Interval, circle graphs and circle graph recognition using split decomposition

Interval, circle graphs and circle graph recognition using split decomposition

Presented by Steven CorreiaKent state university

Nov-18-2011

Email: [email protected]

1Based on :[1][4]

Background and motivation

• Can highly connectivity of protein in cells be found easily? • How should I manage routing of wires in VLSI ?• How should I do memory management in small devices

like PDA and cell phones?

2http://pixiedusthealing.blogspot.com/2011/03/earth-hour-internal-versus-external.html

http://pixiedusthealing.blogspot.com/2011/03/earth-hour-internal-versus-external.html

Background and motivation

• Can highly connectivity of protein in cells be found easily? • How should I manage routing of wires in VLSI ?• How should I do memory management in small devices

like PDA and cell phones?

Solution

I think intersection model can be used.

3http://pixiedusthealing.blogspot.com/2011/03/earth-hour-internal-versus-external.html

http://pixiedusthealing.blogspot.com/2011/03/earth-hour-internal-versus-external.html

4

Outline

DefinitionInterval graph

Circle graph

Forbidden interval and circle graphs

Related work

Modular decomposition

Detection of circle graph – Split decomposition

Maximum click in graph

Related Graphs

ApplicationReduce the complexity of many problems

Memory management

VLSI design

Max clique applications

Conclusion

References

Image taken from:[4]

Interval graph – intersection model

• An intersection graph of a multi-set of intervals on the real line. A vertex corresponds to an interval where as an edge between every pair of vertices corresponding to intervals that overlaps.

• Let {I1, I2, ..., In} P(R) be a set of intervals. The corresponding ⊂interval graph is G = (V, E), where V = {I1, I2, ..., In}, and {Iα, Iβ} E if and only if Iα ∩ Iβ ≠ .∈ ∅

5based on:[4]

Circle graph – intersection model

• An intersection graph of set of chords of circle

• This is an undirected graph whose vertices can be associated with chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other

• The chromatic number of a circle graph is the minimum number of colors that can be used to color its chords so no two chords intersects that has the same color

6based on:[4]

7

Outline


Circle graph


Related work




Related Graphs


Memory management

VLSI design


Conclusion

References


• Forbidden interval graphs (e.g.: asteroid triple graph)

• Forbidden circle graphs

Forbidden graphs

8Images taken from:[1][4]

9

Outline


Circle graph


Related work




Related Graphs


Memory management

VLSI design


Conclusion

References

Related work

• Booth, Lueker & Habib proved determining if a given graph G = (V, E) is an interval graph can be done in O(|V|+|E|) time by seeking an ordering of the maximal cliques of G.

• Many researchers proved different techniques to for recognition of circle graph. • Earliest polynomial time algorithm described by Bouchet

(1987) which takes time.• Gabor, Hsu and Supowit proposed time algorithm• Jeremy Spinrad proposed algorithm• Best known algorithm by Christophe Paul University

Montpellier II, France March 25, 2009 in quasi-linear time*

*An algorithm is said to run in quasi-linear time if T(n) = for any constant k

10

11

Outline


Circle graph


Related work




Related Graphs


Memory management

VLSI design


Conclusion

References



How should I decompose a graph?

12based on:[2]


How should I decompose a graph?

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I’ll find its Modules!

based on:[2]


Module. AKA:

• Autonomous set• Closed set• Stable set• Clump• Committee• Externally Related Set• Interval• Non simplifiable Sub-networks• Partite Set

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Modular Decomposition : A Module is a set of vertices that are indistinguishable from outside

15based on:[2]

B

C

E

f

g

hjk



16based on:[2]

B

C

E

f

g

hjk



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Not a module!

based on:[2]

B

C

E

f

g

h

k j

Another way to view a module

• Biclique : biclique(complete bipartite graph) is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.

• A complete bipartite graph, G := (V1 + V2, E), is a bipartite graph such that for any two vertices, v1 V∈ 1 and v2 V∈ 2, v1v2 is an edge in G. The complete bipartite graph with partitions of size |V1|=m and |V2|=n, is denoted Km,n.

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Module

based on:[2]

Between two modules

• Another way to see this:

Module No module can contain vertices from both sets!

19based on:[2]


Separating a Module:

20based on:[2]


Separating a Module: Any two disjoint modules form either a biclique or are disconnected

21based on:[2]

The Quotient graph

• When placing a vertex instead of each maximal module, we get the quotient graph

• Modular decomposition is also called Substitution Decomposition, or S-decomposition

Quotient graph

22based on:[2]

The Quotient graph

The quotient graph can again have modules! Thus:

Recursive Structure!

23based on:[2]

The degenerate/prime tree

Modules:• {a,b,c},{d},{e,f,g},{a,b,c,d,e,f,g}

• A node corresponds to the set of all its leaves

• All modules are all: node OR: union of children of D-node

24based on:[2]

P_4 has no nontrivial modules!

C

E

DD

DB

C

E

f

g

hjk

D D

f gD

h

jk

The degenerate/prime tree cont..

For D nodes the quotient graph is without edges or is a clique!

25based on:[2]

26

Outline


Circle graph


Related work




Related Graphs


Memory management

VLSI design


Conclusion

References


• G is a circle graph if and only if every remaining subgraph is a circle graph. • Every remaining subgraph should be indecomposable i.e. Prime graph• If V is partitioned into V1,V2, 2 ≤ |V1| ≤ n-2 Let V1io ={} V2io ={y} So V1 and V2 is called a split of G if every vertex in V1io is adjacent to every vertex in V2io.V1={1,2,3} V2={4,5,6,7} is split of G where V1io={2,3} & V2io={4,5}

The split decomposition

27based on:[1]

• After decomposition we find the exact location on circle arc where the chord could be placed.

• The idea of the algorithm is one can prove in O(n2) time that a graph is indecomposable (prime graph) with respect to split decomposition.

• Algorithm produce circular ordering of vertices in that time and check if that circular ordering correctly represents G.

The split decomposition cont..

• We get a quotient graph such that for each pair of parts, the edges that run between them form a biclique

28based on:[2]

Parts are circle => graph is circle

A B C D

v

u

29based on:[2]

v u

A B C D

v v

uu


30based on:[2]

v u

A B C D

v v

uuu

v


31based on:[2]

v u

A B C D

v v

uu


32based on:[2]

v u

33

Outline


Circle graph


Related work




Related Graphs


Memory management

VLSI design


Conclusion

References


• Interval model can be constructed • Clique corresponds to pair-wise intersection of intervals in intersection graph.• Finding maximum clique in original graph can be done by finding maximum

intersecting intervals in intersection model

Maximum clique

34based on:[9]

35

Outline


Circle graph


Related work




Related Graphs


Memory management

VLSI design


Conclusion

References


• Helly circle graph : A graph G is a Helly circle graph if G is a circle graph and there exists a model of G by chords such that every three pairwise intersecting chords intersect at the same point. No diamond should be present.

• Unit circle graphs : a graph G is a unit circle graph if there is a model L for G such that all the chords are of the same length

• Proper circular-arc graphs: A proper circular arc graph is a circular arc graph that has an intersection model in which no arc properly contains another. They are subclass of circle graphs. The representation in arcs can be trivially transformed in the model in chords.

Related graphs

36based on:[1][10]

37

Outline


Circle graph


Related work




Related Graphs


Memory management

VLSI design


Conclusion

References


• Some NP-Hard problems easily solved on Circle Graphs: Independent Set solvable using O(n2) dynamic programming

• Many problem that are NP-complete on general graph have polynomial solution when restricted to circle graph– Treewidth of a circle graph can be determined , in O(n3) time and thus

an optimal tree decomposition constructed in polynomial time– Chordal graph can be found in O(n3) time– Maximum clique of a circle graph can be found in O(nlog2 n) time

• Circular-arc graph can help to utilize storage in small digital devices.

Applications

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• The network of protein interactions- proteins as nodes and protein interactions as undirected edges.

• Aim our analysis was to identify highly connected sub graphs (clusters) that have more interactions within themselves and fewer with the rest of the graph

• Cliques indicate tightly interacting protein network modules

• Used to reveal cellular organization and structure and understanding of cellular modularity

Applications dependent on maximum clique

39Image taken from:[11]

• Wire routing in VLSI design.• In our case routing area is

rectangle.• The perimeter of rectangle

represents terminals.• Goals of wire routing step is to

ensure that different nets stay electrically disconnected.

• If there is crossing then the intersecting part must be laid out in different conducting layer.

• Predict routing complexity and layer design

Applications on VLSI design

40Image taken from:[5]

41

Outline


Circle graph


Related work




Related Graphs


Memory management

VLSI design


Conclusion

References


• A circle graph is an intersection graph of a set of chords of a circle• Splitting the graphs in subgraphs solve reduce the complexity of a problem

and makes the running time faster• Many hard problems can be solved within polynomial time by using circle

graph intersection model• Wire routing design• Find strong bonding in cell structure• Helps to efficiently manage memory storage

Conclusion

42

Thank you

43

Outline


Circle graph


Related work




Related Graphs


Memory management

VLSI design


Conclusion

References


• [1] Spinrad, Jeremy (1994), "Recognition of circle graphs", Journal of Algorithms 16 (2): 264–282

• [2] Graph Decompositions: Modular Decomposition, Split Decomposition, and others Presentation primarily influenced by papers of McConnell, Spinrad and Hsu

• [3] Algorithmic graph theory – Martin Charles Golumbic(2nd edition 2004)• [4] http://en.wikipedia.org• [5] http://www.rulabinsky.com/cavd/text/chap04-3.html• [6] Recognizing Circle Graphs in Polynomial Time CSABA P. GABOR AND KENNETH J.

SUPOWIT Princeton University. Princeton, New Jersey AND WEN-LIAN HSU Northwestern University, Evanston, Illinois

• [7] http://www.rulabinsky.com/cavd/text/chap04-3.html• [8] http://mathworld.wolfram.com/DiamondGraph.html• [9] http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.3619v16.pdf• [10] Some new results on circle graphs, Guillermo Duran• [11 ]http://www.pnas.org/content/100/21/12123.full

References

44

http://en.wikipedia.org/wiki/Circle_graph





http://www.rulabinsky.com/cavd/text/chap04-3.html





http://mathworld.wolfram.com/DiamondGraph.html



http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.3619v16.pdf

http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.3619v16.pdf

http://www.pnas.org/content/100/21/12123.full

http://www.pnas.org/content/100/21/12123.full

Documents

Interval, circle graphs and circle graph recognition using split decomposition