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SPQR Tree - PowerPoint PPT Presentation


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SPQR Tree. v. u. skeleton of S. SPQR-tree. v. Q. u. S. P. Q. R. Q. S. Q. Q. Q. Q. Q. Q. Q. Decomposition of biconected graph with respected to its triconnected components. Applications. SPQR tree represents all embeddings of a graph.

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SPQR Tree


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spqr tree1

v

u

skeleton of S

SPQR-tree

v

Q

u

S

P

Q

R

Q

S

Q

Q

Q

Q

Q

Q

Q

Decomposition of biconected graph with respected to its triconnected components.

slide3

Applications

SPQR tree represents all embeddings of a graph.

  • Online problems connected with planarity and triconnectivity.
  • To maintain the minimum spanning tree and transitive closure of a graph.
  • Orthogonal drawings of planar graphs with minimum number of bends.
  • Symmetric drawings of graphs in three dimension.
  • For cluster planarity testing.
  • Minimum segment drawings of planar graphs.
  • Upward planarity
  • Monotone drawing of graphs.
biconnected graphs
biconnected graphs
  • a cut-vertex is a vertex such that its removal produces a disconnected graph
  • a biconnected graph does not have cut-vertices
  • a separation pair of a biconnected graph is a pair of vertices whose removal produces a disconnected graph
  • a split pair is either a separation pair or a pair of adjacent vertices
split pair
Split pair

v

G1

A split pair

v

u

G2

u

A split pair

A split pair {u, v} of G

v

G

u

A pair of adjacent vertices

maximal split pair
Maximal split pair

v

v’

u

u’

A split pair

b

G

a split pair {u, v}

a

For any other split pair {u’, v’},

a, b, u, v are in the same split component

slide7

A split pair

v

b

v’

v’

a

u

u’

u’

For any other split pair {u’, v’},

a, b, u, v are in the same split component

slide8

v

A maximal split pair

with respect to

a reference edge (a, b)

u

A split pair

b

G

a split pair {u, v}

a

For any other split pair {u’, v’},

a, b, u, v are in the same split component

slide9

v’

(= u’)

A split pair

b

v

G

a split pair {u, v}

a

u

slide10

A maximal split pair

with respect to

a reference edge (a, b)

A split pair

v’

v’

b

a split pair {u, v}

v

a

u

(= u’)

u

(= u’)

For other split pair {u’, v’},

a, b, u, v are not in the same split component

spqr tree2
SPQR-tree

v

With a reference edge

u

spqr tree8
SPQR-tree

Q

S

P

Q

S

Q

Q

Q

spqr tree9
SPQR-tree

Q

S

P

Q

R

Q

S

Q

Q

Q

Q

Q

Q

Q

spqr tree10

v

u

skeleton of S

SPQR-tree

v

  • each internal node of the tree is associated with a skeleton representing its configuration
  • the graph represented by node  into its parent  is called the pertinent of 

Q

u

S

P

Q

R

Q

S

Q

Q

Q

Q

Q

Q

Q

slide20

c

s

s

e

t

t

One or more

cut vertices.

An SPQR-tree

two split components

(One of them consist of e.)

s

e

t

biconnected series-

parallel graph G

slide21

s

S

x

t

x1

x2

pertinent graph of x

An SPQR-tree

two split components

(One of them consist of e.)

c

G1

s

s

s

e

e

G2

t

t

t

Series case

biconnected series-

parallel graph G

pole

slide22

c

G1

s

e

c

e

G2

t

An SPQR-tree

c

G1

s

e

G2

t

S

x

x1

x2

slide23

c

two parallel edge

s

e

S

c

x

s

Q

x2

pertinent graph of x1

An SPQR-tree

c

s

G1

e

Trivial case

S

x

x1

x2

pole

slide24

c

S

x

P

x2

t

xa

xb

pertinent graph of x2

An SPQR-tree

three or more split components

c

c

e

e

Ga

Gb

G2

t

t

Parallel case

S

pole

x

x2

slide25

None of the case above

t

Skeleton of R node

s

R-node

slide26

Q

S

R

Q

P

R

Q

Q

S

S

R

Q

Q

Q

R

Q

Q

Q

Q

Q

Q

Q

Q

series parallel graphs
Series-parallel graphs

G1

series

connection

G1

sink

source

G2

G2

A series-parallel graphs

a single edge

source

sink

slide28

G1

parallel

connection

G1

sink

source

G2

G2

A series-parallel graphs

a single edge

source

sink

spq tree
SPQ-tree

P

S

S

S

P

S

S

An SPQ-tree of G

: an edge

: series connection

S

: parallel connection

P

An SPQ-tree

c

b

j

d

k

a

i

e

l

h

g

f

biconnected series-

parallel graph G

slide30

An SPQ-tree

P

c

S

S

S

b

j

d

P

k

c

g

e

d

h

k

l

f

a

i

e

S

S

l

h

g

f

j

b

a

i

biconnected series-

parallel graph G

An SPQ-tree of G

: an edge

: series connection

S

: parallel connection

P

slide31

An SPQ-tree

P

c

S

S

S

b

j

d

P

k

c

g

e

d

h

k

l

f

a

i

e

S

S

l

h

g

f

j

b

a

i

biconnected series-

parallel graph G

An SPQ-tree of G

: an edge

: series connection

S

: parallel connection

P

slide32

An SPQ-tree

P

c

S

S

S

b

j

d

P

k

c

g

e

d

h

k

l

f

a

i

e

S

S

l

h

g

f

j

b

a

i

biconnected series-

parallel graph G

An SPQ-tree of G

: an edge

: series connection

S

: parallel connection

P

slide33

An SPQ-tree

P

c

S

S

S

b

j

d

P

k

c

g

e

d

h

k

l

f

a

i

e

S

S

l

h

g

f

j

b

a

i

biconnected series-

parallel graph G

An SPQ-tree of G

: 0

The number

of children

: 2 or more

S

: 2(non-root) or 3(root)

P