Discrete Math Round, Round, Get Around… I Get Around Mathematics 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

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

EVEN or ODD We will distinguish the vertices by the even and odd degrees A B C D E F G H

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 #2, 4, 6, 10, 12, 14, 16

Graph Models

Back to Konigsberg

Problems Page #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 VerticesConclusion 0Euler Circuit 2Euler 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