Inductive tools for connected delta-matroids and multimatroids. (English) Zbl 1365.05040
Summary: We prove a splitter theorem for tight multimatroids, generalizing the corresponding result for matroids, obtained independently by T. H. Brylawski [Trans. Am. Math. Soc. 171, 235–282 (1972; Zbl 0224.05007)] and P. Seymour [J. Comb. Theory, Ser. B 22, 289–295 (1977; Zbl 0385.05021)]. Further corollaries give splitter theorems for delta-matroids and ribbon graphs.
MSC:
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
References:
| [1] | Bouchet, A., Greedy algorithm and symmetric matroids, Math. Program., 38, 147-159 (1987) · Zbl 0633.90089 |
| [2] | Bouchet, A., Maps and delta-matroids, Discrete Math., 78, 59-71 (1989) · Zbl 0719.05019 |
| [3] | Bouchet, A., Multimatroids I. Coverings by independent sets, SIAM J. Discrete Math., 10, 626-646 (1997) · Zbl 0886.05042 |
| [4] | Bouchet, A., Multimatroids II. Orthogonality, minors and connectivity, Electron. J. Combin., 8, R8 (1998) · Zbl 0885.05050 |
| [5] | Bouchet, A., Multimatroids III. Tightness and fundamental graphs, European J. Combin., 22, 657-677 (2001) · Zbl 0982.05034 |
| [6] | Bouchet, A.; Duchamp, A., Representability of delta-matroids over \(G F(2)\), Linear Algebra Appl., 146, 67-78 (1991) · Zbl 0738.05027 |
| [7] | Brijder, R.; Hoogeboom, H., The group structure of pivot and loop complementation on graphs and set systems, European J. Combin., 32, 1353-1367 (2011) · Zbl 1230.05197 |
| [8] | Brijder, R.; Hoogeboom, H., Nullity and loop complementation for delta-matroids, SIAM J. Discrete Math., 27, 492-506 (2013) · Zbl 1268.05040 |
| [9] | Brijder, R.; Hoogeboom, H., Interlace polynomials for multimatroids and delta-matroids, European J. Combin., 40, 142-167 (2014) · Zbl 1300.05051 |
| [10] | Brylawski, T. H., A decomposition for combinatorial geometries, J. Combin. Theory Ser. B, 171, 235-282 (1972) · Zbl 0224.05007 |
| [11] | Chmutov, S., Generalized duality for graphs on surfaces and the signed Bollobás-Riordan polynomial, J. Combin. Theory Ser. B, 99, 617-638 (2009) · Zbl 1172.05015 |
| [14] | Ellis-Monaghan, J.; Moffatt, I., Graphs on Surfaces: Dualities, Polynomials, and Knots (2013), Springer · Zbl 1283.57001 |
| [15] | Geelen, J.; Iwata, S.; Murota, K., The linear delta-matroid parity problem, J. Combin. Theory Ser. B, 88, 377-398 (2003) · Zbl 1021.05019 |
| [16] | Gross, J.; Tucker, T., Topological Graph Theory (1987), Wiley-interscience Publication · Zbl 0621.05013 |
| [17] | Moffatt, I., Partial duals of plane graphs, separability and the graphs of knots, Algebr. Geom. Topol., 12, 1099-1136 (2012) · Zbl 1245.05030 |
| [18] | Moffatt, I., Separability and the genus of a partial dual, European J. Combin., 34, 355-378 (2013) · Zbl 1254.05047 |
| [19] | Oxley, J., Matroid Theory (2011), Oxford University Press: Oxford University Press New York · Zbl 1254.05002 |
| [20] | Seymour, P., A note on the production of matroid minors, J. Combin. Theory Ser. B, 22, 289-295 (1977) · Zbl 0385.05021 |
| [21] | Tutte, W., Connectivity in matroids, Canad. J. Math., 18, 1301-1324 (1966) · Zbl 0149.21501 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
