Presentation on Chordal and Interval graphs

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Presentation on Chordal and Interval graphs

• Presented by
• Bansal Tarun
• Bhatewara Shreyas
• Vardhan Hars
• Also available online at
• http//www.utdallas.edu/shreyas.bhatewara/combin
  atorics.html
• OR http//utdallas.edu/tarun/graph.html

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Perfect Graphs

• Clique
• Chromatic number
• Perfect graphs each subset has chromatic number
  equal to size of maximal clique

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Chordal graphs

• Chordal graph Any cycle of four or more nodes
  has a chord in it
• Also called rigid circuit graphs

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Chordal graphs contd.

• Perfect Elimination order necessary and
  sufficient condition for Chordal graphs
• Simplicial vertexes along with their neighbors
  form a clique

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Interval Graphs

• Intersection graph of set of intervals on a real
  line.
• R ? I1, I2, I3....
• V I1, I2, I3....In
• (Ix, Iy) ? E ltgt Ix n Iy ? ?
• Adjacent intervals are overlapping

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Interval graphs contd.

• Perfect ? Chordal ? Interval graphs
• Interval model representation R-blocks and
  L-blocks
• Complement of Interval graphs are called
  comparability graph
• Comparability graph is also known as containment
  graph

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Properties and conditions

• Asteroidal triple free chordal graphs are
  interval graphs.

• Comparability graphs do not have odd cycles
• Elimination of simplicial vertices from chordal
  graph renders a chordal graph

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Applications and motivation to study

• Scheduling and storage problems
• Bioinformatics finding if gene is mutant
• Archaeology studying sequence dating (seriation)
• Ecology Food chain of organisms
• Maximum clique and Isomorphism are less complex
  in chordal graphs.

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Perfect Elimination Order (PEO)

• An ordered sequence of all vertices V1, V2,,
  Vn
• Successor (Vi) Vj jgti and (i,j) e E
• Predecessor (Vi) Vj igtj and (i,j) e E
• Sequence of vertices such that for each vertex
  Vi, successors of Vi form a clique
• A graph is chordal if and only if it has a
  Perfect Elimination Order (PEO) Fulkerson 1965

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Simplicial Vertex

• A vertex whose all neighbors form a clique
• Every chordal graph has at least one simplical
  vertex
• The first vertex in PEO is simplicial
• If you remove the simplicial vertex, then the
  graph induced by remaining nodes is also a
  chordal graph (hence must have a simplicial
  vertex of its own)

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Simple method to find PEO

• Given G (V, E)
• For (i1 to N)
• Vi Simplicial vertex (G)
• G Graph induced by vertex set (V-Vi)
• What if no simplicial vertex exists at some
  point?
• Complexity O(NN2) O(N3)

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Finding a PEO (Maximum Cardinality Method)

• Rose and Tarjan Rose 1975
• Cardinality Number Number of neighbors picked
  up
• Step1 Pick the node whose maximum number of
  neighbors have already been chosen
• Reverse the order
• Step 2 In the end, verify the ordering obtained
  is indeed PEO

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Example
Seq 6
Seq 2
F (1)
F (0)
E (0)
E (1)
F (2)
Seq 8

G(2)
G(0)
G(1)
A(0)
D(0)
D(2)
H (1)
H (3)
D(1)
H (0)
H (2)
Seq 1
Seq 3
Seq 7
The PEO is G, H, F, B, C, D, E , A
B(0)
B (1)
B (2)
B (3)
C(0)
C(1)
C(2)
Seq 5
Seq 4
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Observations

• Except for the first node, any node being picked
  up has at least one neighbor which has already
  been picked
• Set of picked nodes always form a connected graph

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Proof of Correctness

• If a PEO does not exist, then algorithm does not
  generate a PEO
• If the PEO exists, then algorithm outputs some PEO

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Proof of Correctness (Part 2)

• Let V1, V2, V3,,Vi, , Vj,, Vk,,Vn be the
  ordering generated.
• Let Vi is not in sequence. Then Vj and Vk be the
  two successors of Vi that are not connected.
• Case 1
• Then Vj and Vk are parts of two graphs which are
  connected only through Vi
• The order in which the nodes were picked is
• Vn, , Vk, ., Vj,, Vi,,V3, V2, V1
• Then we claim that label sequence must have been
  different

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Proof (Contd.)

• Case 2
• Consider the path from k to j on which the nodes
  were picked up.
• All such nodes should be connected to i except
  those nodes, which are directly connected to j.
• Let m be such a vertex on that path. m must have
  some vertices which are not connected to i.

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Proof (contd.)

• Let w be such a vertex. Therefore w must be
  connected to k. (through a path not involving i).
  
• If i is not connected to w, then we have a non
  chordal cycle i-k-w-m-j-i.
• i must be connected to w as well. Therefore there
  is no way that m can be picked before I since i
  is connected to one extra node than m (Vk)

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Complexity Analysis (Step 2)

• Check if the generated ordering is perfect
• Trivial algorithm
• Number of vertices whose successors have to be
  checked N
• Time complexity to check if all the successors of
  a particular node are connected NC2 N2
• Total complexity N3

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Step 2 Efficient Implementation

• Efficient Way
• Given ordering V1, V2, V3, , Vi, , Vj,,
  Vk,,Vn
• 1. for i 1 to n do
• 2. if Vi has successors
• 3. Let u be the first successor of
  Vi
• 4. For all w ? Successor (Vi), w ?u
• 5. Add (u, w) to TEST
• 6. Test whether all vertex-pairs in TEST are
  adjacent.

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Why?

• If the algorithm detects ordering as non-perfect,
  then ordering must be non-perfect
• If the algorithm detects ordering as PEO, then
  ordering cannot be imperfect (By contradiction)
• complexity if implemented carefully O ( N E)
• Maximum size of set TEST can be O(E)
• Instead of O (N3)

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Implementation of step 1 (generating perfect
ordering)

• Trivial Method
• Maintain a heap of cardinality numbers
• Complexity of each for loop
• Find the maximum cardinality node and removing
  it O(lg N)
• Number of updates for each node picked N
• modifying the cardinality number and
  rearranging the heap N lg N
• Total Complexity N (lg N N lg N) N2 lg N
• Can also be shown as E lg N

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Efficient Implementation (Step 1)

• With complexity O( N E)
• For each node, maintain list of neighbors.
  (Already given in this format)
• Maintain linked list of vertices (in order of
  cardinality number)
• For each node, maintain its location in the
  linked list
• Picking the vertex with max. cardinality number
  O(1)
• Finding all the neighbors and updating their
  cardinality number and their location in linked
  list O(E)

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Polynomial Algorithms for chordal graphs Gavril
1972

• Minimum-Coloring Algorithm
• Greedy Algorithm
• Scan the vertices in the reverse order of PEO and
  color each vertex with the smallest color not
  used among its successors
• Gives Optimal coloring
• Complexity Polynomial (given that PEO is
  already known)

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Polynomial Algorithms for chordal graphs

• Maximum Independent Set
• Greedy algorithm
• Scan the vertices in the order of PEO, and for
  each Vi, add Vi to I if none of its predecessor
  has been added to I
• Complexity Polynomial

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Another definition for chordal graphs

• Intersection Graph A graph that represents the
  intersection of sets
• Subtree Graph
• Break a tree into multiple subtrees (overlapping)
• Now make a graph whose each vertex represents one
  subtree. An edge between two vertices if they
  have a common vertex (between the subtrees they
  represent)
• The graph thus obtained is known as Subtree Graph
• As Gavril 1974 showed, the subtree graphs are
  exactly the chordal graphs. So a chordal graph
  can be represented as an intersection graph of
  subtrees.

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Bibliography

• Gavril 1974 Gavril, Fanica, "The intersection
  graphs of subtrees in trees are exactly the
  chordal graphs, Journal of Combinatorial Theory,
  Series B 16, 1974
• Gavril 1972 Gavril, Algorithms for Minimum
  Coloring, Maximum Clique, Minimum Independent Set
  of a Chordal Graph, SIAM J. Comp., Vol 1, 1972 
• Rose 1975 Rose, Tarjan, Algorithmic Aspects
  of Vertex Elimination, Proc. 7th annual ACM
  Symposium on Theory of Computing (STOC), 1975

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Bibliography

• Therese Biedl, Lecture notes of a graduate
  course, University of Waterloo (Sept 2005),
  http//www.student.cs.uwaterloo.ca/cs762/Notes/bo
  ok.pdf
• Arnborg 1989 Arnborg et. al., Linear time
  algorithms for NP-hard problems restricted to
  partial k-trees, Discrete Applied Mathematics,
  Vol. 23, Issue 1, 1989
• Wiki-Tree-Decomposition http//en.wikipedia.org
  /wiki/Treewidth
• Fulkerson 1965 Fulkerson, D. R. Gross, O. A.
  (1965). "Incidence matrices and interval graphs,
  Pacific J. Math 15 835855.

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Tree Decomposition

• Each subtree associates a graph vertex with a set
  of tree nodes. To define this formally, we
  represent each tree node as the set of vertices
  associated with it. Thus, given a graph G (V,
  E), a tree decomposition is a pair (X, T), where
  X X1, ..., Xn is a family of subsets of V,
  and T is a tree whose nodes are the subsets Xi,
  satisfying the following properties
• The union of all sets Xi equals V. That is, each
  graph vertex is associated with at least one tree
  node.
• For every edge (v, w) in the graph, there is a
  subset Xi that contains both v and w. That is,
  vertices are adjacent in the graph only when the
  corresponding subtrees have a node in common.
• If Xi and Xj both contain a vertex v, then all
  nodes Xz of the tree in the (unique) path between
  Xi and Xj contain v as well. That is, the nodes
  associated with vertex v form a connected subset
  of T.

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Example taken from Wiki-Tree-Decomposition
Graph and the corresponding tree decomposition
Tree width 3-1 2
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Tree decomposition (properties)

• A graph can have multiple tree decompositions
• Width of tree decomposition Size of largest Xi
  minus 1
• The minimum width among all tree decompositions
  is known as tree width
• The graphs with treewidth at most k are also
  called partial k-trees
• For chordal graphs, the treewidth is size of
  biggest clique minus 1

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Uses

• Algorithms work much faster on trees than graphs
• Use dynamic programming on decomposed tree to
  solve problems of the original graph (must have
  bounded treewidth) Arnborg 1989