We have already seen several generalizations of tolerance and bounded tolerance graphs in this book. In defining bounded bitolerance graphs (Chapter 5), we allowed the assignment of different tolerances to the right and left sides of the intervals. In defining NeST tolerance graphs (Chapter 11), we replaced the real line by a tree and the intervals were replaced by neighborhood subtrees. The class of bounded bitolerance graphs properly contains the bounded tolerance graphs, and class of NeST tolerance graphs properly contains the class of tolerance graphs.

We have also presented a number of restrictions of tolerance graphs such as the subclasses of unit tolerance graphs (Chapter 2), probe graphs (Chapter 4), and threshold tolerance graphs (Section 11.8). All of these, like interval graphs and permutation graphs, are properly contained in the class of tolerance graphs.

In all of our tolerance representations so far, an edge is added to the graphs when the size of the intersection of two intervals is large enough to “bother” one of them. In the case of tolerance, ij ∈ E ⇔ |Ii ∩ Ij| ≥ min{ti, tj}. In this chapter, we turn our attention to variations of this condition where the operation “min” is replaced by another binary function φ, for example “max” or “sum”.

Introduction

Let φ be a symmetric binary function, positive valued on positive arguments.