In this chapter we discuss the definitions, results and open questions that appear in Golumbic and Monma (1982) and Golumbic, Monma, and Trotter (1984), the papers which introduced the topic of tolerance graphs. We also present consequences of these results and related topics from more recent literature.

Notation and observations

Recall from Section 1.3 that a graph G is a tolerance graph if each vertex ν ∈ V(G) can be assigned a closed interval Iν and a tolerance tν ∈ R+ so that xy ∈ E(G) if and only if |Ix ∩ Iy| ≥ min{tx, ty}. If graph G has a tolerance representation with tν ≤ |Iν| for all ν ∈ V(G), then G is called a bounded tolerance graph.

Many important graph properties are inherited by all induced subgraphs and thus called hereditary properties. Given a (bounded) tolerance representation 〈I, t〉 of a graph G, for any subset of vertices W ⊆ V(G) the intervals {Iw | w ∈ W} and tolerances {tw | w ∈ W} give a representation of GW. Thus, induced subgraphs of tolerance graphs are also tolerance graphs and induced subgraphs of bounded tolerance graphs are also bounded tolerance graphs. We record this as a remark.

Remark 2.1. The property of being a tolerance graph (resp. bounded tolerance graph) is hereditary.

In a tolerance representation of a graph G, we may have intervals of the form Ix = [ax, ax].

In this chapter we consider the classes of bounded bitolerance orders arranged in a hierarchy in Figure 10.1. We begin by describing the notation and conventions used in Figure 10.1 and justifying the inclusions and equivalences in the hierarchy. In Section 10.3, we restrict attention to bipartite orders. In that setting, the hierarchy collapses and most of the classes are equivalent. Section 10.4 provides the details to show that any example that appears along an edge between two classes provides a separating example between those two classes.

Introduction

In Section 5.2, we defined subclasses of bounded bitolerance orders by adding restrictions on interval lengths, tolerant points p(ν) and q(ν), and left and right tolerances. These restrictions are summarized in Table 10.1. The restrictions are listed so that the top entry of each column is the most restrictive, and they are less restrictive as you travel down the column.

Each of the three categories of restrictions is independent. Thus, the restrictions can be combined, by taking one from each column, to give 18 classes of bitolerance orders, some of which turn out to be equivalent. In this chapter we often refer to a class by its abbreviation, for example, (1aii) is the class of unit point-core bitolerance orders, and (3ci) is the class of (bounded) tolerance orders.

Any transitive orientation of the edges of a comparability graph G = (V, E) gives an ordered set P = (V, ≺), and we say that G is the comparability graph of P. A graph can have many different transitive orientations, so there may be many different orders with the same comparability graph. In Figure 7.1, orders P, Q, and R (and their duals) all have the comparability graph G shown, and they represent all six transitive orientations of G. Determining the number of transitive orientations of a comparability graph was studied by Shevrin and Filippov (1970) and Golumbic (1977) (see also Section 5.3 of Golumbic, 1980).

Interval orders illustrate an interesting invariance property. If G has a transitive orientation F which gives an interval order P, then every transitive orientation of G gives an interval order. This can be seen as follows. Since P has an interval representation, this same representation demonstrates that G is an interval graph. Suppose F′ is another transitive orientation of G whose ordered set P′ is not an interval order. Then P′ must contain a 2 + 2 (Theorem 1.6) in which case G contains an induced C4, a contradiction (Theorem 1.3).

In this chapter, we investigate a variety of order-theoretic properties and parameters which exhibit this kind of invariance. We present a standard technique for proving invariance based on a theorem of Gallai, and illustrate its use on the dimension of an order. We then turn our attention to tolerance properties.

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.

Physical mapping of DNA

The use of interval models in molecular biology dates back to the original studies on the linearity of genes by the well-known biologist Seymore Benzer. In Benzer (1959), interval graphs were defined in order to study overlap data on sub-elements inside the gene. The question at that time was whether the overlap data was consistent with the hypothesis that genes are linear structures with the sub-elements being intervals on that line. If perfect and complete data were to be available for of all pairs of these sub-elements, then the problem would be that of recognizing an interval graph. However, with only partial data, as is always the case in experimental genetics, the problems involve embedding the data in a larger consistent set (e.g., interval graph completion). This will generally require inferring additional intersections (e.g., selectively adding chords to a cycle), and using additional biological information or assumptions to test consistency and propose the possible linear orderings.

During the ensuing years, research in genetics has involved many combinatorial problems on intervals. One of these is DNA physical mapping, in which one wishes to find the linear order of segments (physically contiguous units) of the chromosome. It is an essential part of most sequencing, gene locating, and cloning projects. One of the main goals set for the Human Genome Project is to obtain a detailed mapping of all human chromosomes.