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Maps and \(\Delta \)-matroids revisited. (English) Zbl 1454.05021

Summary: Using Tutte’s combinatorial definition of a map we define a \(\Delta\)-matroid purely combinatorially and show that it is identical to Bouchet’s topological definition.

MSC:

05B35 Combinatorial aspects of matroids and geometric lattices
05C10 Planar graphs; geometric and topological aspects of graph theory
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
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References:

[1] A. Bouchet, Greedy algorithm and symmetric matroids,Math. Programming38(1987), 147- 159, doi:10.1007/BF02604639. · Zbl 0633.90089
[2] A. Bouchet, Maps and4-matroids,Discrete Math.78(1989), 59-71, doi:10.1016/ 0012-365X(89)90161-1.
[3] C. Godsil and G. Royle,Algebraic Graph Theory, Graduate Texts in Mathematics, Springer New York, 2001,https://books.google.com.au/books?id=pYfJe-ZVUyAC. · Zbl 0968.05002
[4] T. Pisanski and B. Servatius,Configurations from a graphical viewpoint, Birkh¨auser Advanced Texts: Basler Lehrb¨ucher. [Birkh¨auser Advanced Texts: Basel Textbooks], Birkh¨auser/Springer, New York, 2013, doi:10.1007/978-0-8176-8364-1. · Zbl 1277.05001
[5] W. T. Tutte, What is a map?, in:New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich., 1971), Academic Press, New York, pp. 309-325, 1973. · Zbl 0258.05105
[6] H. Whitney, Congruent Graphs and the Connectivity of Graphs,Amer. J. Math.54(1932), 150- 168, doi:10.2307/2371086 · JFM 58.0609.01
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