Whitney Connectivity of Matroids

Inukai, Thomas; Weinberg, Louis

doi:10.1137/0602014

Cited by 9 publications

(9 citation statements)

References 6 publications

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“…Moreover, under additional hypotheses on the limiting value of nr/r, we can make considerably stronger statements concerning the connectivity of Mr. To do this, we use the notion of vertical m-connectivity, the matroid generalization of the graph-theoretic concept of m-connectivity. Vertical m-connectivity was introduced by Tutte [20,21] and has been studied by authors, including Cunningham [3], Inukai and Weinberg [12], and Oxley [17]. Theorem 4.4 provides that if nr/r approaches a limit large enough to satisfy a certain inequality, then with probability 1 eventually Mr is vertically r-connected.…”

Section: Summary Of Resultsmentioning

confidence: 99%

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On Random Representable Matroids

Kelly¹,

Oxley

1984

Stud Appl Math

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Results are obtained on the likely connectivity properties and sizes of circuits in the column dependence matroid of a random r × n matrix over a finite field, for large r and n. In a sense made precise in the paper, it is shown to be highly probable that when n is less than r such a matroid is the free matroid on n points, while if n exceeds r it is a connected matroid of rank r. Moreover, the connectivity can be strengthened under additional hypotheses on the growth of n and r, using the notion of vertical connectivity; and the values of k for which circuits of size k exist can be determined in terms of n and r.

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Section: Summary Of Resultsmentioning

confidence: 99%

“…The third of these will be used in the proof of the next theorem. Proofs of this result are given independently in Theorem 1 of Cunningham [3], Theorem 2 of Inukai and Weinberg [12], and Theorem 2 of [17]. PROPOSITION…”

Section: Connectivity and Vertical Connectivitymentioning

confidence: 93%

On Random Representable Matroids

Kelly¹,

Oxley

1984

Stud Appl Math

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“…However, one may require that the vertical connectivity κ(M ) of M be higher. (Vertical connectivity, also known as Whitney connectivity, is discussed in [2] and Section 8.2 of [7]; our terminology follows [7].) The condition we impose in Theorem 5.2 is that κ(M ) is r(M ); this holds if M is a projective geometry.…”

Section: Results For Rank-5 Geometriesmentioning

confidence: 99%

“…(One can give a geometric argument to show that the best bound for d = 4 is eight. Note that U 4,5 , U 4,6 , and AG (3,2) show that the upper bounds 5, 6, and 8 are sharp for these values of d.) For the rest of the proof, we assume d ≥ 5.…”

Section: Lemma 42mentioning

confidence: 99%

Simple Matroids with Bounded Cocircuit Size

Bonin¹,

Reid²

2000

Combinator. Probab. Comp.

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We examine the specialization to simple matroids of certain problems in extremal matroid theory that are concerned with bounded cocircuit size. Assume that each cocircuit of a simple matroid M has at most d elements. We show that if M has rank 3, then M has at most d + ⌊ √ d⌋ + 1 points and we classify the rank-3 simple matroids M that have exactly d + ⌊ √ d⌋ + 1 points. We show that if M is a connected matroid of rank 4 and d is q 3 with q > 1, then M has at most q 3 + q 2 + q + 1 points; this upper bound is strict unless q is a prime power, in which case the only such matroid with exactly q 3 + q 2 + q + 1 points is the projective geometry PG(3, q). We also show that if d is q 4 for a positive integer q and if M has rank 5 and is vertically 5-connected, then M has at most q 4 + q 3 + q 2 + q + 1 points; this upper bound is strict unless q is a prime power, in which case PG(4, q) is the only such matroid that attains this bound.

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“…~J contains the graph consisting of a single edge and its endpoints. The rather contorted condition HG2 is to take account of the fact that connectedness in the cycle matroid corresponds to 2-vertex-connectedness in the graph (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]). If the first alternative in HG2 always holds, then ~a is said to be strongly hereditary.…”

Section: Graphic and Voltage Graphic Geometriesmentioning

confidence: 99%

Numerically regular hereditary classes of combinatorial geometries

Kung

1986

Geom Dedicata

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A hereditary class 24 ° of combinatorial geometries (or simple matroids) is a collection of geometries closed under minors and direct sums. A geometry G in ~ is extremal if no proper extension of G of the same rank is in Yr. The size function h(n) of ~ is defined by h(n)-max{[G[:GEgf and rank(G)-n}, where [G] is the number of points in G. A hereditary class is numerically regular if for every extremal geometry G in ,;/t ~, [ G] = h (rank(G)). We determine all the numerically regular hereditary classes for which the set {h(n) -h(n -1): 1 < n < oo} of positive integers does not have an upper bound: they are all varieties. We also give several examples of numerically regular hereditary classes which are not varieties.ra(X ) + r~(Y) = rG(X w Y) + rG(X c~ Y).

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Whitney Connectivity of Matroids

Cited by 9 publications

References 6 publications

On Random Representable Matroids

On Random Representable Matroids

Simple Matroids with Bounded Cocircuit Size

Numerically regular hereditary classes of combinatorial geometries

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