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Chapter 9: Graphs

Applications of

Depth-First Search

Mark Allen Weiss: Data Structures and Algorithm Analysis in Java

Lydia Sinapova, Simpson College

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Graph Connectivity

Connectivity

Biconnectivity

Articulation Points and BridgesConnectivity in Directed Graphs

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Connectivity

Definition:

An undirected graph is said to be connected

if for any pair of nodes of the graph, the twonodes are reachable from one another (i.e. there

is a path between them).

If starting from any vertex we can visit all other

vertices, then the graph is connected

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Biconnectivity

A graph is biconnected, if there are no

vertices whose removal will disconnect the

graph.

A B

C D

E

A B

CD

E

biconnected not biconnected

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Articulation Points

Definition: A vertex whose removal

makes the graph disconnected is called

an articulation point orcut-vertex

A B

CD

E

Cis an articulation point

We can compute articulation points

using depth-first search and a

special numbering of the vertices in

the order of their visiting.

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Bridges

Definition:

An edge in a graph is called a bridge, if its

removal disconnects the graph.

Any edge in a graph, that does not lie on a cycle, is a bridge.

Obviously, a bridge has at least one articulation point at its end,

however an articulation point is not necessarily linked in a bridge.

A B

CD

E

(C,D) and (E,D) are bridges

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Example 1

A B

C

EF

D

C is an articulation point, there are no bridges

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Example 2

A B

C

EF

D

Cis an articulation point, CB is a bridge

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Example 3

A

BC

E

F

G

D

B and C are articulation points, BC is a bridge

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Example 4

AB C D

E

B and C are articulation points. All edges are bridges

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Example 5A

B

C

D E

FG

Biconnected graph - no articulation points and no bridges

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Connectivity in DirectedGraphs (I)

Definition: A directed graph is said to be

strongly connectedif for any pair of nodes there is a path

from each one to the other

A B

C D

E

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Connectivity in DirectedGraphs (II)

Definition: A directed graph is said to be

unilaterally connected if for any pair ofnodes at least one of the nodes is reachable

from the other

A B

C D

E

Each strongly connected

graph is also unilaterally

connected.

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Connectivity in DirectedGraphs (III)

Definition: A directed graph is said to be

weakly connected if the underlying

undirected graph is connected

A B

C

D

E

There is no path

between B and D

Each unilaterally

connected graph is also

weakly connected