scholarly journals Excluding any graph as a minor allows a low tree-width 2-coloring

2004 ◽  
Vol 91 (1) ◽  
pp. 25-41 ◽  
Author(s):  
Matt DeVos ◽  
Guoli Ding ◽  
Bogdan Oporowski ◽  
Daniel P. Sanders ◽  
Bruce Reed ◽  
...  
Keyword(s):  
10.37236/7691 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Daniel Weißauer

By the Grid Minor Theorem of Robertson and Seymour, every graph of sufficiently large tree-width contains a large grid as a minor. Tree-width may therefore be regarded as a measure of 'grid-likeness' of a graph. The grid contains a long cycle on the perimeter, which is the $\mathbb{F}_2$-sum of the rectangles inside. Moreover, the grid distorts the metric of the cycle only by a factor of two. We prove that every graph that resembles the grid in this algebraic sense has large tree-width: Let $k, p$ be integers, $\gamma$ a real number and $G$ a graph. Suppose that $G$ contains a cycle of length at least $2 \gamma p k$ which is the $\mathbb{F}_2$-sum of cycles of length at most $p$ and whose metric is distorted by a factor of at most $\gamma$. Then $G$ has tree-width at least $k$.


2010 ◽  
Vol 19 (4) ◽  
pp. 579-591 ◽  
Author(s):  
GUOLI DING ◽  
STAN DZIOBIAK

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.


1997 ◽  
Vol 2 (4) ◽  
pp. 1-3
Author(s):  
James B. Talmage

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]


2018 ◽  
Vol 23 (4) ◽  
pp. 9-10
Author(s):  
James Talmage ◽  
Jay Blaisdell

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.


Author(s):  
Katherine Guérard ◽  
Sébastien Tremblay

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.


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

1978 ◽  
Vol 23 (5) ◽  
pp. 376-377 ◽  
Author(s):  
HENRI TAJFEL
Keyword(s):  

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