Decomposing symmetric exchanges in matroid bases

Abstract

LetB,B′ be bases of a matroid, withX ⊂ B, X′ ⊂ B′. SetsX,X′ are asymmetric exchange if(B − X′) ⋃ X′ and(B′ − X′) ⋃ X are bases. SetsX,X′ are astrong serial B-exchange if there is a bijectionf: X → X′, where for any ordering of the elements ofX, sayx _i ,i = 1, ⋯, m, bases are formed by the sets B₀ = B, B_i = (B_i−1 − x_i) ⋃f(x _i), fori = 1, ⋯, m. Any symmetric exchangeX,X′ can be decomposed by partitioning X = ⋃ ^m_i=1 Y_i, X′ = ⋃ ^m_i=1 Y_i, X′, where (1) bases are formed by the setsB ₀ =B, B _i = (B _i−1 −Y _i )⋃ Y ^′ _i ; (2) setsY _i ,Y ^′ _i are a strong serialB _i−1 -exchange; (3) properties analogous to (1) and (2) hold for baseB′ and setsY ^′ _i ,Y _i .