Download Definition A graph G consists of a set of vertices V and a set of edges

Definition
A graph G consists of a set of vertices V and a set of
edges E, where an edge is an unordered pair of vertices.
Examples
1. Computer Networks
V = {computers}
E = {{A, B} | computers A and B are networked}
2. Social Networks
V = {Alice, Bob, Chris, Daniel, ...}
E = {{A, B} | A and B know each other}
3. V = {v1, v2, v3, v4, v5} and E = {e1, e2, e3, e4, e5, e6}.
v1
e4
v5
v2
e2
e1
e3
v4
e5
v3
e6
Directed Graphs
A directed graph or digraph G consists of a set of
vertices V and a set of directed edges E, where an edge
is an ordered pair of vertices.
Examples
1. Functional Digraphs. Let f be a function from X to
Y.
V = X ∪Y
E = {(x, y) ∈ X × Y | f (x) = y}
10
•
•1
•
6
•13
•
3
•
12
•
5
14
•
•
9
2•
•7
16
•
•
11
•15
•
8
•
4
2. Game Theory
V = {possible game states}
E = {(A, B) | state B can be achieved from state A}
Problems
• Eulerian Path - Can we draw a given graph without
lifting our pencil from the paper and without repeating edges?
• Hamiltonian Path - Can we visit each vertex once
and only once using only the edges of a graph?
• Isomorphism Problem - When are two graphs the
“same”?
a
a
b
e
e
b
f
c
d
h
g
d
g
c
f
e
• Shortest Route - Find the shortest path between
two vertices
B
A
• Assignment Problem - How do you assign tasks
to individuals so that each person can do the task assigned to them? How can you minimize time required
to complete all tasks?
Alice
10
Bill
8
7
9
Chris
7
9
6
Job 1
8
8
Job 2
Job 3
Job 4
• Network Flow Problems - Determine the least expensive way to transport product from surplus sites
to demand sites.
4
a
3
4
10 e
b
1
2
1
4
d
-8
2
c
6
-12
• Planar Graphs - Can a given graph be drawn without any pair of edges crossing? Applications to chip
design.
• Chromatic Numbers - What is the minimum number of distinct colors needed to label the vertices of
a graph so that no two adjacent vertices are colored
the same? Four Color Theorem.
• Traveling Salesman Problem - In what order should
you visit cities c1, c2, . . . , cn so that you minimize the
distance traveled?
• Ramsey Numbers - What is the minimum number
of people required to guarantee that there are m people who all know each other or n none of whom know
each other
R(m, 1)
R(2, 2)
R(2, n)
R(3, 3)
R(4, 4)
R(5, 5)
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=
=
=
=
=