The transition matroid of a 4-regular graph: An introduction

The mutually enriching relationship between graphs and matroids has motivated discoveries in both fields. In this paper, we exploit the similar relationship between embedded graphs and deltamatroids. There are well-known connections between geometric duals of plane graphs and duals of matroids. We obtain analogous connections for various types of duality in the literature for graphs in surfaces of higher genus and delta-matroids. Using this interplay, we establish a rough structure theorem for delta-matroids that are twists of matroids, we translate Petrie duality on ribbon graphs to loop complementation on delta-matroids and we prove that ribbon graph polynomials, such as the Penrose polynomial, the characteristic polynomial and the transition polynomial, are in fact delta-matroidal. We also express the Penrose polynomial as a sum of characteristic polynomials.

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“…Chun et al. established in that

R (G)

is delta‐matroidal in the sense that it is determined by

D (G)

(this fact also follows from Traldi ). For a delta‐matroid

D

and subset

A

of its elements, let

D false| A : = D ∖ A^{c}

.…”

Section: The Penrose and Characteristic Polynomialsmentioning

confidence: 83%

On the interplay between embedded graphs and delta-matroids

Chun

Moffatt

Noble

et al. 2018

Proc. London Math. Soc.

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“…We recall that a circle graph is an interlacement graph of some 4-regular graph with respect to some Euler system. Proposition 32 ( [15]). Let G be the interlacement graph of a 4-regular graph F with respect to some Euler system C.…”

Section: Cycle Matricesmentioning

confidence: 99%

A Characterization of Circle Graphs in Terms of Multimatroid Representations

Brijder

2020

Electron. J. Combin.

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A graph G has an associated multimatroid Z3(G), which is equivalent to the isotropic system of G studied by Bouchet. In previous work it was shown that G is a circle graph if and only if for every field F, the rank function of Z3(G) can be extended to the rank function of an Frepresentable matroid. In the present paper we strengthen this result using a multimatroid analogue of total unimodularity. As a consequence we obtain a characterization of matroid planarity in terms of this totalunimodularity analogue.Definition 4. If G is a looped simple graph then Z 3 (G) is the 3-matroid (W (G), Ω, r), where Ω is the set of vertex triples of vertices of G and r is given by the GF (2)-rank of sets of columns of IAS(G). Also,The multimatroids Z 2 (G) and Z 3 (G) are equivalent to the delta-matroid and isotropic system of G, respectively. That is, if G and H are looped simple graphs then Z 2 (G) and Z 2 (H) are isomorphic 2-matroids if and only if the deltamatroids of G and H are isomorphic; and Z 3 (G) and Z 3 (H) are isomorphic 3matroids if and only if the isotropic systems of G and H are isomorphic. Despite

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“…Theorem 34. ( [41]) Let C and D be any two Euler systems of a 4-regular graph F . Then the matrices IAS(I(C)) and IAS(I(D)) represent the same binary matroid on T(F ).…”

Section: The Transition Matroidmentioning

confidence: 99%