Graphs, Part I: Basic Notions

Abstract

Graphs are mathematical structures that have many applications in computer science , electrical engineering, and more widely in engineering as a whole, but also in sciences such as biology, linguistics, and sociology, among others. For example, relations among objects can usually be encoded by graphs. Whenever a system has a notion of state and a state transition function, graph methods may be applicable. Certain problems are naturally modeled by undirected graphs whereas others require directed graphs. Let us give a concrete example. Suppose a city decides to create a public transportation system. It would be desirable if this system allowed transportation between certain locations considered important. Now, if this system consists of buses, the traffic will probably get worse so the city engineers decide that the traffic will be improved by making certain streets one-way streets. The problem then is, given a map of the city consisting of the important locations and of the two-way streets linking them, finding an orientation of the streets so that it is still possible to travel between any two locations. The problem requires finding a directed graph, given an undirected graph. Figure 3.1 shows the undirected graph corresponding to the city map and Figure 3.2 shows a proposed choice of one-way streets. Did the engineers do a good job or are there locations such that it is impossible to travel from one to the other while respecting the one-way signs? The answer to this puzzle is revealed in Section 3.3. There is a peculiar aspect of graph theory having to do with its terminology. Indeed, unlike most branches of mathematics, it appears that the terminology of graph theory is not standardized yet. This can be quite confusing to the beginner who has to struggle with many different and often inconsistent terms denoting the same concept, one of the worse being the notion of a path. Our attitude has been to use terms that we feel are as simple as possible. As a result, we have not followed a single book. Among the many books on graph theory , we have been inspired by the classic texts, Harary [4], Berge [1], and Bollobas 165