scholarly journals The topology of the external activity complex of a matroid

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Federico Ardila ◽  
Federico Castillo ◽  
Jose Samper
Keyword(s):  
Linear Extension ◽  
Activity Complex ◽  
International Audience ◽  
Independence Complex ◽  
A Minor ◽  
Internal Order

International audience We prove that the external activity complex Act<(M) of a matroid is shellable. In fact, we show that every linear extension of Las Vergnas's external/internal order <ext/int on M provides a shelling of Act<(M). We also show that every linear extension of Las Vergnas's internal order <int on M provides a shelling of the independence complex IN(M). As a corollary, Act<(M) and M have the same h-vector. We prove that, after removing its cone points, the external activity complex is contractible if M contains U3,1 as a minor, and a sphere otherwise.

scholarly journals The Topology of the External Activity Complex of a Matroid

10.37236/5042 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Federico Ardila ◽  
Federico Castillo ◽  
José Alejandro Samper
Keyword(s):  
Linear Extension ◽  
Activity Complex ◽  
Independence Complex ◽  
A Minor ◽  
Internal Order

We prove that the external activity complex $\textrm{Act}_<(M)$ of a matroid is shellable. In fact, we show that every linear extension of LasVergnas's external/internal order $<_{ext/int}$ on $M$ provides a shelling of $\textrm{Act}_<(M)$. We also show that every linear extension of LasVergnas's internal order $<_{int}$ on $M$ provides a shelling of the independence complex $IN(M)$. As a corollary, $\textrm{Act}_<(M)$ and $M$ have the same $h$-vector. We prove that, after removing its cone points, the external activity complex is contractible if $M$ contains $U_{1,3}$ as a minor, and a sphere otherwise.

scholarly journals Upper bounds on the non-3-colourability threshold of random graphs

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Nikolaos Fountoulakis ◽  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Upper Bounds ◽  
Average Degree ◽  
Expected Number ◽  
International Audience ◽  
Full Analysis ◽  
A Minor ◽  
Minor Improvement

International audience We present a full analysis of the expected number of 'rigid' 3-colourings of a sparse random graph. This shows that, if the average degree is at least 4.99, then as n → ∞ the expected number of such colourings tends to 0 and so the probability that the graph is 3-colourable tends to 0. (This result is tight, in that with average degree 4.989 the expected number tends to ∞.) This bound appears independently in Kaporis \textitet al. [Kap]. We then give a minor improvement, showing that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989.

scholarly journals Minor-monotone crossing number

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Drago Bokal ◽  
Gašper Fijavž ◽  
Bojan Mohar
Keyword(s):  
Topological Structure ◽  
Crossing Number ◽  
Graph Invariant ◽  
Basic Properties ◽  
International Audience ◽  
Graph Families ◽  
A Minor ◽  
Monotone Graph

International audience The minor crossing number of a graph $G$, $rmmcr(G)$, is defined as the minimum crossing number of all graphs that contain $G$ as a minor. We present some basic properties of this new minor-monotone graph invariant. We give estimates on mmcr for some important graph families using the topological structure of graphs satisfying \$mcr(G) ≤k$.

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 Dyck tilings, linear extensions, descents, and inversions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Jang Soo Kim ◽  
Karola Mészáros ◽  
Greta Panova ◽  
David B. Wilson
Keyword(s):  
Linear Extension ◽  
Lower Boundary ◽  
Linear Extensions ◽  
Young Diagrams ◽  
Dyck Paths ◽  
International Audience

International audience Dyck tilings were introduced by Kenyon and Wilson in their study of double-dimer pairings. They are certain kinds of tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths. We give two bijections between "cover-inclusive'' Dyck tilings and linear extensions of tree posets. The first bijection maps the statistic (area + tiles)/2 to inversions of the linear extension, and the second bijection maps the "discrepancy'' between the upper and lower boundary of the tiling to descents of the linear extension. Les pavages de Dyck ont été introduits par Kenyon et Wilson dans leur étude du modèle des "double-dimères''. Ce sont des pavages des diagrammes de Young gauches avec des tuiles en forme de rubans qui ressemblent à des chemins de Dyck. Nous donnons deux bijections entre les pavages de Dyck ``couvre-inclusive'' et les extensions linéaires de posets dont le diagramme de Hasse est un arbre. La première bijection transforme la statistique (aire + tuiles) / 2 en inversions de l'extension linéaire, et la deuxième bijection transforme la "discordance'' entre la limite supérieure et inférieure du pavage en descentes de l'extension linéaire.

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.

The Effect of Path Length and Display Size on Memory for Spatial Information

2012 ◽  
Vol 59 (3) ◽  
pp. 147-152 ◽  
Author(s):  
Katherine Guérard ◽  
Sébastien Tremblay
Keyword(s):  
Path Length ◽  
Potential Role ◽  
Spatial Information ◽  
Recall Performance ◽  
Minor Role ◽  
Length Effect ◽  
Serial Memory ◽  
Fixed Reference ◽  
A Minor

In serial memory for spatial information, some studies showed that recall performance suffers when the distance between successive locations increases relatively to the size of the display in which they are presented (the path length effect; e.g., Parmentier et al., 2005) but not when distance is increased by enlarging the size of the display (e.g., Smyth & Scholey, 1994). In the present study, we examined the effect of varying the absolute and relative distance between to-be-remembered items on memory for spatial information. We manipulated path length using small (15″) and large (64″) screens within the same design. In two experiments, we showed that distance was disruptive mainly when it is varied relatively to a fixed reference frame, though increasing the size of the display also had a small deleterious effect on recall. The insertion of a retention interval did not influence these effects, suggesting that rehearsal plays a minor role in mediating the effects of distance on serial spatial memory. We discuss the potential role of perceptual organization in light of the pattern of results.

Psychoanalysis—A Minor Classic

1956 ◽  
Vol 1 (12) ◽  
pp. 366-367
Author(s):  
EPHRAIM ROSEN
Keyword(s):  
A Minor
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