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.

Let M be a 3-connected binary matroid; M is called internally 4-connected if one side of every 3-separation is a triangle or a triad, and M is (4, 4, S)-connected if one side of every 3-separation is a triangle, a triad, or a 4-element fan. Assume M is internally 4-connected and that neither M nor its dual is a cubic Möbius or planar ladder or a certain coextension thereof. Let N be an internally 4-connected proper minor of M . Our aim is to show that M has a proper internally 4-connected minor with an N -minor that can be obtained from M either by removing at most four elements, or by removing elements in an easily described way from a special substructure of M . When this aim cannot be met, the earlier papers in this series showed that, up to duality, M has a good bowtie, that is, a pair, {x 1 , x 2 , x 3 } and {x 4 , x 5 , x 6 }, of disjoint triangles and a cocircuit, {x 2 , x 3 , x 4 , x 5 }, where M \x 3 has an Nminor and is (4, 4, S)-connected. We also showed that, when M has a good bowtie, either M \x 3 , x 6 has an N -minor; or M \x 3 /x 2 has an N -minor and is (4, 4, S)-connected. In this paper, we show that, when M \x 3 , x 6 has an Nminor but is not (4, 4, S)-connected, M has an internally 4-connected proper minor with an N -minor that can be obtained from M by removing at most three elements, or by removing elements in a well-described way from one of several special substructures of M . This is a significant step towards obtaining a splitter theorem for the class of internally 4-connected binary matroids. TOWARDS A SPLITTER THEOREM VI 7 PreliminariesIn this section, we give some basic definitions mainly relating to matroid connectivity. The subsequent section contains some straightforward properties of connectivity along with a lemma concerning bowties that distinguishes various cases whose analysis is fundamental to completing our work on the splitter theorem. The main result of this paper completely resolves what happens in one of these cases. In Section 3, we outline the proof of the main result, Theorem 1.3.Let M and N be matroids. We shall sometimes write N M to indicate that M has an N -minor, that is, a minor isomorphic to N . Now let E be the ground set of M and r be its rank function. The connectivity function λ M of M is defined on all subsets X of E by λ M (X) = r(X) + r(E − X) − r(M ). Equivalently, λ M (X) = r(X) + r * (X) − |X|. We will sometimes abbreviate λ M as λ. For a positive integer k, a subset X or a partition (X,If n is an integer exceeding one, a matroid is n-connected if it has no k-separations for all k < n. This definition [13] has the attractive property that a matroid is n-connected if and only if its dual is. Moreover, this matroid definition of n-connectivity is relatively compatible with the graph notion of nconnectivity when n is 2 or 3. For example, when G is a graph with at least four vertices and with no isolated vertices, M (G) is a 3-connected matroid if and only if G is a 3-connected simple graph. But the link between n-connectivity for matroids and graphs brea...

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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