5 The Mathematics of Getting Around

2. 5 The Mathematics of Getting Around 5.1 Euler Circuit Problems 5.2 What Is a Graph? 5.3 Graph Concepts and Terminology 5.4 Graph Models 5.5 Euler’s Theorems 5.6 Fleury’s Algorithm 5.7 Eulerizing Graphs

3. Modeling One of Euler’s most important insights was the observation that certain types ofproblems can be conveniently rephrased as graph problems and that, in fact, graphsare just the right tool for describing many real-life situations. The notion of using amathematical concept to describe and solve a real-life problem is one of the oldestand grandest traditions in mathematics. It is called modeling.

4. Example 5.15 The Seven Bridges of Königsberg: Part 2 The Königsberg bridges question discussed in Example 5.3 asked whether it waspossible to take a stroll through the old city of Königsberg and cross each of theseven bridges once and only once. To answer this question one obviously needs totake a look at the layout of the old city. A stylized map of the city of Königsberg isshown in Fig. 5-13(a).

5. Example 5.15 The Seven Bridges of Königsberg: Part 2 This map is not entirely accurate–the drawing is not toscale and the exact positions and angles of some of the bridges are changed. Doesit matter?

6. Example 5.15 The Seven Bridges of Königsberg: Part 2 The shape and size of the islands, thewidth of the river, the lengths of the bridges–none of these things really matter. So, then, what is it that does matter? Surprisingly little.

7. Example 5.15 The Seven Bridges of Königsberg: Part 2 The only thing that trulymatters to the solution of this problem is the relationship between land masses (islands and banks) and bridges. Which land masses are connected to each otherand by how many bridges? This information is captured by the red edges in Fig.5-13(b).

8. Example 5.15 The Seven Bridges of Königsberg: Part 2 Thus, when we strip the map of all its superfluous information, we endup with the graph model shown in Fig. 5-13(c). The four vertices of the graphrepresent each of the four land masses; the edges represent the seven bridges.

9. Example 5.15 The Seven Bridges of Königsberg: Part 2 Inthis graph an Euler circuit would represent a stroll around the town that crosseseach bridge once and ends back at the starting point; an Euler path would represent a stroll that crosses each bridge once but does not return to the startingpoint.

10. Example 5.15 Walking the ‘Hood”: Part 2 In Example 5.1 we were introduced to the problem of the security guard whoneeds to walk the streets of the Sunnyside neighborhood [Fig. 5-14(a)].

11. Example 5.16 Walking the ‘Hood”: Part 2 The graphin Fig. 5-14(b)–where each edge represents a block of the neighborhood andeach vertex an intersection–is a graph model of this problem.

12. Example 5.16 Walking the ‘Hood”: Part 2 Does the graphhave an Euler circuit? An Euler path? Neither? (These are relevant questionsthat we will learn how to answer in the next section.)

13. Example 5.17 Delivering the Mail: Part 2 Recall that unlike the security guard, the mail carrier (see Example 5.2) mustmake two passes through every block that has homes on both sides of the street(she has to physically place the mail in the mailboxes), must make one passthrough blocks that have homes on only one side of the street, and does not haveto walk along blocks where there are no houses. In this situation an appropriategraph model requires two edges on the blocks that have homes on both

14. Example 5.17 Delivering the Mail: Part 2 sides ofthe street, one edge for the blocks that have homes on only one side of the street, and no edges for blocks having no homes on either side of the street. The graphthat models this situation is shown in Fig.5-14(c).