Golumbic M.C., Trenk A.N. Tolerance Graphs

Издательство Cambridge University Press, 2004, -274 pp.At the 13th Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, 1982), a mathematical model of tolerance, called tolerance graphs, was introduced by Golumbic and Monma in order to generalize some of the well known applications associated with interval graphs. Their motivation was the need to solve scheduling problems in which resources such as rooms, vehicles, support personnel, etc. may be required on an exclusive basis, but where a measure of flexibility or tolerance would be allowed for sharing or relinquishing the resource when total exclusivity prevented a solution. An example of such an application opens our Chapter 1.
During the ensuing years, properties of tolerance graphs have been studied, and quite a number of variations have appeared in the literature, including bitolerance graphs, bounded tolerance orders, NeST graphs, 0-tolerance graphs, tolerance digraphs and others. This continues to be an interesting and active area of investigation. At the 30th Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, 1999), Ann delivered an invited survey talk on the subject, and together we organized a special session on tolerance graphs and related topics. The following year, Marty gave a largely complementary survey talk at the Fields Institute Workshop on Structured Families of Graphs (Toronto, 2000). In July 2001, DIMACS sponsored a workshop on Intersection Graphs and Tolerance Graphs.
It seems to us that the time is ripe to collect and survey the major results on tolerance graphs, presenting them in one volume. Many mathematical scientists around the world have carried out the research which has enabled us to do this, and we salute them. Yet there are still various basic unanswered questions concerning tolerance graphs. Tolerance graphs have not yet been characterized, nor are there recognition algorithms. Other open questions appear in Chapter
14. We hope that this book helps to inspire others to pursue these topics further.
What started as a survey paper has grown into a three year project and a 300-page manuscript. Even so, we have had neither time nor space to include all the topics we would have liked. In particular, we have not covered interval digraphs or the literature on tolerance competition graphs, or very recent results which have not had the opportunity to appear in a journal.
This book is intended primarily for researchers and graduate students, although some of it should be accessible to advanced undergraduates. We have included exercises to facilitate the use of this book in a seminar course. Algorithms and applications are presented in addition to the theory of tolerance graphs. In general, we have tried to include proofs whenever possible, omitting them only when they already appear in other books or sometimes when they are quite long. In several chapters we include hierarchies of structured families of graphs. Naturally, we have tried to catch all errors. We hope our readers will be tolerant of those that inevitably remain, and will report these errors to us.Early work on tolerance graphs.
Trees, cotrees and bipartite graphs.
Interval probe graphs and sandwich problems.
Bitolerance and the ordered sets perspective.
Unit and 50% tolerance orders.
Comparability invariance results.
Recognition of bounded bitolerance orders and trapezoid graphs.
Algorithms on tolerance graphs.
The hierarchy of classes of bounded bitolerance orders.
Tolerance models of paths and subtrees of a tree.
0-tolerance graphs.
Directed tolerance graphs.
Open questions and further directions of research.