Introduction

We began this book by introducing the class of tolerance graphs, which generalize the intersection graphs of intervals on the line (interval graphs), adding an edge between two vertices in the tolerance graph when the size of the intersection of their intervals exceeds at least one of the tolerances. Subsequently, we studied a further generalization defined by allowing separate right and left tolerances on the intervals (bitolerance graphs).

In this chapter, we present a totally different approach to generalizing tolerance graphs by replacing the real line by a tree and replacing the role of intervals by either paths or other types of subtree. Several classical results are known for classes of intersection graphs of paths and subtrees of a tree, which we review in the next three sections. We then present results on tolerance versions.

Intersection models

Let T be a tree and let T = {Ti} be a collection of subtrees (connected subgraphs) of T. We may think of the host tree T either as a continuous model of a tree embedded in the plane, thus generalizing the real line from the one dimensional case, or as a finite discrete model of a tree, namely, a connected graph of vertices and edges having no cycles, thus generalizing the path Pk from the one-dimensional case. Making a distinction between these two models will become important when we measure the size of the intersection of two subtrees.