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On a generalization of the matroid. (English) Zbl 0551.05031

A semimatroid is a pair \(H=<X,{\mathcal B}>\) where X is a non-empty finite set and \({\mathcal B}\) is a non-empty collection of pairwise incomparable subsets of X. The members of \({\mathcal B}\) are called the bases of H. This paper considers the class of \(e^*\)-semimatroids, a class of semimatroids in which the bases satisfy a certain exchange condition involving pairs of elements. The authors note that all matroids are \(e^*\)-semimatroids and prove that the bases of an \(e^*\)-semimatroid are equicardinal. They also answer a question of L. Szamkołowicz [Recent Adv. Graph Theory, Proc. Symp. Prague 1974, 501-505 (1975; Zbl 0326.05120)] concerning a class of \(e^*\)-semimatroids that arise from graphs.
Reviewer: J.G.Oxley

MSC:

05B35 Combinatorial aspects of matroids and geometric lattices

Citations:

Zbl 0326.05120
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References:

[1] C Berge: Graphs and Hypergraphs. North-Holland Publishing Company, 1973. · Zbl 0254.05101
[2] L. Szamkolowicz: On problems of the elementary theory of graphical matroids. Recent Advances in Graph Theory, Praha, 1975, 501-505. · Zbl 0326.05120
[3] W. T. Tutte: Lectures on Matroids. Journal of Research of the National Bureau of Standards-B. Mathematics and Mathematical Physics, Vol. 69B, Nos. 1 and 2, January-June 1965, 1-46. · Zbl 0151.33801
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