Thinking Mathematically, 10.7 & 15.1–15.3 Excursions in Mathematics, 5.1–5.7
ORIGINS OF GRAPH THEORY
ORIGINS OF
GRAPH THEORY
Königsberg, Germany
Seven Bridges
• Salesman’s Bridge
• Green Bridge
• Slaughter Bridge
• Blacksmith’s Bridge
• Timber Bridge
• High Bridge
• Honey Bridge
Four City Areas
• Alstadt & Löbnicht
• Kneiphof Island
• Vorstadt
• Lomse Island
A
K
V
L
Five Bridges Today
• Estacada Bridge
• Timber (October) Bridge
• High Bridge
• Honey Bridge
• NEW: Emperor Bridge
Kaliningrad, Russia
Origins of Graph Theory
Königsberg, Germany in early 1700s had seven bridges Residents of city took walks on Sunday, developed a game
“Can a person cross all bridges only once and return to start?” 1735 Leonhard Euler was given the
problem and determined impossible
Leonhard Euler
Swiss mathematician
St. Petersburg Academy of Sciences Started “geometris situs” which became graph theory
DEFINITIONS
vertex – point (NOTE: plural is vertices)
edge – line segment or curve that starts at one vertex
and ends at another vertex
odd vertex – vertex with an odd number of attached edges
even vertex – vertex with an even number of attached edges
traversable graph – a graph for which all edges can be
traced once and only once
EXAMPLE
Consider the graph. Is it traversable?
That is, can each line of the graph be traced once and
only once without lifting your pencil?
RULES
Euler’s Rules of Traversability
1. A graph with all even vertices is traversable.
One can start at any vertex and end at same vertex.
2. A graph with two odd vertices is traversable.
One must start at one odd vertex and
end at the other odd vertex.
BACK TO EXAMPLE
Consider the graph. Is it traversable?
That is, can each line of the graph be traced once and
only once without lifting your pencil?
Yes, the graph is traversable!
Odd
(3)
Odd
(3)
Even
(2)
There are two, and only two, odd
vertices so it was traversable.
EXAMPLE
Consider the graph. Is it traversable?
Yes, the graph is traversable!
Even
(2)
Even
(2)
Even
(2)
Even
(2)
Even
(4)
Recall Euler’s Rules of Traversability:
1. A graph with all even vertices is traversable.
Let’s model the Seven Bridges of Königsberg as a graph.
EXAMPLE
Consider the graph. Is it traversable?
No, the graph is not traversable!
Odd
(3)
Odd
(3)
Odd
(3)
Odd
(5)
Recall Euler’s Rules of Traversability:
3. A graph with more than two odd vertices is NOT traversable.
The Bridges of Königsberg problem was not a
EXAMPLE
Consider the graph. Is it traversable? If so, find the path.
EXAMPLE
Consider the graph. Is it traversable? If so, find the path.
EXAMPLE
Consider the graph. Is it traversable? If so, find the path.
GRAPHS,
PATHS, & CIRCUITS
DEFINITIONS
vertex – point (NOTE: plural is vertices)
edge – line segment or curve that starts at one vertex
and ends at another vertex
loop – curve that starts and ends at the same vertex
point must be marked with a dot; denoted with letter
denoted with two vertex letters
EXAMPLE
Consider the graph below.
Identify the vertices, edges, and loops.
How many vertices?
4
How many edges?
4
This is NOT a vertex!
edge AD edge DB
edge BC edge CA
How many loops?
1
loop CC
Not every point where two edges cross is a vertex!
If the point is not marked with a dot, it is not a vertex (think of it as one line is above the other).
What is C’s degree?
4
DEFINITION
EXAMPLE
Which are equivalent graphs?
4
1
2
3
EXAMPLE
EXAMPLE
Consider the map of the Rocky Mountain states.
Model the states that share a common border.
EXAMPLE
Can you cross the border of each state once and only
once (that is, is the graph traversable)?
Odd(3)
Odd(3)
Even (2)
Even (4)
Even (2)
Recall Euler’s Rules of Traversability:
2. A graph with two odd vertices is traversable.
One must start at one odd vertex and end at the other odd vertex.
EXAMPLE
Find the path that traverses the graph.
Find the path on the map that crosses each border
exactly one time.
Odd(3)
Odd(3)
Even (2)
Even (4)
Even (2)
EXAMPLE
EXAMPLE
Can you cross the border of each state once and only
once (that is, is the graph traversable)?
Even (2)
Even (2)
Even (2)
Even (4)
Odd (3)
Odd (1)
Yes, it is traversable.
Recall Euler’s Rules of Traversability:
2. A graph with two odd vertices is traversable.
EXAMPLE
Find the path that traverses the graph.
Find the path on the map that crosses each border
exactly one time.
Even (2)
Even (2)
Even (2)
Even (4)
EXAMPLE
Consider the floor plan of a four-room house.
(Consider all the outdoors as one location.)
Model the ways to walk from one area to another.
EXAMPLE
Consider the floor plan of a four-room house.
(Consider all the outdoors as one location.)
EXAMPLE
Consider the bridges of Madison County.
EXAMPLE
Consider this neighborhood. A mail carrier delivers the
mail by walking to each house. If houses are on both
sides of the street, he walks it twice. Model this.
Houses are purple squares.
Roads are pink lines.
DEFINITIONS
adjacent vertices – vertices that are connected directly and
thus share at least one edge
D is adjacent to C
C is adjacent to A & D
B is adjacent to A & E
A is adjacent to B, C, E, & F
E is adjacent to A & B
F is adjacent to A
F is NOT adjacent to E
DEFINITIONS
path – a sequence of adjacent vertices and the edges
connecting them, denoted by list of vertices in order
path: A,B,F,G,H,M
not a path:
A,B,F,G,H,M,H,D
NOTE: A path can use a vertex more than once but can use each edge only once. However, the entire graph
does not have to be touched to have a path.
DEFINITIONS
circuit – a path that begins and ends at the same vertex
circuit: A,B,F,G,L,K,J,E,A
NOTE: Every circuit is a path.
DEFINITIONS
connected graph – a graph in which there is at least one
path connecting any two vertices
disconnected graph – a graph in which there is no path
connecting any two vertices
disconnected graph
DEFINITIONS
bridge – an edge that, if removed, would make a connected
graph into a disconnected graph
edge BC is a bridge
EXAMPLE
Find the bridge(s) in the graph.
edge AB is a bridge
HOMEWORK
From the Cow book