This book focuses on the classic Steiner Problem and illustrates how results of the problem's development have generated the Theory of Minimal Networks, that is systems of "rubber" branching threads of minimal length. This theory demonstrates a brilliant interconnection among differential and computational geometry, topology, variational calculus, and graph theory. All necessary preliminary information is included, and the book's simplified format and nearly 150 illustrations and tables will help readers develop a concrete understanding of the material. All nontrivial statements are proved, and plenty of exercises are included.
Some Necessary Results from Graph Theory and Geometry. The Steiner Problem and Its Modifications. Local Structure of Minimal Networks. Global Structure of Minimal Networks. Global Minimal Networks on the Plane. Planar Local Minimal Networks with Convex Boundaries. Planar Local Minimal Networks with Regular Boundaries. Closed Minimal Networks on Closed Surfaces of Constant Curvature. Minimal Networks in Other Spaces.