Biased graphs. II. The three matroids

Zaslavsky, Thomas

doi:10.1016/0095-8956(91)90005-5

Cited by 123 publications

(128 citation statements)

References 15 publications

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“…It has the properties that G(12)IS = G(QIS) and G(12)/S = G(I2/S) with perhaps some matroid loops deleted. (See [14]. )…”

Section: Def'mitions and General Lemmasmentioning

confidence: 93%

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Biased graphs whose matroids are special binary matroids

Zaslavsky

1990

Graphs and Combinatorics

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Abstract. A biased graph is a graph together with a class of polygons such that no theta subgraph contains exactly two members of the class. To a biased graph ~2 are naturally associated three edge matroids: G(I~), L(12), Lo(D). We determine all biased graphs for which any of these matroids is isomorphic to the Fang plane, the polygon matroid ofK~, K 5, or Ka.3, any of their duals, Bixby's regular matroid Rto, or the polygon matroid of K, for m > 5. In each ease thebias is derived from edge signs. We conclude by finding the biased graphs 12 for which Lo(D ) is not a graphic [or, regular] matroid but every proper contraction is.

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“…It has the properties that G(12)IS = G(QIS) and G(12)/S = G(I2/S) with perhaps some matroid loops deleted. (See [14]. )…”

Section: Def'mitions and General Lemmasmentioning

confidence: 93%

“…A copoint of L(I2) is either a copoint of G(I2), that is to say the complement of a bond ofF, or is a maximal balanced edge set (if f2 is not balanced; if I2 is balanced, there are no copoints of the latter kind). The lift has the properties that L(f2)IS = L(f21S ) and that [14]. )…”

Section: Def'mitions and General Lemmasmentioning

confidence: 98%

Biased graphs whose matroids are special binary matroids

Zaslavsky

1990

Graphs and Combinatorics

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“…We denote by ecycle(G, ⌃) the set of all even cycles of (G, ⌃). The set ecycle(G, ⌃) is the set of cycles of a binary matroid known as the even-cycle matroid [26]. We identify ecycle(G, ⌃) with that matroid.…”

Section: -Connected Even Cycle Matroidmentioning

confidence: 99%

“…Zaslavsky [25,26] introduced the class of signed graphic matroids. Pendavingh and Van Zwam [12] gave a recognition algorithm for the class of near-regular signed-graphic matroids.…”

Section: Recognizing Even Cycle Matroidsmentioning

confidence: 99%

Displaying blocking pairs in signed graphs

Guenin

Pivotto

Wollan

2016

European Journal of Combinatorics

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A signed graph is a pair (G, ⌃) where G is a graph and ⌃ is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of ⌃. A blocking pair of (G, ⌃) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts in an auxiliary graph. We discuss the relevance of this result to the problem of recognizing even cycle matroids and to the problem of characterizing signed graphs with no odd-K 5 minor.

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“…There is another natural class, of Dowling matroids. They are like graphs and originally introduced by Dowling [6] and studied in greater depth by Zaslavsky [44], [45].…”

Section: The Local Structure Of Matroid Tanglesmentioning

confidence: 99%