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Obstructions to branch-decomposition of matroids. (English) Zbl 1094.05011
From the abstract: “A $$(\delta, \gamma)$$-net in a matroid $$M$$ is a pair $$(N, {\mathcal P})$$ where $$N$$ is a minor of $$M$$, $$\mathcal P$$ is a set of series classes in $$N$$, $$| \mathcal P| \geq \delta$$, and the pairwise connectivity, in $$M$$, between any two members of $$\mathcal P$$ is at least $$\gamma$$. We prove that, for any finite field $$\mathbb F$$, nets provide a qualitative characterization for branch-width in the class of $$\mathbb F$$-representable matroids. That is, for an $$\mathbb F$$-representable matroid $$M$$, we prove that (1) if $$M$$ contains a $$(\delta, \gamma)$$-net where $$\delta$$ and $$\gamma$$ are both very large, then $$M$$ has large branch-width, and conversely, (2) if the branch-width of $$M$$ is very large, then $$M$$ or $$M^*$$ contains a $$(\delta, \gamma)$$-net where $$\delta$$ and $$\gamma$$ are both large.”
For graphs, such a qualitative characterization was obtained by N. Robertson and P. D. Seymour [J. Comb. Theory, Ser. B 41, 92–114 (1986; Zbl 0598.05055)].

##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices
##### Keywords:
branch width; matroids; connectivity
Full Text:
##### References:
 [1] Bixby, R.E.; Cunningham, W.H., Matroid optimization and algorithms, () · Zbl 0848.05017 [2] Dharmatilake, J.S., A MIN-MAX theorem using matroid separations, (), 333-342 · Zbl 0954.05014 [3] Geelen, J.F.; Gerards, A.M.H.; Robertson, N.; Whittle, G.P., On the excluded-minors for the matroids of branch-width k, J. combin. theory ser. B, 88, 261-265, (2003) · Zbl 1032.05027 [4] T. Johnson, N. Robertson, P.D. Seymour, Connectivity in binary matroids, handwritten manuscript [5] Kung, J.P.S., Extremal matroid theory, (), 21-62 · Zbl 0791.05018 [6] Oporowski, B., Partitioning matroids with only small cocircuits, Combin. probab. comput., 11, 191-197, (2002) · Zbl 0996.05025 [7] Oxley, J.G., Matroid theory, (1992), Oxford Univ. Press New York · Zbl 0784.05002 [8] Robertson, N.; Seymour, P.D., Graph minors V: excluding a planar graph, J. combin. theory ser. B, 41, 92-114, (1986) · Zbl 0598.05055 [9] Robertson, N.; Seymour, P.D., Graph minors X: obstructions to tree-decomposition, J. combin. theory ser. B, 52, 153-190, (1991) · Zbl 0764.05069 [10] Tutte, W.T., Menger’s theorem for matroids, J. res. nat. bur. standards, B. math. math. phys. B, 69, 49-53, (1965) · Zbl 0151.33802
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