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 Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 2 Exhaustive Routes In this section we will finally answer some of the routing problems raised at the beginning of the chapter Their common thread is the need to find optimal exhaustive routes in a connected graph How is this done Let s first refresh our memories of what this means We will use the term exhaustive route to describe a route that travels along the edges of a graph and passes through each and every edge of the graph at least once Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 3 Exhaustive Routes Such a route could be an Euler circuit if the graph has no odd vertices or an Euler path if the graph has two odd vertices but for graphs with more than two odd vertices an exhaustive route will have to recross some of the edges This follows from Euler s theorems We are interested in finding exhaustive routes that recross the fewest number of edges Why Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 4 Exhaustive Routes In many applications each edge represents a unit of cost The more edges along the route the higher the cost of the route In an exhaustive route the first pass along an edge is a necessary expense part of the requirements of the job Any additional pass along that edge represents a wasted expense these extra passes are often described as deadhead travel Thus an exhaustive route that minimizes cost optimal exhaustive route is one with the fewest number of deadhead edges Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 5 Eulerizing Graphs We are now going to see how the theory developed in the preceding sections will help us design optimal exhaustive routes for graphs with many more than two odd vertices The key idea is that we can turn odd vertices into even vertices by adding duplicate edges in strategic places This process is called eulerizing the graph Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 6 Example 5 22 Covering a 3 by 3 Street Grid The graph represents a 3 block by 3 block street grid consisting of 24 blocks How can we find an optimal route that covers all the edges of the graph and ends back at the starting vertex Our first step is to identify the odd vertices This graph has eight odd vertices B C E F H I K and L shown in red Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 7 Example 5 22 Covering a 3 by 3 Street Grid When we add a duplicate copy of edges BC EF HI and KL we get this graph This is a eulerized version of the original graph its vertices are all even so we know it has an Euler circuit Moreover it s clear we couldn t have done this with fewer than four duplicate edges Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 8 Example 5 22 Covering a 3 by 3 Street Grid This figure shows one of the many possible Euler circuits with the edges numbered in the order they are traveled The Euler circuit represents an exhaustive closed route along the edges of the original graph with the four duplicate edges BC EF HI and KL indicating the deadhead blocks where a second pass is required Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 9 Example 5 22 Covering a 3 by 3 Street Grid The total length of this route is 28 blocks 24 blocks in the grid plus 4 deadhead blocks and this route is optimal no matter how clever you are or how hard you try if you want to travel along each block of the grid and start and end at the same vertex you will have to pass through a minimum of 28 blocks Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 10 Example 5 23 Covering a 4 by 4 Street Grid The graph in the figure represents a 4 block by 4 block street grid consisting of 40 blocks The 12 odd vertices in the graph are shown in red We want to eulerize the graph by adding the least number of edges Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 11 Example 5 23 Covering a 4 by 4 Street Grid This figure shows how not to do it This graph violates the cardinal rule of eulerization you can only duplicate edges that are part of the original graph Edges DF and NL are new edges not duplicates so this figure is out Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 12 Example 5 23 Covering a 4 by 4 Street Grid This figure shows a legal eulerization but it is not optimal as it is obvious that we could have accomplished the same thing by adding fewer duplicate edges Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 13 Example 5 23 Covering a 4 by 4 Street Grid This figure shows an optimal eulerization of the original graph one of several possible Once we have an optimal eulerization we have the blueprint for an optimal exhaustive closed route on the original graph Regardless of the specific details we now know that the route will travel along 48 blocks the 40 original blocks in the grid plus 8 deadhead blocks Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 14 Open Route In some situations we need to find an exhaustive route but there is no requirement that it be closed the route may start and end at different points In these cases we want to leave two odd vertices on the graph unchanged and change the other odd vertices into even vertices by duplicating appropriate edges of the graph Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 15 Semi eulerization This process is called a semi eulerization of the graph When we strategically choose how to do this so that the number of duplicate edges is as small as possible we can obtain an optimal exhaustive open route In this case the route will start at one of the two odd vertices and end at the other one Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 5 7 16 Example 5 24 Parade Route Let s consider once again the 4 by 4 street grid Imagine that your job is to design a good route for a Fourth of July parade that must pass through each of the 40 blocks of the street grid The key is that when routing a parade you do not want the parade …