An extension of elementary bipartite graphs to matroid pairs and its characterization (Q1317049)

The author presents a generalization of elementary bipartite graphs to a certain pair of matroids which is called elementary (this pair is connected and for every element there exists a common base containing it). The main result is the following theorem: For a pair of matroids \((M_ 1,M_ 2)\) on \(E\), the following conditions are equivalent: (1) \((M_ 1,M_ 2)\) is an elementary pair; (2) \(r_ 1(E)= r_ 2(E)\), \(r_ 1(X)+ r_ 2(E- X)> r_ 1(E)\) for any \(\varnothing\neq X\subseteq E\); (3) \((M_ 1,M_ 2)\) is an irreducible pair, i.e. \(L(r_ 1,r_ 2)= \{\varnothing,E\}\).