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.

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.

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.

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

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.

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.

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 $N$.

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.

Rating Post-Operative Backs (Part I)

1997 ◽  
Vol 2 (4) ◽  
pp. 1-3
Author(s):  
James B. Talmage
Keyword(s):  
Spinal Stenosis ◽  
Foot Drop ◽  
Neurogenic Claudication ◽  
Root Damage ◽  
Fourth Edition ◽  
Minor Strain ◽  
Injury Model ◽  
Strain Type ◽  
A Minor ◽  
Anterior Tibialis

Abstract The AMA Guides to the Evaluation of Permanent Impairment, Fourth Edition, uses the Injury Model to rate impairment in people who have experienced back injuries. Injured individuals who have not required surgery can be rated using differentiators. Challenges arise when assessing patients whose injuries have been treated surgically before the patient is rated for impairment. This article discusses five of the most common situations: 1) What is the impairment rating for an individual who has had an injury resulting in sciatica and who has been treated surgically, either with chemonucleolysis or with discectomy? 2) What is the impairment rating for an individual who has a back strain and is operated on without reasonable indications? 3) What is the impairment rating of an individual with sciatica and a foot drop (major anterior tibialis weakness) from L5 root damage? 4) What is the rating for an individual who is injured, has true radiculopathy, undergoes a discectomy, and is rated as Category III but later has another injury and, ultimately, a second disc operation? 5) What is the impairment rating for an older individual who was asymptomatic until a minor strain-type injury but subsequently has neurogenic claudication with severe surgical spinal stenosis on MRI/myelography? [Continued in the September/October 1997 The Guides Newsletter]

Pelvic Impairments

2018 ◽  
Vol 23 (4) ◽  
pp. 9-10
Author(s):  
James Talmage ◽  
Jay Blaisdell
Keyword(s):  
Pelvic Ring ◽  
Reproductive Organs ◽  
High Impact ◽  
Pelvic Fractures ◽  
Motor Dysfunction ◽  
Impact Event ◽  
Concomitant Injuries ◽  
Rating Process ◽  
Pass Through ◽  
A Minor

Abstract Pelvic fractures are relatively uncommon, and in workers’ compensation most pelvic fractures are the result of an acute, high-impact event such as a fall from a roof or an automobile collision. A person with osteoporosis may sustain a pelvic fracture from a lower-impact injury such as a minor fall. Further, major parts of the bladder, bowel, reproductive organs, nerves, and blood vessels pass through the pelvic ring, and traumatic pelvic fractures that result from a high-impact event often coincide with damaged organs, significant bleeding, and sensory and motor dysfunction. Following are the steps in the rating process: 1) assign the diagnosis and impairment class for the pelvis; 2) assign the functional history, physical examination, and clinical studies grade modifiers; and 3) apply the net adjustment formula. Because pelvic fractures are so uncommon, raters may be less familiar with the rating process for these types of injuries. The diagnosis-based methodology for rating pelvic fractures is consistent with the process used to rate other musculoskeletal impairments. Evaluators must base the rating on reliable data when the patient is at maximum medical impairment and must assess possible impairment from concomitant injuries.

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