3eme Cours : Chordal Graphs MPRI 2012 2013

3eme Cours : Chordal Graphs MPRI 2012–2013

3eme Cours : Chordal GraphsMPRI 2012–2013

Michel [email protected]

http://www.liafa.univ-Paris-Diderot.fr/~habib

Chevaleret, octobre 2012

http://www.liafa.univ-Paris-Diderot.fr/~habib


Schedule

Partition refinement II

Chordal graphs

Representation of chordal graphs



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.



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



This technique is very powerfull not only for graph algorithms.First used by Corneil for Isomorphism Algorithms 1970Hopcroft Automaton 1971Crochemore string sorting 1981. . .



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 –



Vertex splitting

Also called vertex partitionningWhen the neighborhood N(x) is used as a pivot set.



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.


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).


Chordal graphs

Two Basic facts

1. Chordal graphs are hereditary2. Interval graphs are chordal


Chordal graphs

Chordal graph

5

1 4 38

6 7 2

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


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


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 .


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 .


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.


Chordal graphs

Proof of the theorem


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


Chordal graphs

1

76

5

4

3

2


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.

1 8

7

6

5

4

3

2


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.


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)



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



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 . . .



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



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/



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 :



(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.



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.



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.



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 .



Two subtrees intersect iff they have at least one vertex in common.By no way, these representations can be uniquely defined !



An example



Back to the theorem

For a connected graph, the following statements are equivalentand characterize chordal graphs :



(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.



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



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.



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.

Documents

3eme Cours : Chordal Graphs MPRI 2012 2013