Discrete Mathematics
(CSC 1204)
9.4 Connectivity
9.5 Euler and Hamilton Paths
1

Paths
2

Paths
n
.
3

Paths in Directed Graph
•
Same as in undirected graphs, but the
path must go
in the direction of the arrows
.
4

Connectedness in Undirected Graphs
•
An
undirected graph
is called
connected
if there is a
path between
every pair of distinct vertices
of the
graph
5

6
.

Example 5 (p.624)
7
Figure 2:
The Graphs G
1
and G
2
The graph
G
l
in Figure 2 is
connected
, because for every pair
of distinct vertices there is a path between them.
However, the graph
G
2
in Figure 2 is
not connected
. For
instance, there is no path in G2 between vertices a and d.

8
•Question: Which of the following graphs are connected?
Connectivity
1
2
3
4

9

Connected Components
•
A
connected component
of a graph G is a connected
subgraph of G that is not proper subgraph of another
connected subgraph of G.
•
A
connected component
of a graph G is a maximal
connected subgraph of G.
–
A graph G that is not connected has two or more
connected components that are disjoint and have
G as their union
10

11
Connected Components
•
Definition
:
A
connected component
in a graph
G
is a set of
vertices such that all vertices in the set are connected to
each other and every possible connected vertex is included.
•
Question
: What are the connected components of the
following graph?
6
2
4
3
5
1
7
8

Connected Components

Connectedness in Directed Graphs