate value of every class of graphs of bounded local tree-width is at most 6, and (ii) the degenerate value of the class of Kn-minor-free graphs is at most n + 1. These results give rise to P-time approximation algorithms for colouring any graph in these classes within an error of at most 7 and n + 2 of its chromatic number, respectively.

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