THE STRUCTURE OF BASES IN BICIRCULAR MATROIDS

In this paper we introduce a partial order on the elements of a matroid based on its fundamental circuits. The partial order is used to define and classify fundamental and secondary equivalence classes of a bicircular matroid. These classes form the basic building blocks of bicircuLlar generalized networks, i.e., generalized networks containing no unit cycles. Here, we use the classes to analyze the connectedness and cycle structure of such networks. In two subsequent articles, we use these theoretical results to develop a polynomial-time algorithm for transforming the constraint matrix of a linear program to the generalized node-arc incidence matrix of a bicircular generalized network, when such a transformation is possible