Submodular functions and electrical networks
This 650-page-long monograph contains the detailed description of many properties of matroids, submodular functions and the principal partition, for their application in the efficient analysis of electric networks. Since his Ph.D. thesis in 1974, the author has published some twenty articles in this area, but this is the first comprehensive treatise on the subject. Certain areas (like Dilworth truncation of submodular functions and its relation to the principal lattice of partitions) probably appear for the first time in book form. Electrical engineers with a solid background in algebra and graph theory can use this book as well as combinatorialists interested in the applications. In order to reach both types of audience the author gives all the necessary background from graphs, matroids and electrical networks before entering into the description of the new results. He points out that the book ``is meant for self study---so it contains a large number of exercises and problems with solutions.'' One can also use it in a classroom but the order of the presentation of the material is sometimes unusual, hence the instructor must be careful. (For example, in Chapter 4 the very first exercise is to ``show that the structures in the above examples are indeed matroids.'' At the end of the chapter, as the ``solution'' for one of the examples, we are referred to a theorem in Chapter 11.) In spite of this, the volume is very clearly structured, with a large number of examples, figures, and a very rich bibliography containing a large number of Indian and Japanese references as well. This latter is also very important since the author feels that the ``west, except for (some cases), has been largely immune to the principal partition virus.'' This volume will clearly contribute to the spreading of this and related viruses.
scientific article; zbMATH DE number 1032971