Representations of bicircular matroids and the complexity of recognizing a class of generalized network flow matrices

John Guy del Greco, Purdue University

Abstract

If G is a graph and C ${\buildrel \Delta\over =}$ $\{$E(B) $\mid$ B is a bicycle of G$\}$, then C is the collection of circuits of a matroid on the edge set of G called the bicircular matroid of G and is denoted B(G). This thesis contains three main results on bicircular matroids. First, if G is a graph such that B(G) is 2-connected, the set (G) $\sb{\bf B}$ ${\buildrel \Delta\over =}$ $\{$G$\sp\prime\vert$B(G$\sp\prime$) = B(G)$\}$ is characterized. Second, if G and G$\sp\prime$ are graphs such that B(G$\sp\prime$) = B(G), then, with some exceptions, there exists row equivalent matrices N and N$\sp\prime$ such that M(N) = B(G) and N(N$\sp\prime$) is a generalized node/arc incidence matrix for G(G$\sp\prime$). Generalized node/arc incidence matrices of graphs are also called generalized network flow matrices. Third, let A be a full row rank matrix such that M(A), the matric matroid of A, is 2-connected and for which there exists a nonsingular matrix T such that TA is a generalized network flow matrix and (D,w) has no unit gain cycles where (D,w) is an associated weighted digraph of TA. Under these assumptions, an algorithm is described that produces such a matrix T in polynomial-time.

Degree

Ph.D.

Advisors

Wagner, Purdue University.

Subject Area

Industrial engineering

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