The Seven Bridges of K onigsberg -...
Transcript of The Seven Bridges of K onigsberg -...
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The Seven Bridges of Konigsberg
I In 1735, the city of Konigsberg (present-day Kaliningrad)was divided into four districts by the Pregel River.1
I The four districts were connected by seven bridges.
1Source for Konigsberg maps: MacTutor History of Mathematics archive,www-history.mcs.st-and.ac.uk
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The Seven Bridges of Konigsberg
Is it possible to design a walking tour of Konigsberg in whichyou cross each of the seven bridges exactly once?
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The Seven Bridges of Konigsberg
Is it possible to design a walking tour of Konigsberg in whichyou cross each of the seven bridges exactly once?
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The Seven Bridges of Konigsberg
Is it possible to design a walking tour of Konigsberg in whichyou cross each of the seven bridges exactly once?
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The Mathematics of Networks
The mathematical models we need to solve the Konigsbergproblem is a graph.
I designing travel routes (Chapters 5, 6)
I connecting networks efficiently (Chapter 7)
I scheduling tasks (Chapter 8)
I coloring regions of maps (Mini-Excursion 2)
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Source: http://commons.wikimedia.org/wiki/File:Caffeine_3d_structure.png
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Source: upload.wikimedia.org/wikipedia/commons/2/20/AA_route_map.PNG
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Source: http://commons.wikimedia.org/wiki/Atlas_of_Denmark
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Source: http://commons.wikimedia.org/wiki/File:Phylogenetic_tree.svg
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Graphs
Things that can be modeled with graphs include
I maps
I molecules
I flow charts
I family trees
I Internet (web pages connected by links)
I Facebook/Google+ (people connected by friendship)
I Six Degrees of Kevin Bacon
I . . .
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The Konigsberg Bridge Problem
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Euler and the Konigsberg Bridge Problem
The great Swiss mathematician Leonhard Euler (1707–1783)became interested in the Konigsberg problem around 1735 andpublished a solution (“Solutio problematis ad geometriam situspertinentis”) in 1741.
Euler’s intuition: The physical map doesn’t matter.What matters mathematically is just the list of whichregions are connected by bridges.
Euler’s solution opened up an entire new branch ofmathematics, now known as graph theory.
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Euler and the Konigsberg Bridge Problem
The great Swiss mathematician Leonhard Euler (1707–1783)became interested in the Konigsberg problem around 1735 andpublished a solution (“Solutio problematis ad geometriam situspertinentis”) in 1741.
Euler’s intuition: The physical map doesn’t matter.What matters mathematically is just the list of whichregions are connected by bridges.
Euler’s solution opened up an entire new branch ofmathematics, now known as graph theory.
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Euler and the Konigsberg Bridge Problem
The great Swiss mathematician Leonhard Euler (1707–1783)became interested in the Konigsberg problem around 1735 andpublished a solution (“Solutio problematis ad geometriam situspertinentis”) in 1741.
Euler’s intuition: The physical map doesn’t matter.What matters mathematically is just the list of whichregions are connected by bridges.
Euler’s solution opened up an entire new branch ofmathematics, now known as graph theory.
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The Bridges of Konigsberg
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The Bridges of Konigsberg
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The Bridges of Konigsberg
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The Bridges of Konigsberg
E
S
N
C
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The Bridges of Konigsberg
E
S
N
C
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Euler and the Konigsberg Bridge Problem
“. . . this type of solution bears little relationship tomathematics, and I do not understand why youexpect a mathematician to produce it, rather thananyone else, for the solution is based on reasonalone, and its discovery does not depend on anymathematical principle.”
(Leonhard Euler, letter of April 1736)
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Euler and the Konigsberg Bridge Problem
“This question is so banal, but seemed to me worthyof attention in that neither geometry, nor algebra,nor even the art of counting was sufficient to solveit.”
(Leonhard Euler, letter of March 1736)
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Routing Problems
The Bridges of Konigsberg is an example of a routingproblem.
Other examples:
I Walking tour: must cross every bridge once
I Garbage collector: must visit every house once
I Airline traveler: get from Medicine Hat to Nairobi ascheaply as possible
Existence Question: Is an actual route possible?
Optimization Question: Which of all possible routes is thebest? (I.e., the shortest, cheapest, most efficient, etc.)
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Unicursal Tracings
Which of these figures can you draw without ever lifting yourpen from the page, or retracing a previous line?
For which ones can you finish with your pen at the same pointit started?
Mathematically, finding a unicursal tracing is equivalent tosolving the Konigsberg Bridge Problem!
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Unicursal Tracings
Which of these figures can you draw without ever lifting yourpen from the page, or retracing a previous line?
For which ones can you finish with your pen at the same pointit started?
Mathematically, finding a unicursal tracing is equivalent tosolving the Konigsberg Bridge Problem!
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Graphs
Definition: A graph consists of one or more vertices,attached by edges.
Frequently, we draw the vertices as points and the edges asline segments or curves.
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This Is Not A Graph In Math 105
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This Is Not A Graph In Math 105
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Graphs
Definition: A graph consists of one or more vertices,attached by edges.
Frequently, we draw the vertices as points and the edges asline segments or curves
However, it does not matter where the vertices arelocated on the page or what the edges are shaped like.
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Different Representations of Graphs
C
DB
A
E G
FH
These five figures all represent the same graph!
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Different Representations of Graphs
C
DB
A
E G
FH
H
G
F
E
B
A C
D
These five figures all represent the same graph!
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Different Representations of Graphs
C
DB
A
E G
FH
H
G
F
E
B
A C
D
H
G
F
E
B
A C
D
These five figures all represent the same graph!
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Different Representations of Graphs
C
DB
A
E G
FH
H
G
F
E
B
A C
D
H
G
F
E
B
A C
D
A C
E G
H
DB
F
These five figures all represent the same graph!
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Different Representations of Graphs
C
DB
A
E G
FH
H
G
F
E
B
A C
D
H
G
F
E
B
A C
D
A C
E G
H
DB
F
A
E G
H
F
B
CD
These five figures all represent the same graph!
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Different Representations of Graphs
C
DB
A
E G
FH
H
G
F
E
B
A C
D
H
G
F
E
B
A C
D
A C
E G
H
DB
F
A
E G
H
F
B
CD
These five figures all represent the same graph!
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Graph Terminology
B
C
DE
A
F
AB
CD
CD
BC
BB
DE
BE
AD
Vertex set:V = {A,B ,C ,D,E ,F}
Edge set:E = {AB ,BB ,BC ,
CD,CD,DE ,BE ,AD}
The order that we write the two vertices in an edge doesn’tmatter: DE and ED mean the same thing.
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Graph Terminology
B
C
DE
A
F
AB
CD
CD
BC
BB
DE
BE
AD
BA =
= DA
Vertex set:V = {A,B ,C ,D,E ,F}
Edge set:E = {AB ,BB ,BC ,
CD,CD,DE ,BE ,AD}
• The order that we write the two vertices in an edgedoesn’t matter: DE and ED mean the same thing.
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Graph Terminology
B
C
DE
A
F
AB
CD
CD
BC
BB
DE
BE
AD
loop
Vertex set:V = {A,B ,C ,D,E ,F}
Edge set:E = {AB ,BB ,BC ,
CD,CD,DE ,BE ,AD}
• An edge can attach a vertex to itself (like BB); this iscalled a loop.
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Graph Terminology
B
C
DE
A
F
AB
CD
CD
BC
BB
DE
BE
AD
multipleedges
Vertex set:V = {A,B ,C ,D,E ,F}
Edge set:E = {AB ,BB ,BC ,
CD,CD,DE ,BE ,AD}
• There can be multiple edges between the same end-points (like CD, which is a double edge).
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Graph Terminology
B
C
DE
A
F
AB
CD
CD
BC
BB
DE
BE
AD
irrelevant!
Vertex set:V = {A,B ,C ,D,E ,F}
Edge set:E = {AB ,BB ,BC ,
CD,CD,DE ,BE ,AD}
• It doesn’t matter if edges cross each other; crossingpoints do not count as vertices.
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Graph Terminology
B
C
DE
A
F
AB
CD
CD
BC
BB
DE
BE
AD
isolatedvertex
Vertex set:V = {A,B ,C ,D,E ,F}
Edge set:E = {AB ,BB ,BC ,
CD,CD,DE ,BE ,AD}
• Not every vertex has to have an edge attached to it. Avertex with no edges is called an isolated vertex.
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Graphs as Models
Graphs can be used as models for zillions of differentstructures arising in the real world.
I Facebook: vertices = people, edges = friendships
I Internet: vertices = web pages, edges = links
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Vertices = regions of Konigsberg; edges = bridges
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Vertices = atoms; edges = bonds
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Vertices = cities; edges = airplane routes
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Vertices = regions of Denmark; edges = common borders
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Vertices = states; edges = common borders
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Vertices = states; edges = common borders
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Vertices = states; edges = common borders
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Vertices = groups of species; edges = biological kinship
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Graph Terminology: Adjacency and Connectedness
I Two vertices are called adjacent if they are attacheddirectly by at least one edge.
I Two vertices are called connected if they are connectedby a sequence of edges.
I Adjacent vertices are always connected, but connectedvertices are not necessarily adjacent.
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Kansas and Colorado are adjacent
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Kansas and Oregon are connected, but not adjacent
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Kansas and Alaska are neither adjacent nor connected
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Graph Terminology: Degree
The degree of a vertex v is the number of edges attachedto v . (A loop counts as two edges.)
B
C
DE
A
F
An odd vertex is a vertex whose degree is odd.An even vertex is a vertex whose degree is even.
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Graph Terminology: Degree
The degree of a vertex v is the number of edges attachedto v . (A loop counts as two edges.)
B
C
DE
A
F
4
5
3
Degree
2
0
2
An odd vertex is a vertex whose degree is odd.An even vertex is a vertex whose degree is even.
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Graph Terminology: Degree
The degree of a vertex v is the number of edges attachedto v . (A loop counts as two edges.)
B
C
DE
A
F
4
5
3
Degree
2
0
2
An odd vertex is a vertex whose degree is odd.An even vertex is a vertex whose degree is even.
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Graph Terminology: Degree
Example: Konigsberg.Vertices = regions of cityDegree of a vertex = number of bridges that go to that region
E
S
N
C
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Graph Terminology: Degree
Example: Konigsberg.Vertices = regions of cityDegree of a vertex = number of bridges that go to that region
E
S
N
C
degree(N) = 3
degree(E) = 3
degree(S) = 3
degree(C) = 5
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Graph Terminology: Degree
Example: Map of USA.Vertices = states; degree = number of bordering states
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Graph Terminology: Degree
Example: Map of USA.Vertices = states; degree = number of bordering states
deg(AK) = 0
deg(CA) = 3
deg(MN) = 4 deg(ME) = 1
deg(KS) = 4
deg(TN)= 8
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Paths and Circuits
I In many graph models (maps, road networks, KevinBacon, . . . ), we want to travel from one vertex toanother by walking along the edges (“taking a trip”).
I Rule: A trip cannot use the same edge more than once,but it may pass through the same vertex more than once.
I A trip is called a path if its starting and ending verticesare different. It is called a circuit if the starting andending vertices are the same.
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Trip #1 (a path)
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Trip #2 (another path)
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Trip #3 (a circuit)
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A circuit without beginning or end
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Connectedness
A graph is called connected if any two vertices can be linkedby a path. Otherwise, it is disconnected.
DisconnectedConnected
Disconnected
Isolated vertices No isolated vertices
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Connectedness
A graph is called connected if any two vertices can be linkedby a path. Otherwise, it is disconnected.
DisconnectedConnected
Disconnected
Isolated vertices No isolated vertices
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Bridges
Removing a single edge from a connected graph can make it
disconnected. Such an edge is called a bridge.
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Bridges
Removing a single edge from a connected graph can make it
disconnected. Such an edge is called a bridge.
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Bridges
Removing a single edge from a connected graph can make it
disconnected. Such an edge is called a bridge.
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Bridges
Loops cannot be bridges, because removing a loop from agraph cannot make it disconnected.
deleteloop e
e
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Bridges
If two or more edges share both endpoints, then removing anyone of them cannot make the graph disconnected. Therefore,none of those edges is a bridge.
edgesA
C
D
B
multiple
delete
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Bridges
If two or more edges share both endpoints, then removing anyone of them cannot make the graph disconnected. Therefore,none of those edges is a bridge.
edgesA
C
D
B
multiple
delete