Post on 24-Dec-2015
Routing Problems• Finding routes to deliver goods
• Does a route exist?
• If one does exist, what is the fastest (best) route?
Unicursal Tracings• Pass through all paths without crossing one twice, and never
lift your pencil• Closed- starts and ends at the same point• Open- starts and ends at different points
Graph Theory• Vertices- the dots on the graph (stops or crossroads)
• Edges- lines on the graph (bridges or paths)
• Vertices- A,B,C,D,E,F
• Edges-AB,BC,CD,AD,DE,EB,CD,BB
• Loop- when an edge starts and ends at the same vertex
• Double Edge (Multiple Edge)- edges that connect to same vertices.
• No direction- AB or BA
Friend Connection• Mary is friends with Ken, Bob, Amy, Sally, and Juan• Ken is friends with Mary, Bob, and Amy• Bob is friends with Ken, Mary, and Amy• Amy is friends with Bob, Ken, and Mary• Sally is only friends with Mary• Juan is friends with Mary and Jay• Jay is friends with Juan, Sasha, and Max• Max is friends with Jay, Sasha, Peter, and Ben• Sasha is friends with Jay, Max, and Peter• Peter is friends with Sasha and Max• Ben is only friends with Max
• Length- the number of edges in a path
• Connected Graph- a graph that any vertex can be reached by any path
• Disconnected Graph- a graph that any vertex cannot be reached by a path
• Components- A disconnected graph is made up of multiple components
• Euler Path- A path that travels through every edge once and only once. Starts and ends in different places.
• Euler Circuit- A path that travels through every edge once and only once and ends in the place it starts.
Graph Models• Taking a word problem and creating an algebraic expression or
geometric figure.
• There are tickets to go to the movie theater. Each movie ticket costs $12.50. If 18 people are going to see Man of Steel, how much did it cost for everyone to go.
Euler’s Circuit Theorem• If a graph is connected and every vertex is even, then there is
at least one Euler Circuit.
Euler’s Path Theorem• If a graph is connected and has exactly two odd vertices, then
it has at least one Euler Path.• The path must start at an odd vertex, and end at the other odd
vertex.• If it has more than 2 odd vertices then it does not have an Euler
Path.
Back to the 7 Bridges• Is there an Euler • Circuit?• No
• Path?• No
• What is the shortest • Circuit?• 9• Path?• 8
Euler’s Sum of Degrees Theorem• The sum of the degrees of all the vertices of a graph, equals
twice the number of edges. (This will always be even)
• A graph will always have an even number of odd vertices.
Number of Odd Vertices Conclusion
0 Euler Circuit
2 Euler Path
4,6,8,… Neither
1,3,5,7,… Check again you messed up
Fleury’s Algorithm• “Do Not Burn Your Bridges Behind You”
• The bridges are the last edges you are to cross
• As you move you create more bridges behind you.
Fleury’s Algorithm• Make sure the graph is connected• Is there a Euler Circuit (all even) or a Euler Path (2 odd)• Choose your starting point, if Circuit start anywhere, if Path
start at an odd vertex• Choose paths that are not bridges.
Eulerizing Graphs• Exhaustive Route- Route that passes through every edge at
least once
• Euler Circuit if all vertices are even• Euler Path is two vertices are odd• A path that will recross the least number of bridges
Eulerizing Graphs• Deadheads- A recrossed edge
• Eulerizing- Adding edges to odd vertices to turn them even so that we can create an Euler Circuit
• Semi-Eulerizing- Leaving two vertices odd so that we can create an Euler Path