Chapter 1. Graphs
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For years, mathematicians have affected the growth and development of computer science. In the beginning they helped design computers for the express purpose of simplifying large mathematical computations. However, as the role of computers in our society changed, the needs of computer scientists began affecting the kind of mathematics being done. Graph theory is a prime example of this change in thinking. Mathematicians study graphs because of their natural mathematical beauty, with relations to topology, algebra and matrix theory spurring their interest. Computer scientists also study graphs because of their many applications to computing, such as in data representation and network design. These applications have generated considerable interest in algorithms dealing with graphs and graph properties by both mathematicians and computer scientists. Today, a study of graphs is not complete without at least an introduction to both theory and algorithms. This text will attempt to convince you that this is simply the nature of the subject and, in fact, the way it was meant to be treated. Graphs arise in many settings and are used to model a wide variety of situations. Perhaps the easiest way to adjust to this variety is to see several very different uses immediately. Initially, let's consider several problems and concentrate on finding models representing these problems, rather than worrying about their solutions. Suppose that we are given a collection of intervals on the real line, say C = { I 1 , I 2 ,. .. , I k }. Any two of these intervals may or may not have a nonempty intersection. Suppose that we want a way to display the intersection relationship among these intervals. What form of model will easily display these intersections? One possible model for representing these intersections is the following: Let each interval be represented by a circle and draw a line between two circles if, and only if, the intervals that correspond to these circles intersect. For example, consider the set The model for these intervals is shown in Figure 1.1.1.
[1] D. R. Fulkerson. UPSETS IN ROUND ROBIN TOURNAMENTS , 1965 .
[2] David S. Johnson,et al. Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .
[3] H. Ryser. Combinatorial Properties of Matrices of Zeros and Ones , 1957, Canadian Journal of Mathematics.
[4] D. R. Fulkerson,et al. Some Properties of Graphs with Multiple Edges , 1965, Canadian Journal of Mathematics.
[5] Gert Sabidussi,et al. Graph multiplication , 1959 .