1. Introduction.- 2. Degeneracy problems in mathematical optimization.- 2.1. Convergence problems in the case of degeneracy.- 2.2 Efficiency problems in the case of degeneracy.- 2.3 Degeneracy problems within the framework of postoptimal analysis.- 2.4. On the practical meaning of degeneracy.- 3. Theory of degeneracy graphs.- 3.1. Fundamentals.- 3.2 Theory of ? × n-degeneracy graphs.- 3.3. Theory of 2 × n-degeneracy graphs.- 4. Concepts to explain simplex cycling.- 4.1. Specification of the question.- 4.2 A pure graph theoretical approach.- 4.3 Geometrically motivated approaches.- 4.4 A determinant approach.- 5. Procedures for constructing cycling examples.- 5.1 On the practical use of constructed cycling examples.- 5.2 Successive procedures for constructing cycling examples.- 5.3 On the construction of general cycling examples.- A. Foundations of linear algebra and the theory of convex polytopes.- B. Foundations of graph theory.- C. Problems in the solution of determinant inequality systems.- References.