245:
A weak form of Tutte's homotopy theorem states that any closed path is homotopic to the trivial path. A stronger form states a similar result for paths not meeting certain "convex" subsets.
146:
are called 0-flats, the minimal non-empty flats that are not 0-flats are called 1-flats, the minimal nonempty flats that are not 0-flats or 1-flats are called 2-flats, and so on.
39:, and states roughly that closed paths can be written as compositions of elementary closed paths, so that in some sense they are homotopic to the trivial closed path.
357:, Modern Analytic and Computational Methods in Science and Mathematics, vol. 37, New York: American Elsevier Publishing Company, pp. 72–77,
306:
260:
142:(this is the language used in Tutte's original paper, however in modern usage the flats of a matroid mean something different). The elements of
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if one can be obtained from the other by the operations of adding or removing elementary paths inside a path, in other words changing a path
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is a finite sequence of 0-flats such that any two consecutive elements of the path lie in some 1-flat.
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Tutte, William Thomas (1958), "A homotopy theorem for matroids. II",
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379:, Encyclopedia of Mathematics and its Applications, vol. 29,
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219:can be composed in the obvious way to give a path
307:Transactions of the American Mathematical Society
261:Transactions of the American Mathematical Society
414:
258:(1958), "A homotopy theorem for matroids. I",
55:is specified by a class of non-empty subsets
31:), generalises the concept of "path" from
138:that are unions of circuits are called
415:
355:Introduction to the theory of matroids
368:
349:
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28:
215:is the same as the first 0-flat of
16:On composition of paths in matroids
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14:
434:
1:
248:
211:such that the last 0-flat of
42:
7:
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439:
381:Cambridge University Press
200:all lying in some 2-flat.
67:, such that no element of
71:contains another, and if
389:10.1017/CBO9781107325715
376:Combinatorial geometries
373:, in White, Neil (ed.),
21:Tutte homotopy theorem
371:"Unimodular matroids"
256:Tutte, William Thomas
238:or vice versa, where
226:Two paths are called
107:, then there is some
369:White, Neil (1987),
160:is one of the form (
19:In mathematics, the
383:, pp. 40–52,
398:978-0-521-33339-9
123:and contained in
430:
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320:10.2307/1993244
274:10.2307/1993243
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242:is elementary.
158:elementary path
134:The subsets of
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436:
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423:Matroid theory
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351:Tutte, W.T.
115:containing
103:but not in
363:0231.05027
249:References
203:Two paths
328:0002-9947
282:0002-9947
228:homotopic
63:, called
51:on a set
43:Statement
417:Category
353:(1971),
119:but not
65:circuits
37:matroids
407:0921064
344:0101526
336:1993244
298:0101526
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332:JSTOR
286:JSTOR
140:flats
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393:ISBN
324:ISSN
278:ISSN
207:and
151:path
91:and
75:and
29:1958
385:doi
359:Zbl
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