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.

The concept of a bounded tolerance order

A set of real intervals {Iν | ν ∈ V} can be viewed as a representation of the interval graph G = (V, E) where xy ∈ E ⇔ Ix ∩ Iy ≠ Ø. It can also be interpreted as representing an interval order P = (V, ≺) where x ≺ y if and only if Ix is completely to the left of Iy (which we denote Ix ≪ Iy). The graph G and the order P are related in that G is the incomparability graph of P, that is, xy ∈ E(G) ⇔ x ∥ y in P. Thus, results about interval graphs have counterparts in the world of ordered sets. For example, note the similarity in the characterization theorems below.

Theorem 5.1.(Gilmore and Hoffman, 1964) A graph G is an interval graph if and only if it is a cocomparability graph with no induced C4.

Theorem 5.2.(Fishburn, 1970) An order P is an interval order if and only if it has no induced2 + 2.

The graph C4 is the incomparability graph of the order 2 + 2, so the same 4-element structure is forbidden in both theorems. Theorem 5.1 has the extra condition that G be a cocomparability graph. This is not needed in Theorem 5.2 since the relation in any ordered set is transitive, and thus the incomparability graph of any ordered set is always a cocomparability graph.

We saw in Chapter 1 that the purpose of channel coding is to introduce redundancy to information messages so that errors that occur in the transmission can be detected or even corrected. In this chapter, we formalize and discuss the notions of error-detection and error-correction. We also introduce some well known decoding rules, i.e., methods that retrieve the original message sent by detecting and correcting the errors that have occurred in the transmission.

Communication channels

We begin with some basic definitions.

Definition 2.1.1 Let A = {a1, a2, …, aq} be a set of size q, which we refer to as a code alphabet and whose elements are called code symbols.

1. (i) A q-ary word of length n over A is a sequence w = w1w2 … wn with each wi ∈ A for all i. Equivalently, w may also be regarded as the vector (w1, …, wn).

2. (ii) A q-ary block code of length n over A is a nonempty set C of q-ary words having the same length n.

3. (iii) An element of C is called a codeword in C.

4. (iv) The number of codewords in C, denoted by ∣C∣, is called the size of C.

5. (v) The (information) rate of a code C of length n is defined to be (logq ∣C∣)/n.

6. (vi) A code of length n and size M is called an (n, M)-code.

Background and motivation

Our mathematical adventure begins with a collection of intervals on the real line. The intervals may have come from an application, for example, they could represent the durations of a set of events on a time line, or fragments of DNA on the genome, or sectors of consecutive elements of a linearly ordered set. Some of the intervals may intersect one another, and others may be disjoint. No matter what they may represent, intervals are familiar to us as mathematical entities. There are many relationships between these intervals that we could study. In this book, we deal mostly with intersection.

When two intervals intersect, we might interpret this positively as their having something important in common, like an opportunity to share information. For example, if each interval represented the time period during which a group of school children would be visiting a science museum, then two groups whose intervals intersect could participate in a joint activity. We might then ask, how many times would we need to flash the new Artificial Bolt of Lightning so that each group would get to see it? Or we might interpret intersection negatively as having a major conflict, like competing for a resource that cannot be shared. For example, in a one-television household, when a parent wants to watch the News and at the same time a teenager wants to watch an old movie on a different channel, we have a temporal conflict.

The early papers, Golumbic and Monma (1982) and Golumbic, Monma, and Trotter (1984), leave several open questions which have shaped research in the field. The questions of characterizing the classes of tolerance graphs and bounded tolerance graphs remain open and no efficient recognition algorithms are yet known.

Theorem 2.8 gives a way to find graphs that are not bounded tolerance graphs, specifically by choosing graphs that are not cocomparability graphs. For example, the graph G shown in Figure 3.1 is not a bounded tolerance graph because its complement (also shown in Figure 3.1) is not transitively orientable (Exercise 1.6(c)). Yet, G is a tolerance graph as seen by the following representation: Ia = [12, 15], Ib = [3, 7], Ic = [23, 27], Id = [1, 21], Ie = [5, 25], If = [10, 30], ta = ∞, tb = tc = td = tf = 1, te = 17.

In Golumbic, Monma, and Trotter (1984), the authors ask if there are other types of separating examples for the classes of tolerance graphs and bounded tolerance graphs besides graphs that are not cocomparability graphs, that is,

Question 3.1. Is there a cocomparability graph that is a tolerance graph but not a bounded tolerance graph?

This question is still open. By Theorem 2.20 and the containments in Figure 2.8, if such a graph exists, it must be a trapezoid graph.

The preceding chapter covered the subject of general cyclic codes. The structure of cyclic codes was analyzed, and two simple decoding algorithms were introduced. In particular, we showed that a cyclic code is totally determined by its generator polynomial. However, in general it is difficult to obtain information on the minimum distance of a cyclic code from its generator polynomial, even though the former is completely determined by the latter. On the other hand, if we choose some special generator polynomials properly, then information on the minimum distance can be gained, and also simpler decoding algorithms could apply. In this chapter, by carefully choosing the generator polynomials, we obtain several important classes of cyclic codes, such as BCH codes, Reed–Solomon codes and quadratic-residue codes. In addition to their structures, we also discuss a decoding algorithm for BCH codes.

BCH codes

The class of Bose, Chaudhuri and Hocquenghem (BCH) codes is, in fact, a generalization of the Hamming codes for multiple-error correction (recall that Hamming codes correct only one error). Binary BCH codes were first discovered by A. Hocquenghem [8] in 1959 and independently by R. C. Bose and D. K. Ray-Chaudhuri [1] in 1960. Generalizations of the binary BCH codes to q-ary codes were obtained by D. Gorenstein and N. Zierler [5] in 1961.

Definitions

We defined the least common multiple lcm(f1(x), f2(x)) of two nonzero polynomials f1(x), f2(x) ∈ Fq[x] to be the monic polynomial of the lowest degree which is a multiple of both f1(x) and f2(x) (see Chapter 3).

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, ϕ-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.

Interval relations play a significant role in many resource allocation, temporal reasoning, biological and scheduling problems. We saw this in Sections 1.1 and 4.1 in our motivating examples for interval graphs, tolerance graphs and interval probe graphs. Intervals can represent events in time, which may conflict or may be compatible. They can represent certain tasks to be performed according to a timetable which must be assigned distinct processors or people. Or they may represent fragments of DNA, which are compatible or incompatible.

For many optimization problems, such as graph coloring or finding maximum stable sets, there are efficient algorithms that give solutions when the set of graphs under consideration is restricted to a structured family. Many applications reduce to solving optimization problems on such families of graphs. Indeed, at the very beginning of this book, a 4-coloring of the tolerance graph in Figure 1.3 provided an assignment of four meeting rooms for that motivating example. In a similar application, with say only one room available for a given collection of meetings (intervals) with tolerances, a maximum stable set would provide the largest number of meetings from the collection that can be scheduled. In this chapter, we investigate these algorithmic aspects of tolerance graphs.

Narasimhan and Manber (1992) were the first to study the chromatic number, clique and stable set problems for representations of tolerance graphs.

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.