Isotropic Matroids II: Circle Graphs

Brijder, Robert

doi:10.37236/5223

Cited by 6 publications

(10 citation statements)

References 34 publications

(57 reference statements)

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“…The local equivalence class of W 5 is important in Bouchet's famous characterization of circle graphs by obstructions [7]. In sequels to the present paper [18,19] we extend Proposition 50 and provide several new characterizations of circle graphs.…”

mentioning

confidence: 82%

“…Corollary 44 has an interesting consequence, having as a special case a result regarding bicycle spaces of planar graphs [24,Theorem 17.3.5]. The connection with planar graphs arises from the fact that medial graphs of planar graphs are associated with bipartite circle graphs; see the sequel to the present paper [18] for details. Proof.…”

Section: Disjoint Transversals and Bipartite Graphsmentioning

confidence: 94%

“…(Looped circle graphs constitute a proper subfamily of G cographic .) We discuss this important family in sequels to the present paper [18,19].…”

Section: Matroid Minors and Vertex-minorsmentioning

confidence: 99%

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Isotropic Matroids I: Multimatroids and Neighborhoods

Brijder

2016

Electron. J. Combin.

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Several properties of the isotropic matroid of a looped simple graph are presented. Results include a characterization of the multimatroids that are associated with isotropic matroids and several ways in which the isotropic matroid of G incorporates information about graphs locally equivalent to G. Specific results of the latter type include a characterization of graphs that are locally equivalent to bipartite graphs, a direct proof that two forests are isomorphic if and only if their isotropic matroids are isomorphic, and a way to express local equivalence indirectly, using only edge pivots.

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mentioning

confidence: 82%

Section: Disjoint Transversals and Bipartite Graphsmentioning

confidence: 94%

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Isotropic Matroids I: Multimatroids and Neighborhoods

Brijder

2016

Electron. J. Combin.

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“…We have described the fundamental relationship between a ribbon graph and its medial graph, which is an embedded four-regular graph; in Section 5.7 we describe how a ribbon graph with one vertex may be represented by a symmetric binary matrix. In [16] Brijder and Traldi describe the construction of the transition matroid of a ribbon graph. We now describe the almost identical construction of the isotropic matroid of a ribbon graph, and discuss the extent to which it determines the ribbon graph.…”

Section: Sketch Of Proofmentioning

confidence: 99%

Matroids, delta-matroids and embedded graphs

Chun

Moffatt

Noble

et al. 2019

Journal of Combinatorial Theory, Series A

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Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic matroids to the setting of embedded graphs. We show that various basic ribbon graph operations and concepts have delta-matroid analogues, and illustrate how the connections between embedded graphs and delta-matroids can be exploited. Also, in direct analogy with the fact that the Tutte polynomial is matroidal, we show that several polynomials of embedded graphs from the literature, including the Las Vergnas, Bollabás-Riordan and Krushkal polynomials, are in fact delta-matroidal. (Carolyn Chun), iain.moffatt@rhul.ac.uk (Iain Moffatt), steven.noble@brunel.ac.uk (Steven D. Noble), ralf@rueckriemen.de (Ralf Rueckriemen)1 Ralf Rueckriemen was financed by the DFG through grant RU 1731/1-1.

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“…In addition to Bouchet's characterization by obstructions, there are also characterizations of circle graphs using binary matroids [10], delta-matroids [18] and monadic second-order logic [11]. The first two of these characterizations involve the field GF (2), and the third involves the even cardinality set predicate.…”

Section: Resultsmentioning

confidence: 99%

Notes on a theorem of Naji

Traldi

2017

Discrete Mathematics

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We present a new proof of an algebraic characterization of circle graphs due to W. Naji. For bipartite graphs, Naji's theorem is equivalent to an algebraic characterization of planar matroids due to J. Geelen and B. Gerards. Naji's theorem also yields an algebraic characterization of permutation graphs.Comment: v1: 22 pages, no figure. v2: minor improvements. 23 pages, no figure v3: minor improvements. 23 pages, no figure v4: substantial improvements reflecting reader's suggestions. 32 pages, no figur

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Isotropic Matroids II: Circle Graphs

Abstract: We present several characterizations of circle graphs, which follow from Bouchet's circle graph obstructions theorem.

Cited by 6 publications

References 34 publications

Isotropic Matroids I: Multimatroids and Neighborhoods

Isotropic Matroids I: Multimatroids and Neighborhoods

Matroids, delta-matroids and embedded graphs

Notes on a theorem of Naji

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