Discrete MathRound, Round, Get Around… I Get AroundMathematics of Getting Around

Routing Problems• Finding routes to deliver goods

• Does a route exist?

• If one does exist, what is the fastest (best) route?

Seven Bridges of Konigsberg

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

• Vertex Set-

• Edge Set-

• Parallel Edges- (CD,CD)

• Multigraph- allows loops and parallel edges

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

Isolation• Isolated Vertices- Vertex with no edges

• Pure Isolation- A with only isolated vertices

Adjacent Vertices• When two vertices are connected by the same edge

Adjacent Edges• When two edges share a common vertex

Degree of a Vertex• The number of edges at each vertex. • A loop counts as 2• Written as deg(V)=

EVEN or ODD• We will distinguish the vertices by the even and odd degrees

ABCDEFGH

Path• A trip that starts and ends at different vertices

Circuit• A trip that starts and ends at the same vertex

• 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

• Bridge- An edge that if it is removed turns a connected graph into a disconnected graph

• 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.

Problems

Page 191-193 #2, 4, 6, 10,

12, 14, 16

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.

Back to Konigsberg

Problems

Page 193-194 #18, 19, 20

Euler’s Theorems

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

Unicursal Tracings• Euler Circuits or Paths?

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

Problems

Pg. 194 #24-28

Fleury’s Algorithm• Algorithm-• A set of rules for solving a problem

• Create an algorithm:

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.

F C D A C E A B D F

J K B C L K H J B A J I H G L E C D E G F E

Problems

Pg. 195 #30-34

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

Eulerizing

Semi-Eulerizing

Problem

Page 196 #38-42