Matroid base polytope decomposition

2009/09/04 by V. Chatelain, Chatelain, V., J. L. Ramirez Alfonsin +1 · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.0909.0840

23 pages

Abstract

Let P(M) be the matroid base polytope of a matroid M. A \em matroid base polytope decomposition of P(M) is a decomposition of the form P(M) = \bigcup_i=1^t P(M_i) where each P(M_i) is also a matroid base polytope for some matroid M_i, and for each 1≤ i ≠ j≤ t, the intersection P(M_i) ∩ P(M_j) is a face of both P(M_i) and P(M_j). In this paper, we investigate \em hyperplane splits, that is, polytope decompositions when t=2. We give sufficient conditions for M so P(M) has a hyperplane split and characterize when P(M₁ ⊕ M₂) has a hyperplane split where M₁ ⊕ M₂ denote the \em direct sum of matroids M₁ and M₂. We also prove that P(M) has not a hyperplane split if M is binary. Finally, we show that P(M) has not a decomposition if its 1-skeleton is the \em hypercube.

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