1. Discrete Math and Management Science
Why is Discrete Math important?

2. Chapter 1: Urban Services 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. What is a routing problem?
How to deliver goods/services to an assortment of destinations the most efficient way. (best,cheapest,shortest route)
This problem comes up in many areas: transportation, communications, and delivery services to name a few.

4. Examples Mail delivery Garbage collection Police patrols
Most importantly – late night pizza deliveries

5. For instance, it is very important for airlines to efficiently schedule flights among the cities served by the airline.
Putting together efficient airline schedules is a routing problem.

6. We will study two classes of routing problems
1. Graph Theory and Euler circuit problems. (this chapter)
2. Hamilton circuit problem (the next chapter)

7. Once upon a time……..
There is a town in Russia whose name today is Kaliningrad.
In the 1700s, it was in Prussia and its name was Königsberg.
Here's a true story.
Once upon a time (OK, in the 18th century), in a place far away (well, Prussia), there once was a town called Königsberg. The townspeople enjoyed walking through their town and passing over the bridges spanning the river which passed through town. The city sits along river Pregel, in which there are two large islands. Seven bridges connected the islands to the mainland , as depicted in the following image. The citizens began to wonder if there was a path that would allow them to cross all seven bridges exactly once. Over the years, a controversy arose and eventually word of it reached a very famous mathematician of the time, Leonhard Euler.

8. Leonhard Euler (1707–1783) (pronounced “oiler”) Swiss Mathematician
Among other discoveries, he was credited with inventing the idea of a graph as well as the concepts of valence and connectedness.
Let's reconstruct Euler's solution to the bridge problem.

9. Seven Bridges of Konigsberg
First, we will take a closer look at the bridges and the river.

10. Now let’s strip away some of the distractions:

11. Make the land circles and the bridges lines.O L

12. Simplifying things once again:R
K O L

13. A final simplification: A Graph
Definition: Graph – a finite set of dots connecting lines.

14. Chapter 1: Euler Circuits Outline/learning Objectives
To understand the meaning of basic graph terminology.
To identify and model Euler circuit and path problems.
To eulerize a graph when necessary

15. Reminder: Graph is a finite set of dots connecting lines.
Which of these are graphs?
All of them 1) 2) G 3) A D O B L F C E M N 6) 4) H H J J 5) P Q X W L L I I R S K K U V

16. Edges: {AB, BC, CD, DE, AE, AD, BE, BD}
Definition: Graph is a finite set of dots (called vertices, single dot vertex) connecting lines (called edges). C
Vertices: {A, B, C, D, E} B D
Edges: {AB, BC, CD, DE, AE, AD, BE, BD} A E
There are a total of 8 edges in this graph.

17. List all vertices and the number of edges for each graph.1) 2) G 3) A D O B L F C E M N
V = {A, B, C}
E = {CB, BA}
2 EDGES
V = {E, D, G, F}
E = {ED, DG, GF, FE}
4 EDGES
V = {L, M, N, O}
E = {LO, ON, LM, MN}
4 EDGES

18. List all vertices and the number of edges for each graph.4) 5) P Q X W 6) H J L R S I K U V
V = {X, W, U, V}
E = {XW, XW, WU, UV, UU, VV}
6 EDGES
V = {H, I ,J, K , L}
E = {HJ, JK, KI, IH}
4 EDGES
V = {S, R, Q, P}
E = {PQ, PQ, QR, RS}
4 EDGES

19. Homework
p , 8
p , 9
(page 7 is figure 1.6)

20. Definition: Path in a graph is a route along the edges that starts at a vertex and ends at a vertex.B
GCDH is a PATH of length 3. F G C D E H

21. Definition: Circuit in a graph is a path that starts and ends at the same vertex.B
BCGHDCAB is a CIRCUIT. F G C D E H

22. Path vs. Circuit
Paths – Paths can start and end at any vertex using the edges given. Examples: NLB, NMRB, or if you moved from New York to London
Circuits – Paths that starts and ends at the same vertex. Examples: MRLM, LRBL, or if you took a vacation to London
Nonstop air routes 22

23. Determine if there is a path and/or a circuit for each graph.
Path ABC, No circuit
Both
Both 1) 2) G 3) A D O B L F C E M N
Both
Both, PQP circuit
Both 6) 6) 4) H H J J 5) P Q X W L L I I R S K K U V

24. Definition: A graph is connected, if for every pair of vertices there is at least one path connecting them.
Connected
Connected
Connected 1) 3) A B 2) G O D L C M F N E
Disconnected
Connected
Connected 6) 6) 4) H H J J 5) P Q X W L L I I R S K K U V

25. Definition: Valence (or degree) of a vertex in a graph is the number of edges meeting at the vertex
Valence of L = 2 O
Val (P) = 3 P
Val (O) = 6 M N

26. Definition: An odd vertex has an odd valence; an even vertex has an even valence.
Val (L) = 2 and is an even vertex L
Val (P) = 3 and V(P) = odd O
Val (O) = 6 and V(O) = even P M N

27. Determine the valence of each vertex for each graph and state whether it is odd or even.1) 2) G 3) A D O B L F C E M N
Val(A)=1 odd
Val(B)=2 even
Val(C)=1 odd
Val(D)=2 even
Val(G)=2 even
Val(E)=2 even
Val(F)=2 even
Val(L)=2 even
Val(O)=2 even
Val(N)=2 even
Val(M)=2 even

28. Determine the valence of each vertex for each graph and state whether it is odd or even.6) 4) H J 5) P Q X W L I R S K U V
Val(H)=2 even
Val(I)=2 even
Val(J)=2 even
Val(K)=2 even
Val(L)=0
Val(P)=2 even
Val(Q)=3 odd
Val(R)=2 even
Val(S)=1 odd
Val(X)=2 even
Val(W)=3 odd
Val(U)=4 even
Val(V)=3 odd

29. Classwork/Homework
p20

30. Graph Theory Yes, start at B, end at C
Can the following graph be drawn with one pen/pencil stroke and only cross each edge once? If so, how? E
A D
B C
Yes, start at B, end at C
Can it be drawn with one stroke starting and ending at the same vertex? NO

31. Graph Theory NO, this is NOT an Euler Circuit.
Euler went on to formulate a general theory which solved this particular problem and created a new branch of mathematics called graph theory.
It can be drawn with one pencil stroke (start at B, end at C), therefore this is a Euler Path. E
A D
B C
Can it be drawn with one stroke starting and ending at the same vertex?
NO, this is NOT an Euler Circuit.

32. Circuit vs. Euler Circuit (Both start and end at same vertex.)
Circuits may retrace edges or not use all the edges.
Euler circuits travel each edge once and cover all edges.

33. Definitions: Euler Path – is a path that passes through every edge of a graph once and only once. Euler circuits is a path that passes through every edge of a graph once and only once and starts and ends at the same vertex.
Find an Euler Path and circuit for these graphs 1) 2) G A D B F C E
Euler Path ABC, no Euler circuit
Euler Path and Euler circuit GFEDG

34. Determine if there is an Euler path and/or Euler Circuit.1) 2) P Q X W 3) H J L R S I U V K
Euler Path VVUUWXW, no Euler circuit
Euler Path QPQRS, no Euler circuit
Euler Path and Euler circuit HJKI

35. 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)
Euler’s Theorem:
An Euler circuit exists if the following statements are true:
All vertices have even valence.
The graph is connected.

36. Graph Theory: Is there an Euler Circuit?
Does it have even valence?
Is the graph connected?
Euler circuit exists if both are “yes.”
Yes
Yes 36

37. Graph Theory: Rules for creating Euler Circuits
Create (Find) an Euler Circuit
Pick a point to start (if none has been given to you).
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. 37

38. Beyond Euler Circuits Chinese Postman Problem
In real life, not all problems will be perfect Euler circuits.
What do you do if there are odd valences?
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. 38

39. Beyond Euler Circuits: Eulerizing a Graph
Eulerize the Graph to Solve Chinese Postman Problem
The edge CG is reused, which would make all vertices appear to have even valence.
A circuit is made by reusing the edge CG. The graph is eulerized (starts and stops at same point and covers all “edges” once — including reused ones). 39

40. Eulerizing a Graph Step 1: Identify odd vertices. 2 Odd vertices,
X and Y 40

41. Eulerizing a Graph
The key is to turn odd vertices into even vertices by adding “duplicate” edges. This process is called eulerization. 41

42. Eulerizing a Graph
Step 2: Duplicate existing edges to make the odd vertices now even.
This is not allowed — must only reuse existing edges.
Reuse existing edges between the odd vertices. 42

43. Eulerizing a Graph Find one Euler circuit.
Don’t forget to number the edges in the order traveled! 43

44. Eulerizing a Graph
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.
What is the best route for a postal worker delivering mail in Soho? 44

45. 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:
1) If the vertex is odd, add an edge by linking it to the next vertex.
2) If this next vertex becomes even, skip it (just keep “walking”).
3) If this next vertex becomes odd, (on a corner) link it to the next vertex.
4) Repeat this rule until you reach the upper left corner again. 45