network graph

- consists of a set of points called vertives(sigular is vertex)connected by lines called edges

- 由一些叫vertices(vertex)(顶点)的点被叫edge(边)线条连接在一起的图

Network Graphs

In the 18th century in the town of Königsberg, Germany, a favorite pastime was walking along the Pregel River and strolling over the town's seven bridges (Fig. 1). During this period a natural question arose: Is it possible to take a walk and cross each bridge only once? Before reading further, can you determine the answer? This question was solved by the Swiss mathematician Leonard Euler. His solution was the beginning of network theory.

Euler represented the four land areas of Königsberg (A, B, C, and D in Figure 2) as four points and the seven bridges as seven lines joining these points. For example, the island of Kneiphoff (A) can be reached by five bridges, and in the diagram in Figure 2 there are five lines to point A. The three lines from point D represent three bridges, etc. This kind of diagram is called a network graph, or more simply, a network. Notice that Euler was concerned not with the size and shape of the bridges and land regions but rather with how the bridges were connected.

A network is a collection of points, called vertices, and a collection of lines, called arcs, connecting these points. A network is traversable if you can trace each arc exactly once by beginning at some point and not lifting your pencil from the paper. The problem of crossing each bridge exactly once reduces to one of traversing the network representing these bridges.

Euler made the remarkable discovery that whether a network is traversable depends on the number of odd vertices. In the Königsberg network, there are an odd number of arcs at point A, so A is called an odd vertex. If the number of arcs meeting at a point is even, the point is called an even vertex. Euler found that the only traversable networks are those that have either no odd vertices or exactly two odd vertices. Since the Königsberg network has four odd vertices, it is not traversable. Therefore, it is not possible to take a walk over the bridges of Königsberg and cross each bridge only once.

Try working the following Examples A-D before looking at the solutions.