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,...

This book deals with the new class of one-dimensional variational problems -- the problems with branching solutions. Instead of extreme curves (mappings of a segment to a manifold) we investigate extreme networks, which are mappings of graphs (one-dimensional cell complexes) to a manifold. Various applications of the approach are presented, such as several generalizations of the famous Steiner problem of finding the shortest network spanning given points of the plane.