The proposed project will tackle a series of fundamental problems in discrete mathematics by studying labeled graphs, a generalization of graphs which readily apply to problems beyond graph theory. To achieve these goals will require both developing new graph theoretic tools and techniques as well as further expounding upon known methodologies.The specific problems to be studied can be grouped into a series of semi-independent projects. The first focuses on signed graphs with applications to a conjecture of Seymour concerning 1-flowing binary matroids and a related conjecture on the intregality of polyhedra defined by a class of binary matrices. The second proposes to develop a theory of minors for directed graphs. Finally, the project looks at topological questions arising from graphs embedding in a surface and the classic problem of efficiently identifying the trivial knot. The range of topics considered will lead to the development of tools and techniques applicable to questions in discrete mathematics beyond those under direct study.The project will create a research group incorporating graduate students and post doctoral researchers lead by the PI. Each area to be studied offers the potential for ground-breaking results at the same time offering numerous intermediate opportunities for scientific progress.