224 research outputs found
Matroids, delta-matroids and embedded graphs
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
The excluded 3-minors for vf-safe delta-matroids
Vf-safe delta-matroids have the desirable property of behaving well under
certain duality operations. Several important classes of delta-matroids are known to be
vf-safe, including the class of ribbon-graphic delta-matroids, which is related to the class
of ribbon graphs or embedded graphs in the same way that graphic matroids correspond
to graphs. In this paper, we characterize vf-safe delta-matroids and ribbon-graphic deltamatroids by finding the minimal obstructions, called excluded 3-minors, to membership in
the class. We find the unique (up to twisted duality) excluded 3-minor within the class of
set systems for the class of vf-safe delta-matroids. In the literature, binary delta-matroids
appear in many different guises, with appropriate notions of minor operations equivalent
to that of 3-minors, perhaps most notably as graphs with vertex minors. We give a direct
explanation of this equivalence and show that some well-known results may be expressed
in terms of 3-minors
On the interplay between embedded graphs and delta-matroids
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
delta-matroids. 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
Dessins, their delta-matroids and partial duals
Given a map on a connected and closed orientable surface, the
delta-matroid of is a combinatorial object associated to which captures some topological information of the embedding. We explore how
delta-matroids associated to dessins d'enfants behave under the action of the
absolute Galois group. Twists of delta-matroids are considered as well; they
correspond to the recently introduced operation of partial duality of maps.
Furthermore, we prove that every map has a partial dual defined over its field
of moduli. A relationship between dessins, partial duals and tropical curves
arising from the cartography groups of dessins is observed as well.Comment: 34 pages, 20 figures. Accepted for publication in the SIGMAP14
Conference Proceeding
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