Network Theory

Abstract

Network optimization needs to use many terms and notions used in graph theory. In this chapter we seek to introduce most of the graph theory terms and notions used in the book. We also introduce some concepts used in the study of algorithms. To explain these concepts, we will make use of a fictitious map of one of the facilities of the fictitious Global Telecom Company (GTC). The map is shown in Figure B.1 and is drawn to scale.

If we want to depict the connectivity among various points in the facility, we can have a schematic representation of the map as shown in Figure B.2. In this figure, the locations of the map have been replaced by points, depicted as labeled circles. In Figure B.2 for example, point 1 represents the Security building in the map, point 2 represents the Reception desk, point 3 represents Warehouse 1, and so on. The lines between the points show that the buildings represented by the points are connected by a direct road segment. Figure B.2 is known as a graph. The points 1, 2,... , 10 are referred to as nodes (or vertices) of the graph, and the lines between the nodes are referred to as edges (or links) in the graph. Hence a graph is just a collection of nodes and edges linking pairs of nodes. In some graphs, we allow more than one edges to connect a pair of nodes. As a special case of such graphs, an edge may connect a node to itself. Graphs with such possibilities are called multigraphs. Graphs that are not multigraphs are called simple graphs.