The Bridges of Konigsberg Problem Can you walk around the town crossing each bridge only once?

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1 The Bridges of Konigsberg Problem Can you walk around the town crossing each bridge only once? Many people had tried the walk and felt that it was impossible, but no one knew for sure. In 1736, Leonard Euler (pronounced oiler), solved the problem creating an important practical branch of mathematics known as graph theory.

2 Chapter 1: Urban Services Management Science Management Science Uses mathematical methods to help find optimal solutions to management problems. Often called Operations Research. Optimal Solutions The best (most favorable) solution Government, business, and individuals all seek optimal results. Optimization problems: Finish a job quickly Maximize profits Minimize costs Urban Services to optimize: Checking parking meters Delivering mail Removing snow 2

3 Euler Circuits Parking-Control Officer Problem Checking parking meters Our job is to find the most efficient route for the parking-control officer to walk as he checks the parking meters. Problem: Check the meters on the top two blocks. Goals for Parking-Control Officer Must cover all the sidewalks without retracing any more steps than necessary. Should end at the same point at which he began. Problem: Start and end at the top lefthand corner of the left-hand block. Street map for part of a town. Euler circuit A circuit that traverses each edge of a graph exactly once and starts and stops at the same point. 3

4 Important Terms in Graph Theory Graph A finite set of dots (vertices) and connecting links (edges). Graphs can represent our city map, air routes, etc. Vertex (pl. vertices) A point (dot) in a graph where the edges meet. Edge A link that joins two vertices in a graph (traverse edges). Path A connected sequence of edges showing a route, described by naming the vertices traveled. Circuit A path that starts and ends at the same vertex. Valence The number of edges touching that vertex (counting spokes on the hub of a wheel). Connectedness You can reach any vertex by traversing the edges given in the graph. Euler circuit Has even-valent vertices and is connected. p. 22 # 1, 3, 4, 5, 6, 7

5 Looking at the Valence of Vertices on a Graph What happens to the total valence of a graph, when you an edge to the graph? Can the total valence of the graph be odd? What do you get when you add even numbers together? What do you get when you add an odd number of odd numbers? What do you get get when you add an even number of odd numbers? Conclusion: You can only have an even number of odd vertices on a graph. p. 23 # 8, 9, 10

6 Can you trace this graph without lifting your pencil? Consider an odd vertex on a graph: If you start at an odd vertex, can you finish there? If you start away from an odd vertex, can you get back to where you started? If a graph has only two odd vertices, then it can be traced without lifting your pencil, because you can start at one odd vertex and end at the other. To have an Euler Circuit, you must have all even vertices.

7 Euler Circuits Simplified graph (b) is enlarged to show the points (vertices) labeled with letters A F which are linked by edges. Simplified graph (a) is superimposed on the streets with parking meters. Graph A finite set of dots (vertices) and connecting links (edges). Graphs can represent our city map, air routes, etc. Vertex (pl. vertices) A point (dot) in a graph where the edges meet. Edge A link that joins two vertices in a graph (traverse edges). Path A connected sequence of edges showing a route, described by naming the vertices traveled. Circuit A path that starts and ends at the same vertex. 7

8 Path vs. Circuit Paths Paths can start and end at any vertex using the edges given. examples: NLB, NMRB, etc. Circuits Paths that starts and ends at the same vertex. Examples: MRLM, LRBL, etc. Euler Circuits Circuit vs. Euler Circuit (Both start and end at same vertex.) Circuits may retrace edges or not use all the edges. Nonstop air routes Euler circuits travel each edge once and cover all edges. p. 23 # 11 8

9 Finding Euler Circuits Two Ways to Find an Euler Circuit Trial and error Keep trying to create different paths to find one that starts and ends at the same point and does not retrace steps. Mathematical approach (better method) An Euler circuit exists if the following statements are true: All points (vertices) have even valence. The graph is connected. Leonhard Euler ( ) Among other discoveries, he was credited with inventing the idea of a graph as well as the concepts of valence and connectedness. 9

10 Finding Euler Circuits Valence The number of edges touching that vertex (counting spokes on the hub of a wheel). Connectedness You can reach any vertex by traversing the edges given in the graph. Euler circuit Has even-valent vertices and is connected. If vertices have odd valence, it is not an Euler circuit. An Euler circuit starting and ending at A Proving Euler s Theorem If a graph has an Euler circuit, it must have only even-valent vertices and it must be connected. This can be proved by pairing up edges at each vertex, thus proving all vertices have paired edges and further proving there is an even number of edges at each vertex, X. Thus, every edge at X has an incoming edge (arriving at vertex X) and an outgoing edge (leaving from vertex X). Example: At vertex B, you can pair up edges 2 and 3 and edges 9 and

11 Finding Euler Circuits Is there an Euler Create (Find) an Euler Circuit Circuit? Pick a point to start (if none has been given to you). Does it have even valence? (Yes) Is the graph connected? (Yes) Euler circuit exists if both yes. Number the edges in order of travel, showing the direction with arrows. Cover every edge only once, and end at the same vertex where you started. As you travel the graph, make sure the remaining part stays connected. 11

12 Beyond Euler Circuits Chinese Postman Problem In real life, not all problems will be perfect Euler circuits. If no Euler circuit exists (odd valences), you want to minimize the length of the circuit by carefully choosing the edges to be retraced. For our purposes, we assume all edges have the same length simplified Chinese postman problem. Chinese mathematician Meigu Guan first studied this problem in 1962, hence the name. The blue dots indicate parking meters along the street. The graph represents edges with parking meters. Notice only vertices C and G have odd valence. 12

13 Beyond Euler Circuits Eulerize the Graph to Solve Chinese Postman Problem For graphs that are connected but have vertices with odd valence, we will want to reuse (duplicate) the minimum number of edges until all vertices appear to have even valence. Only existing edges can be duplicated (or added). Each edge that is duplicated (added) will later be the edge that will be reused during eulerization. The edge CG is reused, which would make all vertices appear to have even valence. A circuit is made by reusing the edge CG. Below, the graph is eulerized (starts and stops at same point and covers all edges once including reused ones. 13

14 Beyond Euler Circuits Eulerizing a Graph On the graph, add edges by duplicating existing ones, until you arrive at a graph that is connected and even-valent. The graph below is an efficient eulerization because the fewest number of edges were added. Find an Euler circuit on the eulerized graph. Traverse every original and added edge once, as you find a circuit that starts and ends at the same vertex. Squeeze this Euler circuit from the eulerized graph onto the original graph by replacing the added edge with an arrow showing it was retraced. Only reuse (add) edge BC. Squeeze the eulerized circuit onto the graph. 14

15 Beyond Euler Circuits Hints for Eulerizing a Graph For the most efficient eulerization, look for the fewest edges to add to make all vertices even. Typically, locate odd valence vertices and try to reuse (add) the connecting edge between the vertices. Sometimes vertices are more than one edge apart; in this case, reuse edges between vertices (see graph below). Remember: Only duplicate (add to) the existing edges. Odd vertices, X and Y, are more than one edge apart. This is not allowed must only reuse existing edges. Reuse existing edges between the odd vertices. p. 24 # 12, 13, 15, 16, 18 15

16 Beyond Euler Circuits Rectangular Networks This is the name given to a street network composed of a series of rectangular blocks that form a large rectangle made up of so many blocks high by so many blocks wide. Eulerizing rectangular networks: Edge Walker Start in upper left corner (at A). Travel (clockwise) around the outer boundary. As you travel, add an edge by the following rules: If the vertex is odd, add an edge by linking it to the next vertex. If this next vertex becomes even, skip it (just keep walking ). If this next vertex becomes odd, (on a corner) link it to the next vertex. Repeat this rule until you reach the upper left corner again. 16

17 Urban Graph Traversal Problem Euler Circuits and Eulerizing Graphs: Practical Applications Checking parking meters (discussed) Collecting garbage Salting icy roads Inspecting railroad tracks Special Requirements May Need to Be Addressed Traffic directions Number of streets/lanes (divided routes) Parking time restriction Theory Modifications Can Address Special Requirements A digraph (directed graph) is used to show one-way street. Arrows show restriction in traversal possibilities (not part of circuits). Territories may need to be divided into multiple routes. p. 25 #1, 4, 11, 12, 17, 21, 24, 25, 26, 33, 35, 36 17

18 Summary Management Science Optimal Solutions for Urban Services Euler Circuits Parking-Control Officer Problem Finding Euler Circuits Qualifications: Even Valence and Connectedness Beyond Euler Circuits Chinese Postman Problem Eulerizing a Graph Urban Graph Traversal Problems More practical applications and modifications For All Practical Purposes Mathematical Literacy in Today s World, 7th ed. 2006, W.H. Freeman and Company 18