This book is quite different from most books reviewed in this place. It has been published by the National Speleological Society, and has the half-true subtitle “applying graph theory to cave exploration.” I say that the subtitle is half-true, since it correctly describes the second half of the book.
The first half of the book is a historical overview of why and how people get lost in caves, and why and how they do or do not find their way out, and why and how they are or are not rescued. There is not much mathematics in this part, though there are a fair number of statistical tables.
After this, there is a brief introduction to graph theoretical terminology, which readers who took an undergraduate class in combinatorics will not need. Then comes the second half of the book, which is what the subtitle promised. The author discusses a large number of methods for a lost person to get out of a cave. The advantages and drawbacks of each method are analyzed, both in words and in a quantitative fashion. Both the worst-case and the average-case scenario is considered for each method, and the expected time of getting out of the cave is computed for certain graphs. After this, the author treats the similar question of how rescuers should search for a lost person in the cave.
This is mathematics at its most applied. The assumptions made for each method are very reasonable, and the graph theory will be understandable for an undergraduate who is concurrently taking a combinatorics class. The book is good material for a reading project.
Miklós Bóna is Professor of Mathematics at the University of Florida.