Graph Decompositions: Modular Decomposition, Split Decomposition, and others

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Graph Decompositions Modular Decomposition,
Split Decomposition, and others

• Or How to Lefarek LeGraphim et Hatzura.
• Presentation primarily influenced by papers of
  McConnell, Spinrad and Hsu

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Lecture Objectives

• See what graph decompositions look like
• See what we can do with them
• Get a feeling on how problems can be solved with
  them
• See many kool graph classes (Circle Graphs
  roolez!!11!!1!1)

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Heres a graph for ya, Joe!
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Heres a graph for ya, Joe!
Joe scratches his head. What shall I do, what
shall I do. Its so big! Im going to be working
all night long!
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Heres a graph for ya, Joe!
Maybe Ill Divide it into connected components?
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Heres a graph for ya, Joe!
No! Ill partition it to blocks and
bridges! 34-connected-components? 76-edge-connecte
d-components?
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Heres a graph for ya, Joe!
No, I know
Ill find its Modules!
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Modular Decomposition

• A Module is a set of vertices that are
  indistinguishable from outside.

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Modular Decomposition

• A Module is a set of vertices that are
  indistinguishable from outside.

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Modular Decomposition

• A Module is a set of vertices that are
  indistinguishable from outside.

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Modular Decomposition

• A Module is a set of vertices that are
  indistinguishable from outside.

NOT A MODULE!
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Modular Decomposition

• Module. AKA
• Autonomous set
• Closed set
• Stable set
• Clump
• Committee
• Externally Related Set
• Interval
• Nonsimplifiable Subnetworks
• Partitive Set

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Another way to view a module
MODULE
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A Peek to Split Decomposition
SPLIT
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Modular Decomposition

• Operation
• Seperating a Module

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Modular Decomposition

• Seperating a Module

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Partitioning V into Modules

• V is always a module
• so are the singletons

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Partitioning V into Modules

• V is always a module
• so are the singletons
• Look at a partition into non-trivial modules

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Between two modules

• Any two disjoint modules form either a biclique
  or are disconnected

biclique
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Between two modules

• Another way to see this

No module can contain vertices from both sets!
MODULE
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The Quotient Graph

• Any two disjoint modules form either a biclique
  or are disconnected

QUOTIENT GRAPH
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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

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The Quotient Graph

• The quotient graph can again have modules! Thus
• Recursive Structure!

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The Degenerate/Prime Tree

• A node corresponds to the set of all its leaves
• All modules are all
• node
• OR
• union of children of D-node

• Modules
• a,b,c,d,e,f,g
• a,b,c,e,f,g,a,b,c,d,e,f,g
• a,b,b,c,a,c,a,b,c,d,a,b,c,e,f,g,d,e,f,g
  

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Modular Decomposition
D
X
Y
X
Y
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Modular Decomposition
A
C
D
B
D
D
A
B
C
E
E
D
X
Y
X
Y
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Modular Decomposition
A
C
f
B
D
h
j
D
k
D
g
A
B
C
E
E
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Modular Decomposition
A
C
f
B
D
h
j
D
k
D
g
D
D
C
E
E
D
g
h
f
j
k
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Modular Decomposition
A
For D nodes the quotient graph is without edges
or is a clique!
C
f
B
D
h
j
D
k
D
g
D
D
C
E
E
D
g
h
f
j
k
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Modular Decompositions with prime nodes
P_4
P_4 has no nontrivial modules!
prime node!
The Module Tree
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Relation to graph classes

• The cographs are the graphs with no prime nodes.
  Equivalently, these are the graphs without an
  induced P_4

For split decomp,, the distance-hereditary graphs
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Relation to graph classes

• The cographs are the graphs with no prime nodes.
  Equivalently, these are the graphs without an
  induced P_4
• A permutation graph has a single representation
  iff it is prime

For split decomp,, the distance-hereditary graphs
For split decomp, the circle graphs
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Relation to graph classes

• The cographs are the graphs with no prime nodes.
  Equivalently, these are the graphs without an
  induced P_4
• A permutation graph has a single representation
  iff it is prime
• A graph is prime iff ???

For split decomp,, the distance-hereditary graphs
For split decomp, the circle graphs
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Connections to partitive family theory

• The Modules is a family of sets over V. ??, all
  singletons, and V, are modules. Let A, B be
  modules. Then
• A?B is a module
• If A?B? then A?B is a module
• If B?A then A\B is a module

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Connections to partitive family theory

• From these properties we get that given a module
  M, the smallest module containing M is uniquely
  determined
• A module is called strong if it does not overlap
  any module
• The nodes of the tree are the strong modules
• These fact are why everything works. This is the
  decomposition frame

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Computing the Modular Decomposition Tree

• Especially nice linear algorithm of Dahlhaus,
  Gustedt and McConnell, that uses
  partially-complemented representations of graphs,
  and algorithms on them
• Exercise Compute this in poly time using very
  simple methods.

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When we get a problem

• Two approaches
• Look at a simple decomposition

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When we get a problem

• Two approaches
• Look at a simple decomposition
• Look at a partition to modules, and the quotient
  graph

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When we get a problem

• Two approaches
• Look at a simple decomposition
• Look at a partition to modules, and the quotient
  graph

Then you generalize to the tree
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• Some uses of The Modular Decomposition
• Finding Transitive Orientation of a graph
• Recognizing permutation graphs, isomorphism of
  permutation graphs, and of other graph classes
• Solving NP-Hard problems on well-decomposing
  graphs

combinatorial
algorithmic
Solving NP-Hard Problems
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How to solve NP-Hard Problems

• On Graphs with good decompositions we can solve
  some NP-Hard problems easily
• Example Minimum Weighted independent set.

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Solving MWIS on the graph by solving MWIS on the
Quotient
Remember Disjoint modules are unconnected or
form bicliques!!!
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Solving MWIS - Outline

• Decompose.

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Solving MWIS - Outline

• Decompose.
• Solve recursively on each part.

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Solving MWIS - Outline

• Decompose.
• Solve recursively on each part.
• Solve on the quotient using the new weights.

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Solving MWIS - Outline

• Decompose.
• Solve recursively on each part.
• Solve on the quotient using the new weights.
• That is your solution.

Using the decomp tree we can do this cleanly
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Solving MWIS bottom-up

• Compute the Decomposition Tree (linear time!)

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Solving MWIS bottom-up

• Compute the Decomposition Tree (linear time!)
• Compute bottom-up, each time taking the quotient
  at the node and computing with it

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Solving MWIS bottom-up

• Compute the Decomposition Tree (linear time!)
• Compute bottom-up, each time taking the quotient
  at the node and computing with it
• For degenerate nodes the quotient is complete or
  empty, so this is trivial.

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Solving MWIS bottom-up

• Compute the Decomposition Tree (linear time!)
• Compute bottom-up, each time taking the quotient
  at the node and computing with it
• For degenerate nodes the quotient is complete or
  empty, so this is trivial.
• Complexity sum of costs of computing MWIS for
  the quotient graphs at prime nodes!

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Other easy NP-Hard problems

• COLORING
• MAX-CUT
• CLIQUE
• Many more
• Condition We need to be able to compute the
  solution for the graph by (Exponential)
  computations on its quotient

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Combinatorial Uses of Modular Decomposition

• Many for recognition of graph classes. Natural
  for permutation graphs
• Any others?

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Circle Graphs
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Circle Graphs
Circle Graphs Intersection Graphs of Chords on
a circle
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Circle Graphs
Circle Graphs Intersection Graphs of Chords on
a circle
d
c
f
e
b
a
b a f c b d c e f d a e
Overlap Graphs Overlap Graphs of Intervals
Circle Graphs
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Circle Graphs

• Important and nice graph class
• Some NP-Hard problems easily solved on Circle
  Graphs Independent Set solvable using O(n2)
  dynamic programming

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The Split Decomposition
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Split Decomposition
SPLIT
such that each side is at least 2 vertices to
be non-trivial
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Split Decomposition
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Split Decomposition

• Cunningham introduced the split decomposition
  (for directed graphs)
• He called it join decomposition

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Split Decomposition

• Cunningham introduced the split decomposition
  (for directed graphs)
• He called it join decomposition
• He proved unique decomposition (up to some highly
  decomposable graphs)
• Split Decomp is finer than the decomposition to
  2-connected components (via cut-vertices)

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Split Decomposition
We get a quotient graph s.t. For each pair of
parts, the edges that run between them form a
biclique
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Split Decomposition

• Is a generalization of modular decomp.
• In comparison to Modular Decomp.
• symmetric
• Finer
• Less structure less to say (and dirtier)
• Harder to compute?

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Split Decomposition

• Can be used to solve some NP-Hard problems
  efficiently
• Structures are dirtier than those of Modular
  Decomposition.

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MWIS with the split decomp

• Less structure, more thinking!

A
B
C
D
Solving this gives the solution
D
C
D
B
A
w_vsol_CD-sol_D
Sol_CD
sol_D
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When we get a problem

• Two approaches
• Look at a simple decomposition
• Look at a partition to modules, and the quotient
  graph

Then you generalize to the tree
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MWIS on the graph using the quotient
Work is double-exponential in the size of the
quotient!
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• Some uses of the split decomposition
• Circle Graph Recognition, in O(n2)
• Hsu gives a connection between circular-arc
  graphs and circle graphs, and gets circular-arc
  recognition, and circular-arc and circle-graph
  isomorphism algorithms, all in O(nm)

Circle Graph Recognition, some highlights
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Split Decomposition and Recognizing Circle Graphs

• All known algorithms for recognizing circle
  graphs use the split decomposition
• They use the following property A graph is a
  circle graph iff all parts of its split
  decomposition are circle graphs

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Split Decomposition and Recognizing Circle Graphs

• All known algorithms for recognizing circle
  graphs use the split decomposition
• They use the following property A graph is a
  circle graph iff all parts of its split
  decomposition are circle graphs
• Proof gt Simple. The parts are induced
  subgraphs. lt

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Parts are Circle gt Graph is circle
u
v
u
v
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Parts are Circle gt Graph is circle
u
v
u
u
v
v
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Parts are Circle gt Graph is circle
u
v
u
u
u
v
v
v
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Parts are Circle gt Graph is circle
u
v
u
u
v
v
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Recognizing Circle Graphs

• Take the graph, and decompose with the split
  decomposition to get prime graphs
• Each such graph is either trivial, or has an
  induced P_4 (as before, an induced P_4 always
  stays on the same side)
• We use the P_4 to find a
• substructure that will add
• restrictions, and do so
• repeatedly

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Show connection between Modular Decomposition and
Permutation Graphs
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Generalizations of the Modular and Split
Decompositions

• Directed Graphs (broad research, similar results)

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Generalizations of the Modular and Split
Decompositions

• Directed Graphs (broad research, similar results)
• Bipartite Graphs (cannot decompose into modules,
  but have a different decomp)

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Generalizations of the Modular and Split
Decompositions

• Directed Graphs (broad research, similar results)
• Bipartite Graphs (cannot decompose into modules,
  but have a different decomp)
• Hypergraphs
• 2-structures and k-structures
• Posets
• Boolean Functions, and other fields!

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Other Graph Decompositions

• P_4-related
• Clique cutset
• Homogeneous
• Twomodules of Mongolfier (influenced by Modules
  in 2-structures)

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Heres a graph for ya, Joe!

• Joe wants a way to decompose the graph
• Such that the properties of the parts reflect
  those of the whole graph
• The decomposition can be recursive
• Decomposition For both Algorithmic purposes and
  Combinatorial purposes!
• This is very much needed for combinatorial
  research, during the work research is easier
  when you are dealing with simple graphs, say
  with high connectivity. Afterwards we can
  sometimes integrate the decomposition into the
  result.

Ill find its Modules!
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Unrelated types of decompositions

• Decomposition into disjoint sets of edges
• Algorithmic Decompositions Structural,
  non-computable decompositions useful for
  algorithmics

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Algorithmic Decompositions tree-width

• Is best in a certain sense
• As opposed to the structured decompositions, the
  tree-width decomposition is chosen to be the best
  for certain algorithms
• Computing the optimal tree-width is NP-Hard
• Linear for fixed k.
• tw(G)minimum val of ?(G)-1 over all
  triangulations of G
• See the paper Treewidth Algorithmic techniques
  and results by Hans L. Bodlaendery

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Other Algorithmic Decompositions

• Pathwidth
• Bandwidth
• Branchwidth
• Cutwidth
• All of these explore structural characteristics
  of the graph by bounding some structural
  property

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My Question

• There seems to be a connection between classes of
  graphs with simple modular/split decompositions
  (permutation graphs, circle graphs, etc.), but I
  do not know of any proven connection between the
  modular decomposition and the treewidth.

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Conclusion

• Types of Decompositions
• Connected Components
• Blocks and bridges
• Connectivity (edge- or vertex-)
• Modular
• Split
• And intractable ones that optimally break a
  problem
• Many others pick the one that suits you!

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Fin.
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Further TODO

• 1)More on treewidth,
• Introduction, including connectivity decomps
• Polynomial Algorithm for finding a module
• The block structure
• Graphics for tree-width, the tree, and how to
  compute something NP-Hard using it
• Find list of probs solvable with Modular Decomp.