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].

From the previous chapter, we know that a code alphabet A is a finite set. In order to play mathematical games, we are going to equip A with some algebraic structures. As we know, a field, such as the real field R or the complex field C, has two operations, namely addition and multiplication. Our idea is to define two operations for A so that A becomes a field. Of course, then A is a field with only finitely many elements, whilst R and C are fields with infinitely many elements. Fields with finitely many elements are quite different from those that we have learnt about before.

The theory of finite fields goes back to the seventeenth and eighteenth centuries, with eminent mathematicians such as Pierre de Fermat (1601–1665) and Leonhard Euler (1707–1783) contributing to the structure theory of special finite fields. The general theory of finite fields began with the work of Carl Friedrich Gauss (1777–1855) and Evariste Galois (1811–1832), but it only became of interest for applied mathematicians and engineers in recent decades because of its many applications to mathematics, computer science and communication theory. Nowadays, the theory of finite fields has become very rich. In this chapter, we only study a small portion of this theory. The reader already familiar with the elementary properties of finite fields may wish to proceed directly to the next chapter.

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.

In the seminal paper ‘A mathematical theory of communication’ published in 1948, Claude Shannon showed that, given a noisy communication channel, there is a number, called the capacity of the channel, such that reliable communication can be achieved at any rate below the channel capacity, if proper encoding and decoding techniques are used. This marked the birth of coding theory, a field of study concerned with the transmission of data across noisy channels and the recovery of corrupted messages.

In barely more than half a century, coding, theory has seen phenomenal growth. It has found widespread application in areas ranging from communication systems, to compact disc players, to storage technology. In the effort to find good codes for practical purposes, researchers have moved beyond block codes to other paradigms, such as convolutional codes, turbo codes, space-time codes, low-density-parity-check (LDPC) codes and even quantum codes. While the problems in coding theory often arise from engineering applications, it is fascinating to note the crucial role played by mathematics in the development of the field. The importance of algebra, combinatorics and geometry in coding theory is a commonly acknowledged fact, with many deep mathematical results being used in elegant ways in the advancement of coding theory.

Coding theory therefore appeals not just to engineers and computer scientists, but also to mathematicians. It has become increasingly common to find the subject taught as part of undergraduate or graduate curricula in mathematics.