<ul><li>How do delivery companies, such as UPS and FedEx, plan the routes they will take to deliver packages? What must they think about as they plan the route? </li></ul><ul><li>Do they want to have to backtrack? Why not? </li></ul><ul><li>A network (route) is traversable (sounds like “traceable”) if you can draw it without lifting your pencil or backtracking. </li></ul>
Network or Vertex-edge Graph <ul><li>A figure made up of vertices and edges that shows how objects can be connected (see communicator). </li></ul><ul><li>The objects can be cities, airport terminals, houses, computers, people, phones, television networks (NBC to smaller companies), rides at Disneyworld, maps, printers, etc. </li></ul><ul><li>The vertices represent the object and the edges represent the paths (wires, roads, etc.) connecting the objects. </li></ul><ul><li>Vertices (vertex plural) are classified as being odd or even depending on the number of edges that go into it. </li></ul>
Network Traveling <ul><li>Can you predict by looking at a network whether it is traversable or not? </li></ul><ul><li>Look at the communicator. Make a chart in your notebook. Count the number of odd and even vertices for each network. Decide whether each network is traversable or not; use the marker to trace the path. </li></ul><ul><li>You can go through a vertex more than once but you can use each edge only once. </li></ul><ul><li>Can you state a rule to predict whether a network is traversable or not? </li></ul><ul><li>If the network is traversable, does it matter where your start? Does it depend upon the type of network? </li></ul>
Euler paths and circuits <ul><li>If a network is traversable it is called an Euler path (more about him later). </li></ul><ul><li>If it has all even vertices, it is traversable! You can start anywhere </li></ul><ul><li>If you start and end at the same point, it is called an Eurler Circuit . </li></ul><ul><li>If it has 2 or less than 2 odd vertices, it is traversable! </li></ul><ul><li>If it has 2 odd vertices, you have to start at one odd vertex and end at the other one. </li></ul>
Cities and towns route their garbage trucks to drive down every street. The best routes are the ones where the trucks drive down each street on;y once, saving the town money in terms of gas and time. Unfortunately not all subdivisions or neighborhoods take that into mind! Like my neighborhood. Careful planning can save time and money later !
Koningberg Bridge Problem, 1700’s <ul><li>The river, Pregel, divides the town of Konigsberg, Germany into four separate land masses. Seven bridges connect the various parts of town. The residents of the town of Konigsberg liked to take Sunday strolls around their town to visit with friends who lived in different parts of the town. In order to do that, they had to cross the bridges. The curious citizens wondered if it were possible to take a journey across all seven bridges without having to cross any bridge more than once. They tried it. They talked about it with their friends. No one was able to find a way to do this! </li></ul>
Euler Paths and Circuits <ul><li>A famous Swiss mathematician, Leonard Euler, visited the town and heard them talking about their problem. After studying the problem, he finally reported in 1735 that is was impossible; there was no path what would enable them to visit each part of the town crossing each bridge exactly once. He compared the town to a graph in which all the vertices are odd, and proved that such a graph could not be traced. Euler reasoned that for such a journey to be possible that each land mass should have an even number of bridges connected to it, or if the journey would begin at one land mass and end at another, then exactly those two land masses could have an odd number of connecting bridges while all other land masses must have an even number of connecting bridges. </li></ul><ul><li>He wrote a paper about the problem and stated: </li></ul><ul><li>A graph has a path traversing each edge exactly once if exactly two vertices have odd degree . </li></ul>