Post on 03-Feb-2022
3eme Cours : Chordal Graphs MPRI 2012–2013
3eme Cours : Chordal GraphsMPRI 2012–2013
Michel Habibhabib@liafa.jussieu.fr
http://www.liafa.univ-Paris-Diderot.fr/~habib
Chevaleret, octobre 2012
3eme Cours : Chordal Graphs MPRI 2012–2013
Schedule
Partition refinement II
Chordal graphs
Representation of chordal graphs
3eme Cours : Chordal Graphs MPRI 2012–2013
Partition refinement II
Tree isomorphism using Partition refinement
Compute the generalized degree partitions of the two graphs Gand H
Folklore Property
iF G and H are isomorphic then their partitions are identical.
Particular case of trees
For trees the converse is also true.
3eme Cours : Chordal Graphs MPRI 2012–2013
Partition refinement II
To compute this partition we can use a variation of the partitionrefinement.DegreeRefine(P, S) :compute the partition of S in parts having same degree with P
3eme Cours : Chordal Graphs MPRI 2012–2013
Partition refinement II
This technique is very powerfull not only for graph algorithms.First used by Corneil for Isomorphism Algorithms 1970Hopcroft Automaton 1971Crochemore string sorting 1981. . .
3eme Cours : Chordal Graphs MPRI 2012–2013
Partition refinement II
Applications
I QUICKSORT : Hoare, 1962.
I Minimal deterministic automaton : Hopcroft O(nlogn) 1971.
I Relational coarset partition : Paige, Tarjan 1987
I Doubly Lexicographic ordering : Paige Tarjan 1987 O(LlogL).using a 2-dimensional refinement technique.
I Interval graph recognition, modular decomposition, manyproblems on graphs (LexBFS . . .). 1990 –
3eme Cours : Chordal Graphs MPRI 2012–2013
Partition refinement II
Vertex splitting
Also called vertex partitionningWhen the neighborhood N(x) is used as a pivot set.
3eme Cours : Chordal Graphs MPRI 2012–2013
Partition refinement II
J.E. Hopcroft, A nlogn algorithm for minimizing states in afinite automaton, Theory of Machine and Computations,(1971) 189-196.
A. Cardon and M. Crochemore, Partitioning a Graph inO(|A|log2|V |), Theor. Comput. Sci., 19 (1982) 85-98.
M. Habib, R. M. McConnell, C. Paul and L. Viennot, Lex-BFSand partition refinement, with applications to transitiveorientation, interval graph recognition and consecutive onestesting, Theor. Comput. Sci. 234 :59-84, 2000.
R. Paige and R. E. Tarjan, Three Partition RefinementAlgorithms, SIAM J. Computing 16 : 973-989, 1987.
3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
Definition
A graph is a chordal graph if every cycle of length ≥ 4 has a chord.Also called triangulated graphs, (cordaux in french)
1. First historical application : perfect phylogeny.
2. Many NP-complete problems for general graphs arepolynomial for chordal graphs.
3. Second application : graph theory. Treewidth (resp.pathwidth) are very important graph parameters that measuredistance from a chordal graph (resp. interval graph).
3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
Two Basic facts
1. Chordal graphs are hereditary2. Interval graphs are chordal
3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
Chordal graph
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A vertex is simplicial if its neighbourhood is a clique.
Simplicial elimination scheme
σ = [x1 . . . xi . . . xn] is a simplicial elimination scheme if xi issimplicial in the subgraph Gi = G [{xi . . . xn}]
ca b
3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
A characterization theorem for chordal graphs
Theorem
Dirac 1961, Fulkerson, Gross 1965, Gavril 1974, Rose, Tarjan,Lueker 1976.For a connected graph G the following items are equivalent :
(0) G is chordal (every cycle of length ≥ 4 has a chord).
(i) G has a simplicial elimination scheme
(ii) Every minimal separator is a clique
3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
Minimal Separators
A subset of vertices S is a minimal separator if Gif there exist a, b ∈ G such that a and b are not connected inG − S .and S is minimal for inclusion with this property .
3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
An example
a
b c ef
d3 minimal separators {b} for f and a, {c} for a and e and {b, c}
for a and d .
3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
If G = (V ,E ) is connected then for every a, b ∈ V such thatab /∈ Ethen there exists at least one minimal separator.But there could be an exponential number of minimal separators.Consider 2 stars a, x1, . . . , xn (centered in a) and b, y1, . . . , yn(centered in b) and then add all the edges xiyi for 1 ≤ i ≤ n.There exist 2n minimal separators for the vertices a and b.
3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
Proof of the theorem
3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
Lexicographic Breadth First Search (LexBFS)
Data: a graph G = (V ,E ) and a start vertex s
Result: an ordering σ of V
Assign the label ∅ to all verticeslabel(s)← {n}for i ← n to 1 do
Pick an unumbered vertex v with lexicographically largest labelσ(i)← vforeach unnumbered vertex w adjacent to v do
label(w)← label(w).{i}end
end
3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
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3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
The reference for a graph algorithm theorem
LexBFS Characterization [Rose, Tarjan et Lueker 1976]
A graph is chordal G iff every LexBFS ordering of G provides asimplicial elimination scheme.
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3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
How can we prove such a theorem ?
1. A direct proof, finding the invariants ?
2. Find some structure of chordal graphs
3. Understand how LexBFS explores a chordal graph
4. We will consider the 3 viewpoints.
3eme Cours : Chordal Graphs MPRI 2012–2013
Chordal graphs
Chordal graphs recognition so far
Chordal graph recognition
1. Apply a LexBFS on G O(n + m)
2. Check if the reverse ordering is a simplicial elimination schemeO(n + m)
3. In case of failure, exhibit a certificate : i.e. a cycle of length≥ 4, without a chord. O(n)
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
About Representations
I Interval graphs are chordal graphs
I How can we represent chordal graphs ?
I As an intersection of some family ?
I This family must generalize intervals on a line
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
Which kind of representation to look for for special classesof graphs ?
I Easy to manipulate (optimal encoding, easy algorithms foroptimisation problems)
I Geometric in a wide meaning (ex : permutation graphs =intersection of segments between two lines)
I Examples : disks in the plane, circular genomes . . .
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
First remark
Proposition
Every undirected graph can be obtained as the intersection of asubset family
Proof
G = (V ,E )Let us denote by Ex = {e ∈ E |e ∩ x 6= ∅} the set of edges adjacentto x .xy ∈ E iff Ex ∩ Ey 6= ∅We could also have taken the set Cx of all maximal cliques whichcontains x .Cx ∩ Cy 6= ∅ iff ∃ one maximal clique containing both x and y
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
Starting from a graph in some application, find its characteristic :
1. 2-intervals on a line (biology), intersection of disks (orhexagons) in the plane (radio frequency), filament graphs,trapezoid graphs . . .
2. A whole book on this subject :J. Spinrad, Efficient Graph Representations, Fields InstituteMonographs, 2003.
3. A website on graph classes :http ://www.graphclasses.org/
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
For chordal graphs the solution is Subtrees in a tree
Using results of Dirac 1961, Fulkerson, Gross 1965, Buneman1974, Gavril 1974 and Rose, Tarjan and Lueker 1976 :
For a connected graph, the following statements are equivalentand characterize chordal graphs :
(i) G has a simplicial elimination scheme
(ii) Every minimal separator is a clique
(iii) G admits a maximal clique tree.
(iv) G is the intersection graph of subtrees in a tree.
(v) Any MNS (LexBFS, LexDFS, MCS) provides asimplicial elimination scheme.
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
Helly Property
Definition
A subset family {Ti}i∈I satisfies Helly property if∀J ⊆ I et ∀i , j ∈ J Ti ∩ Tj 6= ∅ implies ∩iy∈JTi 6= ∅
Exercise
Subtrees in a tree satisfy Helly property.
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
Demonstration.
Suppose not. Consider a family of subtrees that pairwise intersect.For each vertex x of the tree T , it exists at least one subtree of thefamily totally contained in one connected component of T − x .Else x would belong to the intersection of the family, contradictingthe hypothesis.Direct exactly one edge of T from x to this part.We obtain a directed graph G , which has exactly n vertices and ndirected edges. Since T is a tree, it contains no cycle, therefore itmust exist a pair of symmetric edges in G , which contradicts thepairwise intersection.
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
VIN : Maximal Clique trees
A maximal clique tree (clique tree for short) is a tree T thatsatisfies the following three conditions :
I Vertices of T are associated with the maximal cliques of G
I Edges of T correspond to minimal separators.
I For any vertex x ∈ G , the cliques containing x yield a subtreeof T .
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
Two subtrees intersect iff they have at least one vertex in common.By no way, these representations can be uniquely defined !
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
An example
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
Back to the theorem
For a connected graph, the following statements are equivalentand characterize chordal graphs :
(i) G has a simplicial elimination scheme
(ii) Every minimal separator is a clique
(iii) G admits a maximal clique tree.
(iv) G is the intersection graph of subtrees in a tree.
(v) Any MNS (LexBFS, LexDFS, MCS) provides asimplicial elimination scheme.
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
Proof of the chordal characterization theorem
I Clearly (iii) implies (iv)
I For the converse, each vertex of the tree corresponds to aclique in G .But the tree has to be pruned of all its unnecessary nodes,until in each node some subtree starts or ends. Then nodescorrespond to maximal cliques.
I We need now to relate these new conditions to chordal graphs.(iii) implies (i) since a maximal clique tree yields a simplicialelemination scheme.(iv) implies chordal since a cycle without a chord generates acycle in the tree.(iv) implies (ii) since each edge of the tree corresponds to aminimal separator which is a clique
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
from (i) to (iv)
Demonstration.
By induction on the number of vertices. Let x be a simplicialvertex of G .By induction G − x can be represented with a family of subtreeson a tree T .N(x) is a clique and using Helly property, the subtreescorresponding to N(x) have a vertex in common α.To represent G we just add a pending vertex β adjacent to α.x being represented by a path restricted to the vertex β, and weadd to all the subtrees corresponding to vertices in N(x) the edgeαβ.
3eme Cours : Chordal Graphs MPRI 2012–2013
Representation of chordal graphs
Playing with the representation
Easy Exercises :
1. Find a minimum Coloring (resp. a clique of maximum size) ofa chordal graph in O(n + m).Consequences : chordal graphs are perfect.At most n maximal cliques.
2. Find a minimum Coloring (resp. a clique of maximum size) ofan interval graph in O(n)using the interval representation.