Graph representations of a bicircular matroid

Neudauer, Nancy Ann

doi:10.1016/s0166-218x(01)00210-4

Cited by 4 publications

(4 citation statements)

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“…Observe that if (G, B) is a balanced biased graph, then F (G, B) is the cycle matroid M (G) of G. We therefore view a graph as a biased graph with all cycles balanced. When no cycles are balanced F (G, ∅) is the bicircular matroid of G investigated by Matthews [7], Wagner [14], and others (for instance, [6,8]). The Dowling geometries [2] arise precisely from those biased graphs for which the bias of cycles may be defined by associating an element of a finite group, and a direction, to each edge (see [18]).…”

Section: The Structure Of Biases In Biased Graphs With a Balancing Vementioning

confidence: 99%

“…Little else is known about representations of general frame matroids by biased graphs. Those biased graphs representing graphic matroids are known [1], and there have been studies on representations of subclasses (for example, [5,8,10,11,12]). In this paper, we determine all biased graph representations of frame matroids that arise from biased graphs having a special structure.…”

mentioning

confidence: 99%

“…1 have been studies on representations of subclasses (for example, [5,8,10,11,12]). In this paper, we determine all biased graph representations of frame matroids that arise from biased graphs having a special structure.…”

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confidence: 99%

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Almost balanced biased graph representations of frame matroids

DeVos¹,

Funk²

2018

Advances in Applied Mathematics

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Given a 3-connected biased graph Ω with a balancing vertex, and with frame matroid F (Ω) nongraphic and 3-connected, we determine all biased graphs Ω with F (Ω ) = F (Ω). As a consequence, we show that if M is a 4connected nongraphic frame matroid represented by a biased graph Ω having a balancing vertex, then Ω essentially uniquely represents M . More precisely, all biased graphs representing M are obtained from Ω by replacing a subset of the edges incident to its unique balancing vertex with unbalanced loops.

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Section: The Structure Of Biases In Biased Graphs With a Balancing Vementioning

confidence: 99%

mentioning

confidence: 99%

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Almost balanced biased graph representations of frame matroids

DeVos¹,

Funk²

2018

Advances in Applied Mathematics

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“…Observe that for a biased graph (G, B), if B contains all cycles in G, then F (G, B) is the cycle matroid M (G) of G. We therefore view a graph as a biased graph with all cycles balanced. At the other extreme, when no cycles are balanced F (G, ∅) is the bicircular matroid of G, introduced by Simões-Pereira [9] and further investigated by Matthews [6], Wagner [10], and others (for instance, [5,7]). Frame matroids also include Dowling geometries [4] (see also [14]).…”

Section: Introductionmentioning

confidence: 99%