The principal minors of a matroid

Abstract

Kishi and Kajitani introduced the concepts of the principal partition of a graph and maximally distant forest pairs. These concepts lead to the determination of the topological degrees of freedom of a graph or, equivalently, the minimum hybrid rank of a graph, where the minimum hybrid rank is equal to the minimum number of equations needed to solve the network represented by the graph. Recently Iri provided a link between linear algebra and combinatorial mathematics by giving essentially analogous results for matrices, where he defines a minimum term rank of a matrix. One of the results of this paper generalizes the work of Kishi and Kajitani and of Iri. We define and develop the matroid concepts of the r-principal minors, the r-maximally distant bases, and the r-minors of a matroid. The principal partition of a matroid is obtained for r = 2. Further unique decompositions of a matroid are given for r > 2. Thus a new important set of invariants of a matroid, a graph, and a matrix become available. Efficient algorithms are given for the generation of the r-maximally distant bases and all the principal minors.