Basis pair graphs of transversal matroids are connected

Abstract

The basis pair graph of a matroid on the ground set S has, as its vertices, ordered triples of the form ( B 1, B 2, B 3), where B 1 and B 2 are disjoint bases and B 3= S⧹( B 1∪ B 2). Two such vertices, ( A 1, A 2, A 3) and ( B 1, B 2, B 3), are adjacent if ( B 1, B 2, B 3) can be obtained from ( A 1, A 2, A 3) by interchanging two elements of S belonging to different components of ( A 1, A 2, A 3). It is known that basis pair graphs of graphic and cographic matroids are connected. We show that this holds for transversal matroids as well.