Theorems on matroid connectivity

Inukai, Thomas; Weinberg, Louis

doi:10.1016/0012-365x(78)90064-x

Cited by 8 publications

(2 citation statements)

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“…Proof of Theorem 4.2. This follows the proof of Theorem 2.1 up to inequalities (11) and (12). Then, by using Lemma 3.2 and its dual, one can sharpen these inequalities to get and Moreover, inequalities (25) and (27) can also be sharpened by one.…”

Section: Corollary 43 Let M Be a 4-connected Matroid With At Least supporting

confidence: 58%

“…If t is a non-negative real number, then [t\ and \t] will denote, respectively, the greatest integer not exceeding t and the least integer not less than t. The restriction on the cardinality of £(M) imposed in the last lemma is frequently applied when one is considering n-connected matroids. It is a very weak restriction [12,23] for the only matroids it excludes are those uniform matroids U r^ such that r € {\k/2\, \k/l\} and * € {0,1,2,....2/1-3}.…”

Section: Lemma 12 Let M Be An N-connected Matroid Having At Least 2mentioning

confidence: 99%

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Some Local Extremal Connectivity Results for Matroids

Akkari

Oxley

1993

Combinator. Probab. Comp.

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Tutte proved that if e is an element of a 3-connected matroid M such that neither M\e nor M/e is 3-connected, then e is in a 3-circuit or a 3-cocircuit. In this paper, we prove a broad generalization of this result. Among the consequences of this generalization are that if X is an (n − 1)-element subset of an n-connected matroid M such that neither M\X nor M/X is connected, then, provided |E(M)| ≥ 2(n − 1)≥ 4, X is in both an n-element circuit and an n-element cocircuit. When n = 3, we describe the structure of M more closely using Δ − Y exchanges. Several related results are proved and we also show that, for all fields F other than GF(2), the set of excluded minors for F-representability is closed under both Δ − Y and Y − Δ exchanges.

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Section: Corollary 43 Let M Be a 4-connected Matroid With At Least supporting

confidence: 58%

Section: Lemma 12 Let M Be An N-connected Matroid Having At Least 2mentioning

confidence: 99%