CS 70
Discrete Mathematics for CS
Fall 2006
Papadimitriou & Vazirani
Lecture 16
Graphs
Formulating a simple, precise specification of a computational problem is often a prerequisite to writing
a computer program for solving the problem.
Many computational problems are best stated in terms of
graphs: a directed graph
G
(
V
,
E
)
consists of a finite set of vertices
V
and a set of (directed) edges or arcs
E
. An edge is an ordered pair of vertices
(
v
,
w
)
and is usually indicated by drawing a line between
v
and
w
,
with an arrow pointing towards
w
. Undirected graphs may be regarded as special kinds of directed graphs,
such that
(
u
,
v
)
∈
E
↔
(
v
,
u
)
∈
E
. Thus, since the directions of the edges are unimportant, an undirected
graph
G
(
V
,
E
)
consists of a finite set of vertices
V
, and a set of edges
E
, each of which is an unordered pair
of vertices
{
u
,
v
}
. As we have defined them, graphs are allowed to have selfloops; i.e. edges of the form
(
u
,
u
)
that go from a vertex to itself. Sometimes it is more convenient to disallow such selfloops.
Graphs are useful for modeling a diverse number of situations. For example, the vertices of a graph might
represent cities, and edges might represent highways that connect them. In this case, the edges would be
undirected:
In the above picture,
V
=
{
SF
,
LA
,
NY
,...
}
, and
E
=
{{
SF
,
LA
}
,
{
SF
,
NY
}
,...
}
.
Alternatively, an edge
might represent a flight from one city to another, and now edges would be directed (a certain airline might
have a nonstop flight from SFO to LAX, but no nonstop flight back from LAX to SFO).
Graphs can also be used to represent more abstract relationships. For example, the vertices of a graph might
represent tasks, and the edges might represent precedence constraints: a directed edge from
u
to
v
says that
task
u
must be completed before
v
can be started. An important problem in this context is scheduling: in
what order should the tasks be scheduled so that all the precedence constraints are satisfied.
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 Fall '08
 HAULLGREN
 Graph Theory, Leonhard Euler, De Bruijn graph, De Bruijn sequence, Eulerian, eulerian tour

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