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Lecture Notes
CMSC 251
Observe that directed graphs and undirected graphs are different (but similar) objects mathematically.
Certain notions (such as path) are deﬁned for both, but other notions (such as connectivity) are only
deﬁned for one.
In a digraph, the number of edges coming out of a vertex is called the
outdegree
of that vertex, and
the number of edges coming in is called the
indegree
. In an undirected graph we just talk about the
degree
of a vertex, as the number of edges which are
incident
on this vertex. By the
degree
of a graph,
we usually mean the maximum degree of its vertices.
In a directed graph, each edge contributes 1 to the indegree of a vertex and contributes one to the
outdegree of each vertex, and thus we have
Observation:
For a digraph
G
=(
V,E
)
,
X
v
∈
V
indeg
(
v
)=
X
v
∈
V
outdeg
(
v
)=

E

.
(

E

means the cardinality of the set
E
, i.e. the number of edges).
In an undirected graph each edge contributes once to the outdegree of two different edges and thus we
have
Observation:
For an undirected graph
G
=(
V,E
)
,
X
v
∈
V
deg
(
v
)=2

E

.
Lecture 21: More on Graphs
(Tuesday, April 14, 1998)
Read: Sections 5.4, 5.5.
Graphs:
Last time we introduced the notion of a graph (undirected) and a digraph (directed). We deﬁned
vertices, edges, and the notion of degrees of vertices. Today we continue this discussion. Recall that
graphs and digraphs both consist of two objects, a set of vertices and a set of edges. For graphs edges
are undirected and for graphs they are directed.
Paths and Cycles:
Let’s concentrate on directed graphs for the moment. A
path
in a directed graph is a
sequence of vertices
h
v
0
,v
1
,...,v
k
i
such that
(
v
i

1
,v
i
)
is an edge for
i
=1
,
2
,...,k
. The
length
of
the path is the number of edges,
k
. We say that
w
is
reachable
from
u
if there is a path from
u
to
w
.
Note that every vertex is reachable from itself by a path that uses zero edges. A path is
simple
if all
vertices (except possibly the ﬁrst and last) are distinct.
A
cycle
in a digraph is a path containing at least one edge and for which
v
0
=
v
k
. A cycle is
simple
if,
in addition,
v
1
,...,v
k
are distinct. (Note: A selfloop counts as a simple cycle of length 1).
In undirected graphs we deﬁne path and cycle exactly the same, but for a
simple cycle
we add the
requirement that the cycle visit at least three distinct vertices. This is to rule out the degenerate cycle
h
u, w, u
i
, which simply jumps back and forth along a single edge.
There are two interesting classes cycles. A
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This note was uploaded on 01/13/2012 for the course CMSC 351 taught by Professor Staff during the Fall '11 term at University of Louisville.
 Fall '11
 Staff

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