scholarly journals Obstructions for Bounded Branch-depth in Matroids

2021 ◽  
Author(s):  
Jochen Pascal Gollin ◽  
Kevin Hendrey ◽  
Dillon Mayhew ◽  
Sang-il Oum
Keyword(s):  
Finite Field ◽  
Natural Analogue ◽  
Large Branch ◽  
Branch Width ◽  
Cycle Matroid ◽  
Fan Graph ◽  
A Minor

DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid Un;2n or the cycle matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. As a corollary, we prove their conjecture for matroids representable over a fixed finite field and quasi-graphic matroids, where the uniform matroid is not an option.

scholarly journals Towards Unavoidable Minors of Binary 4-connected Matroids

2021 ◽  
Author(s):  
◽  
Susan Jowett
Keyword(s):  
Binary Matroid ◽  
Connected Matroid ◽  
Cycle Matroid ◽  
Circular Ladder ◽  
A Minor

<p>We show that for every n ≥ 3 there is some number m such that every 4-connected binary matroid with an M (K3,m)-minor or an M* (K3,m)-minor and no rank-n minor isomorphic to M* (K3,n) blocked in a path-like way, has a minor isomorphic to one of the following: M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, or a rank-n matroid closely related to the cycle matroid of a double wheel, which we call a non graphic double wheel. We also show that for all n there exists m such that the following holds. If M is a 4-connected binary matroid with a sufficiently large spanning restriction that has a certain structure of order m that generalises a swirl-like flower, then M has one of the following as a minor: a rank-n spike, M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, a rank-n non graphic double wheel, M* (K3,n) blocked in a path-like way or a highly structured 3-connected matroid of rank n that we call a clam.</p>

scholarly journals On matroids that are transversal and cotransversal

2021 ◽  
Author(s):  
◽  
Meenu Mariya Jose
Keyword(s):  
Finite Field ◽  
Lattice Path ◽  
Specific Class ◽  
Interesting Phenomenon ◽  
Class A ◽  
Polynomial Time Algorithms ◽  
Undecidable Theory ◽  
Decidable Theories ◽  
Np Complete ◽  
Branch Width

<p>There are distinct differences between classes of matroids that are closed under principal extensions and those that are not Finite-field-representable matroids are not closed under principal extensions and they exhibit attractive properties like well-quasi-ordering and decidable theories (at least for subclasses with bounded branch-width). Infinite-field-representable matroids, on the other hand, are closed under principal extensions and exhibit none of these behaviours. For example, the class of rank-3 real representable matroids is not well-quasi-ordered and has an undecidable theory. The class of matroids that are transversal and cotransversal is not closed under principal extensions or coprincipal coextentions, so we expect it to behave more like the class of finite-field-representable matroids. This thesis is invested in exploring properties in the aforementioned class. A major idea that has inspired the thesis is the investigation of well-quasi-ordered classes in the world of matroids that are transversal and cotransversal. We conjecture that any minor-closed class with bounded branch-width containing matroids that are transversal and cotransversal is well-quasi-ordered. In Chapter 8 of the thesis, we prove this is true for lattice-path matroids, a well-behaved class that falls in this intersection. The general class of lattice-path matroids is not well-quasi-ordered as it contains an infinite antichain of so-called ‘notch matroids’. The interesting phenomenon that we observe is that this is essentially the only antichain in this class, that is, any minor-closed family of lattice-path matroids that contains only finitely many notch matroids is well-quasi-ordered. This answers a question posed by Jim Geelen.  Another question that drove the research was recognising fundamental transversal matroids, since these matroids are also cotransversal. We prove that this problem in general is in NP and conjecture that it is NP-complete. We later explore this question for the classes of lattice-path and bicircular matroids. We are successful in finding polynomial-time algorithms in both classes that identify fundamental transversal matroids. We end this part by investigating the intersection of bicircular and cobicircular matroids. We define a specific class - whirly-swirls - and conjecture that eventually any matroid in the above mentioned intersection belongs to this class.</p>

scholarly journals Towards Unavoidable Minors of Binary 4-connected Matroids

2021 ◽  
Author(s):  
◽  
Susan Jowett
Keyword(s):  
Binary Matroid ◽  
Connected Matroid ◽  
Cycle Matroid ◽  
Circular Ladder ◽  
A Minor

<p>We show that for every n ≥ 3 there is some number m such that every 4-connected binary matroid with an M (K3,m)-minor or an M* (K3,m)-minor and no rank-n minor isomorphic to M* (K3,n) blocked in a path-like way, has a minor isomorphic to one of the following: M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, or a rank-n matroid closely related to the cycle matroid of a double wheel, which we call a non graphic double wheel. We also show that for all n there exists m such that the following holds. If M is a 4-connected binary matroid with a sufficiently large spanning restriction that has a certain structure of order m that generalises a swirl-like flower, then M has one of the following as a minor: a rank-n spike, M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, a rank-n non graphic double wheel, M* (K3,n) blocked in a path-like way or a highly structured 3-connected matroid of rank n that we call a clam.</p>

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 On matroids that are transversal and cotransversal

2021 ◽  
Author(s):  
◽  
Meenu Mariya Jose
Keyword(s):  
Finite Field ◽  
Lattice Path ◽  
Specific Class ◽  
Interesting Phenomenon ◽  
Class A ◽  
Polynomial Time Algorithms ◽  
Undecidable Theory ◽  
Decidable Theories ◽  
Np Complete ◽  
Branch Width

<p>There are distinct differences between classes of matroids that are closed under principal extensions and those that are not Finite-field-representable matroids are not closed under principal extensions and they exhibit attractive properties like well-quasi-ordering and decidable theories (at least for subclasses with bounded branch-width). Infinite-field-representable matroids, on the other hand, are closed under principal extensions and exhibit none of these behaviours. For example, the class of rank-3 real representable matroids is not well-quasi-ordered and has an undecidable theory. The class of matroids that are transversal and cotransversal is not closed under principal extensions or coprincipal coextentions, so we expect it to behave more like the class of finite-field-representable matroids. This thesis is invested in exploring properties in the aforementioned class. A major idea that has inspired the thesis is the investigation of well-quasi-ordered classes in the world of matroids that are transversal and cotransversal. We conjecture that any minor-closed class with bounded branch-width containing matroids that are transversal and cotransversal is well-quasi-ordered. In Chapter 8 of the thesis, we prove this is true for lattice-path matroids, a well-behaved class that falls in this intersection. The general class of lattice-path matroids is not well-quasi-ordered as it contains an infinite antichain of so-called ‘notch matroids’. The interesting phenomenon that we observe is that this is essentially the only antichain in this class, that is, any minor-closed family of lattice-path matroids that contains only finitely many notch matroids is well-quasi-ordered. This answers a question posed by Jim Geelen.  Another question that drove the research was recognising fundamental transversal matroids, since these matroids are also cotransversal. We prove that this problem in general is in NP and conjecture that it is NP-complete. We later explore this question for the classes of lattice-path and bicircular matroids. We are successful in finding polynomial-time algorithms in both classes that identify fundamental transversal matroids. We end this part by investigating the intersection of bicircular and cobicircular matroids. We define a specific class - whirly-swirls - and conjecture that eventually any matroid in the above mentioned intersection belongs to this class.</p>

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.

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