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Section 2.1 : Euler Circuit Problems. Example 2.1.1 : Walking the ‘Hood’.
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Example 2.1.1: Walking the ‘Hood’ • After a rash of burglaries, a private security guard is hired to patrol the streets of the Sunnyside neighborhood shown. The security guard’s assignment is to make an exhaustive patrol, on foot, through the entire neighborhood. Obviously, he doesn’t want to walk any more than necessary. His starting point is the southeast corner across from the school – that’s where he parks his car. • Is it possible to start and end at the starting point, cover every block of the neighborhood, and pass through each block just once? • If some of the blocks will have to be covered more than once, what is an optimal route that covers the entire neighborhood? (Notes)
Example 2.1.2: Delivering the Mail • A mail carrier has to deliver the mail in the same Sunnyside neighborhood. The difference between the mail carrier’s route and the security guard’s route is that the mail carrier’s must make two passes through blocks with houses on both sides of the street and only one pass through blocks with houses on one side of the street; and, where there are no homes on either side of the street, the mail carrier does not have to walk at all. In addition, the mail carrier has no choice as to her starting and ending points – she has to start and end her route at the local post office.
Example 2.1.3: The Seven Bridges of Konigsberg • The figure shows an old map of the city of Konigsberg and its seven bridges. Can a walker take a stroll and cross each of the seven bridges of Konigsberg without crossing any of them more than once? • Try it – 3 minutes
Example2.1.4: A Baseball Schedule • Notes • The graph models the baseball schedule for a week. The vertices represent the teams. Each game played during that week is represented as an edge between two teams. • How many games are scheduled for Pittsburgh during the week? List the teams that are playing. How many times are they playing each of these teams? • Do the positions of New York and Montreal correspond to their geographic locations on a map? If not, is the graph drawn incorrectly? • Notes
Example2.1.5: Relationship Graphs • Imagine that as part of a sociology study we want to describe the network of “friendships” that develops among a group of students through their Facebook sites. We can illustrate this with a graph. • What do the vertices represent? Edges? • On page 172, there is an example of a graph that might be more realistic.
Example2.1.6: Degrees of Separation • Get in groups of 3. • Each group member should select 3-4 famous actors. Determine the relationship between all the actors (you should have 9-16 different actors). Represent the actors as vertices and use edges to show these relationships. Each group should turn in one graph.
Example2.2.1: Draw a graph • List all the adjacent vertices. • List all the adjacent edges. • What is the degree • Of A? • Of F? • Of D? • Which vertices are even? Are odd? • What is the difference between a path and a circuit?
Example 2.2.2(Continued) • Find 3 different paths from A to D. • What is the length of each path? • Find a circuit. • Is the graph connected? • Can you find • an Euler path? • An Euler circuit? • Draw a disconnected graph.
Example2.2.3: The Seven Bridges of Konigsberg • Recall the ‘Seven Bridges of Konigsberg’ problem: The figure shows an old map of the city of Konigsberg and its seven bridges. Can a walker take a stroll and cross each of the seven bridges of Konigsberg without crossing any of them more than once? • Is an Euler path possible? • Is an Euler circuit possible? • If not, what is the optimal path/circuit? • What is the significance of this problem? Why did Euler bother exploring graphs at all?
Example2.3.1: Euler Circuits • Draw a graph with an Euler circuit and one without an Euler circuit. When you have drawn one in your notes, draw it on the board. • What do all the graphs with Euler circuits have in common? • What do all the graphs without Euler circuits have in common?
Euler’s Circuit Theorem • If a graph is connected and every vertex is even, then it has at least one Euler circuit. • If a graph has any odd vertices, then it does not have an Euler circuit.
Example2.3.2: Euler Paths • Draw a graph with an Euler path and one without an Euler path. Once you have drawn one in your notes, draw it on the board. • What do all the graphs with Euler paths have in common? • What do all the graphs without Euler paths have in common?
Euler’s Path Theorem • If a graph is connected and has exactly two odd vertices, then it has at least one Euler path. • If a graph has more than two odd vertices, then it cannot have an Euler path. • What if a graph has exactly one odd vertex? Later…
Example2.3.3: Back to the Konigsberg problem • Look at the graph. Is there an Euler path? An Euler circuit? • Nope!
Example2.3.4: A Crazy Graph • Is there an Euler path? • An Euler circuit?
Euler’s Sum of Degrees Theorem • The sum of the degrees of all the vertices of a graph equals twice the number of edges (and, thus, is even). (Can a graph have one odd vertex?) • A graph always has an even number of odd vertices. Now we know that the graph from Example 5.5.4 has an Euler path. But how do we find it?
Fleury’s Algorithm 0. Make sure that the graph is connected and either (depending on what you’re trying to find) • Has no odd vertices (circuit) or • Has two odd vertices (path). • Choose a starting vertex. (If you’re finding an Euler path, it must be one of the odd vertices.) • At each step, if you can, don’t choose a bridge. • When you can’t travel any more, you’re done!
Example2.4.1: 3 by 3 street grid • How many blocks are there? • How many odd vertices are there? • How can we find an optimal route? • Eulerizing Graphs – turn odd vertices into even vertices by adding duplicate edges.
Example2.4.2: Bridges of Madison County • Madison County, Iowa is the birth place of John Wayne. A novel by Robert James Waller and a 1995 movie directed by Clint Eastwood as based on this quaint place. A beautiful river runs through the county, and there are four islands and 11 bridges joining the islands to both banks of the river and one another. A photographer is hired to take pictures of each of the bridges. She needs to drive across each bridge once for a photo shoot. Moreover, since there is a $25 toll every time an out-of-town visitor drives across a bridge, the photographer wants to minimize the total cost of her trip and to recross bridges only if absolutely necessary. What is the optimal route? Remember that she needs to start and end in the same place.
