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Biconnectivity
1
Biconnectivity
SEA
PVD
MIA
SNA
ORD
FCO
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Biconnectivity
2
Outline and Reading
Definitions
Separation vertices and edges
Biconnected graph
Biconnected components
Equivalence classes
Linked edges and link components
Algorithms
Auxiliary graph
Proxy graph
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Biconnectivity
3
Separation Edges and Vertices
Definitions
Let
G
be a connected graph
A separation edge of
G
is an edge whose removal disconnects
G
A separation vertex of
G
is a vertex whose removal disconnects
G
Applications
Separation edges and vertices represent single points of failure in a
network and are critical to the operation of the network
Example
DFW, LGA and LAX are separation vertices
(DFW,LAX) is a separation edge
ORD
PVD
MIA
DFW
SFO
LAX
LGA
HNL
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Biconnectivity
4
Biconnected Graph
Equivalent definitions of a biconnected graph
G
Graph
G
has no separation edges and no separation vertices
For any two vertices
u
and
v
of
G
, there are two disjoint
simple paths between
u
and
v
(i.e., two simple paths
between
u
and
v
that
share no other vertices or edges)
For any two vertices
u
and
v
of
G
, there is a simple cycle
containing
u
and
v
Example
ORD
PVD
MIA
DFW
SFO
LAX
LGA
HNL
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Biconnectivity
5
Biconnected Components
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 Fall '08
 Staff
 Algorithms, Data Structures, Equivalence relation, Binary relation, Transitive relation, Graph connectivity

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