3ème Cours Cours MPRI 2010 2011 - irif.frhabib/Documents/cours3_2010.pdf · 3 eme Cours Cours MPRI...

3eme Cours Cours MPRI 2010–2011

3eme CoursCours MPRI 2010–2011

Michel Habibhabib@liafa.jussieu.fr

http://www.liafa.jussieu.fr/~habib

Chevaleret, septembre 2010

Schedule

Answer to one exercice

Representation of chordal graphs

More structural insights of chordal graphs

Properties of reduced clique graphs

Interval graphs

Exercises

Helly Property

Definition

A subset family {Ti}i∈I satisfies Helly property if∀J ⊆ I et ∀i , j ∈ J Ti ∩ Tj 6= ∅ implies ∩i ∈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 .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.

An operation research problem

I Storage of products in fridges : each product is given with aninterval of admissible temperatures.Find the minimum number of fridges needed to store all theproducts (a fridge is at a given temperature).

I A solution is given by computing a minimum partition intomaximal cliques.

I Fortunately for an interval graph, this can be polynomiallycomputed

I So knowing that a graph is an interval graph can help to solvea problem.

Back to chordal graphs

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)

Playing with the representation

Exercise :Find a minimum Coloring (resp. a clique of maximum size) of aninterval graph in O(n)using the interval representation.

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 ?

I Easy to manipulate (optimal encoding, easy algorithms foroptimisation problems)

I Geometric in a wide meaning

I Examples : disks in the plane, circular genomes . . .

First remark

Proposition

Every undirected graph can be obtained as the intersection of asubset family

G = (V ,E )Let us denote by Ex = {e ∈ E | e ∩ x 6= ∅} the set of edgesadjacent to x .xy ∈ E iff Ex ∩ Ey 6= ∅We could also have taken the set Cx of all maximal cliques whichcontains x .On peut aussi associer a x, l’ensemble des cliquesmaximales qui contiennent xCx ∩ Cy 6= ∅ iff ∃ one maximal clique containing both x and y

A whole book on this subject :J. Spinrad, Efficient Graph Representations, Fields InstituteMonographs, 2003.

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.

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 :

(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.

Proof of the chordal characterisation 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.

Clique treeclique tree of G = a minimum size tree model of G

for a clique tree T of G :I vertices of T correspond precisely to the maximal cliques of GI for every maximal cliques C ,C ′, each clique on the path in T

from C to C ′ contains C ∩ C ′

I for each edge CC ′ of T , the set C ∩ C ′ is a minimal separator(an inclusion-wise minimal set separating two vertices)

Note : we label each edge CC ′ of T with the set C ∩ C ′.

Theorem

Every minimal separator belongs to every maximal clique tree.

Every minimal separator is the intersection of at least 2 maximalcliques of G .

Clique graphthe clique graph C(G ) of G = intersection graph of maximalcliques of G

G C(G )

Reduced clique graphthe reduced clique graph Cr (G ) of G = graph on maximal cliquesof G where CC ′ is an edge of Cr (G ) ⇐⇒ C ∩ C ′ is a minimalseparator.

Combinatorial structure of Cr (G )

Lemma 1 : M.H and C. Paul 95

If C1,C2,C3 is a cycle in Cr (G ), with S12,S23 and S23 be theassociated minimal separators then two of these three separatorsare equal and included in the third.

Lemma 2 : M.H. and C. Paul 95

Let C1,C2,C3 be 3 maximal cliques, ifC1 ∩ C2 = S12⊂S23 = C2 ∩ C3 then it yields a triangle in Cr (G )

Lemma 3 : Equality case

Let C1,C2,C3 be 3 maximal cliques, if S12 = S23 then :

I either the C1 ∩ C3 = S13 is a minimal separator

I or the edges C1C2 and C2C3 cannot belong together to amaximal clique tree of G .

b,d,e e,g c,e,f

c,hb,e

b,d,e e,g c,e,f

c,hb,e c,e

b,d,e e,g c,e,f

c,hb,e

Theorem (Gavril 87, Shibata 1988, Blayr and Payton 93)

The clique trees of G are precisely the maximum weight spanningtrees of C(G ) where the weight of an edge CC ′ is defined as|C ∩ C ′|.

Theorem (Galinier, Habib, Paul 1995)

The clique trees of G are precisely the maximum weight spanningtrees of Cr (G ) where the weight of an edge CC ′ is defined as|C ∩ C ′|.

Moreover, Cr (G ) is the union of all clique trees of G.

Applications

I All clique trees have exactly the same labels

Others simplicial elimination schemes form the theorem

1. Maximal Cardinality Search (MCS)

2. Maximal Neighbourhood search (MNS) :pick a vertex with a maximal neighbourhood in the verticesalready visited

3. LexBFS and MCS are particular cases of MNS.

Maximal Cardinality Search : MCS

Data: a graph G = (V ,E ) and a start vertex s

Result: an ordering σ of V

Assign the label 0 to all verticeslabel(s)← 1for i ← n a 1 do

Pick an unumbered vertex v with largest labelσ(i)← vforeach unnumbered vertex w adjacent to v do

label(w)← label(w) + 1end

Maximal Neighbourhood Search : MNS

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 a 1 do

Pick an unumbered vertex v with a maximal label (under inclu-sion)σ(i)← vforeach unnumbered vertex w adjacent to v do

label(w)← label(w) ∪ {i}end

Propriete (MNS)

Fort an ordering σ on V , if a < b < c and ac ∈ E and ab /∈ E ,then it must exist a vertex d such that d < b, db ∈ E and dc /∈ E .

Theorem

For a graph G = (V ,E ), an order σ on V is a MNS of G iff σsatisfies property (MNS).

Theorem

G is chordal graph iff any MNS produces a simplicial eliminationscheme.

Similar to that of LexBFS.

Maximum spanning trees

Maximal Cardinality Search can be seen as Prim algorithm forcomputing a maximal spanning tree of Cr (G ).

I How to implement a generic MNS search in linear time ?

I 2 known linear implementations : LexBFS, MCS

I How to compute a clique tree ?

I How to generate all simplicial elimination schemes ?

When applying a search (LexBFS or MNS) one can compute aclique tree, by considering the strictly increasing sequences oflabels.

Simplicial elimination schemes

1. Choose a maximal clique tree T

2. While T is not empty doSelect a vertex x ∈ F − S in a leaf F of T ;F ← F − x ;If F = S delete F ;

Canonical simplicial elimination scheme

1. Choose a maximal clique tree T

2. While T is not empty doChoose a leaf F of T ;Select successively all vertices in F − Sdelete F ;

Exercise

Does there exist other simplicial elimination scheme ?

Size of a maximal clique tree in a chordal graph

I Let G = (V ,E ) be a chordal graph.

I G admits at most |V | maximal cliques and therefore the treeis also bounded by |V | (vertices and edges).

I But some vertices can be repeated in the cliques. If weconsider a simplicial elimination ordering the size of a givenmaximal clique is bounded by the neighbourhood of the firstvertex of the maximal clique.

I Therefore any maximal clique tree is bounded by |V |+ |E |.

Size of CS(G )

Considering a star on n vertices,shows |CS(G )| ∈ O(n2)Not linear in the size of G

CS(G ) is not chordal !

In fact CS(G ) is dually chordal (almost chordal)and CS(CS(G )) is chordal.

Interval graphs

Characterisation Theorem for interval graphs

(0) G = (V ,E ) is interval graph.

(i) G has ...

(ii) It exists a total ordering τ of the vertices of V s.t.∀x , y , z ∈ G with x ≤τ y ≤τ z and xz ∈ E thenxy ∈ E .

(iii) G has a maximal clique path. (A maximal clique pathis just a maximal clique tree T, reduced to a path).

(iv) G is the intersection graph of a family of intervals ofthe real line

Interval graphs

To recognize an interval graph, we just have to compute amaximal clique tree and check if it is a path ?Difficulty : an interval graph has many clique trees among themsome are paths

Interval graphs

Many linear time algorithms already proposed for interval graphrecognition ....using nice algorithmic tools :graph searches, modular decomposition, partition refinement,PQ-trees . . .

Interval graphs

Linear time recognition algorithms for interval graphs

I Booth and Lueker 1976, using PQ-trees.

I Korte and Mohring 1981 using LexBFS and ModifiedPQ-trees.

I Hsu and Ma 1995, using modular decomposition and avariation on Maximal Cardinality Search.

I Corneil, Olariu and Stewart SODA 1998, using a series of 6consecutive LexBFS, published in 2010.

I M.H, McConnell, Paul and Viennot 2000, using LexBFS andpartition refinement on maximal cliques.

Interval graphs

Variations on these representations

I Path graphs = Paths in a treeIn between chordal and interval.

I Directed path graphs = directed paths in a rooted directedtreeappear in some polynomial CSP class.

Interval graphs

State of the Art

I We (Juraj Stacho, M.H.) study chordal representations foralgorithms

I A linear time algorithm for path graph recognition is stillmissing

Interval graphs

Cannonical representation

I For an interval graph, its PQ-tree represents all its possiblemodels and can be taken as a cannonical representation of thegraph (for example for graph isomorphism)

I But even path graphs are isomorphism complete. Therefore acanonical tree representation is not obvious for chordal graphs.

I Cr (G ) is a Pretty Structure to study chordalgraphs.To prove structural properties of all maximal clique trees of agiven chordal graph.

Exercises

Ends of a LexBFS

Many properties can be expressed on the last vertex of a LexBFS.Namely : the last vertex is simplicial if G is a chordal graph.

1. Show that the last maximal clique visited can be taken as theend of some chain of cliques if G is an interval graph.

2. Complexity of the following decision problem :

Data: a graph G = (V ,E ) and a given vertex x ∈ V

Result: Does there exist a LexBFS of G ending in x ?

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