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Algorithmic aspects of modular decomposition Cours MPRI 2009–2010

Algorithmic aspects of modular decompositionCours MPRI 2009–2010

Michel Habib [email protected]://www.liafa.jussieu.fr/~habib

Chateau des rentiers, octobre 2009

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


Basic Definitions on Modules

Modules

Modules

For a graph G = (V ,E ), a module is a subet of vertices A ⊆ Vsuch that∀x , y ∈ A, N(x) − A = N(y)− A

Trivial Modules

∅, {x} and V are modules.

Prime Graphs

A graph is prime if it admits only trivial modules.



Examples

Characterisation of Modules

A subset of vertices M of a graph G = (V ,E ) is a module iff∀x ∈ V \M, either M ⊆ N(x) or M ∩ N(x) = ∅

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Examples of modules

◮ connected components of G

◮ connected components of G

◮ any vertex subset of thecomplete graph (or the stable)



Playing with the definition

Duality

A is a module of G implies A is a module of G .

Easy observations

◮ No prime graph with ≤ 3 vertices.

◮ P4 the path with 4 vertices is the only prime on 4 vertices.

◮ P4 is isomorphic to its complement.



Twins and strong modules

Twins

x , y ∈ V are false- (resp. true-) twins if N(x) = N(y) (resp.N(x) ∪ {x} = N(y) ∪ {y}.x , y are false twins in G iff x , y are true twins in G .

Strong modules

A strong module is a module that does not strictly overlap anyother module.



Modular partition

A partition P of the vertex set of a graph G = (V ,E ) is amodular partition of G if any part is a module of G .

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Let P be a modular partition of a graph G = (V ,E ). Thequotient graph G/P is the induced subgraph obtained by choosingone vertex per part of P.



Lemma (MR84)

Let P be a modular partition of G = (V ,E ).X ⊆ P is a module of G/P iff ∪M∈XM is a module of G .

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Uniqueness decomposition theorem

Modular Decomposition Theorem

Theorem (Gal’67)

Let G = (V ,E ) be a graph with |V | ≥ 4, the three following casesare mutually exclusive :

1. G is not connected,

2. G is not connected,

3. G/M(G) is a prime graph, withM(G ) the modular partitioncontaining the maximal strong modules of G .




As a byproduct, we notice that a prime graph G satisfies :G and G are connected




Modular decomposition tree

Tree

A recursive a this theorem yields a tree T in which :

◮ The root corresponds to V

◮ Leaves are associated to vertices

◮ Each node corresponds to a strong module

There are 3 types of nodes :

Parallel, Series and Prime




Another explanation

The set of strong modules is nested into an inclusion tree (calledthe modular decomposition tree MD(G ) of G ).

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Partitive Families

Partitive Families

Lemma

If M and M ′ are two overlapping modules then

◮ (i) M \M ′is a module

◮ (ii) M ∩M ′ is a module

◮ (iii) M ∪M ′ is a module

◮ (iv) M∆M ′ is a module



Partitive Families

CHM81

The set of all modules of an undirected graph (resp. a directedgraph)constitutes a partitive family (resp. a weak partitive family).


Partitive (resp. weakly partitive) families admit a decompositiontree with two (resp. three) types of nodes :

◮ degenerate (also called fragile)

◮ prime

◮ (resp. linear)



Partitive Families

This tree representation theorem for partitive (resp. weaklypartitive) families F ⊆ 2|X |, yields an encoding of these families inO(|X |).



Partitive Families

Modular Decomposition Theorem for Directed Graphs

Theorem (CHM81)

Let G = (V ,E ) be a directed graph with |V | ≥ 4, the fourfollowing cases are mutually exclusive :

1. G is not connected, Parallel node

2. G is not connected, Series node

3. G ∗ is not strongly connected, Linear node

4. G/M(G) is a prime graph, withM(G ) the modular partitioncontaining the maximal strong modules of G , Prime node



Structural Aspects of Prime graphs

Prime graphs are nested

Folklore Theorem

Let G be a prime graph (|G | ≥ 4), then G contains a P4.

Theorem Schmerl, Trotter, Ille 91 . . .

Let G be a prime graph (|G | = n ≥ 4), then G contains a primegraph on n − 1 vertices or a prime graph on n − 2 vertices.




A simple proof

A stronger statement A. Cournier, P. Ille, 1991

For a prime graph there is at most one vertex not contained in aP4.

Proof

If ∃x ∈ V that does not belong to a P4 then N(x) is a stable set.If ∃x 6= y ∈ V that does not belong to a P4,wlog assume xy /∈ E (else we consider G )But then y ∈ N(x) and therefore : N(y) ⊆ N(x).By symmetry x , y must be twins, a contradiction.




The previous result need no hypothesis on the cardinality of thegraphsuch as the given proof.




In fact if there is such a vertex x ∈ V ,x is adjacent to the middle vertices of a P4.(Such a subgraph is called a bull).

A bull is isomorphic to its complement.


Modular Decomposition Algorithms

Historical Notes

Historical notesThe big list of published algorithms for modular decomposition(N.B. Perhaps some items are missing . . . please give me themissing references)

◮ Cowan, James, Stanton 1972 O(n4)◮ Maurer 1977 O(n4) directed graphs◮ Blass 1978 O(n3)◮ Habib, Maurer 1979 O(n3)◮ Habib 1981 O(n3) directed graphs◮ Corneil, Perl, Stewart 1981, O(n + m) cograph recognition.◮ Cunningham 1982 O(n3) directed graphs◮ Buer, Mohring 1983 O(n3)◮ McConnell 1987 O(n3)◮ McConnell, Spinrad 1989 O(n2) incremental◮ Adhar, Peng 1990 O(log2n),O(nm) proc. parallel, cographs,

CRCW-PRAM



Historical Notes

◮ Lin, Olariu 1991 O(logn),O(nm) proc. parallel, cographs,EREW-PRAM

◮ Spinrad 1992 O(n + malpha(m, n))◮ Cournier, Habib 1993 O(n + malpha(m, n))◮ Ehrenfeucht, Gabow, McConnell, Spinrad 1994 O(n3)

2-structures◮ Ehrenfeucht, Harju, Rozenberg 1994 O(n2) 2-structures,

incremental◮ McConnell, Spinrad 1994 O(n + m)◮ Cournier, Habib 1994 O(n + m)◮ Bonizzoni, Della Vedova 1995 O(n3k−5) Committee

decomposition for hypergraphs◮ Dahlhaus 1995 O(log2n),O(n + m) proc. parallel, cographs,

CRCW-PRAM◮ Dahlhaus 1995 O(log2n),O(n + m) proc. parallel,

CRCW-PRAM



Historical Notes

◮ Habib, Huchard, Sprinrad 1995 O(n + m) inheritance graphs◮ McConnell 1995 O(n2) 2-structures, incremental◮ Capelle, Habib 1997 O(n + m) if a factoring permutation is

given◮ Dahlhaus, Gustedt, McConnell 1997 O(n + m)◮ Dahlhaus, Gustedt, McConnell 1999 O(n + m) directed graphs◮ Habib, Paul, Viennot 1999 O(n + mlogn) via a factoring

permutation◮ McConnell, Spinrad 2000 O(n + mlogn)◮ Habib, Paul 2001 O(n + m) cographs via a factoring

permutation◮ Capelle, Habib, Montgolfier 2002 O(n + m) directed graphs if

a factoring permutation is provided.◮ Shamir, Sharan 2003 O(n + m) cographs, fully-dynamic◮ McConnell, Montgolfier 2003 O(n + m) directed graphs◮ Habib, Montgolfier, Paul 2003 O(n + m) computes a factoring

permutation



Historical Notes

◮ McConnell, de Montgolfier 2003 O(n + m) Interval graphsand dimension k orders

◮ Brestcher, Corneil, Habib, Paul, 2006 O(n + m) Cographs viaLexBFS.

◮ . . .

◮ But also, Uno, Yagura, 2000 O(n) to decompose permutationgraphs when a layout is provided.

◮ Bergeron, Chauve, de Montgolfier, Raffinot 2006

◮ A simple linear time algorithm for modular decomposition . . .M. Tedder, D. Corneil, M. Habib, C. Paul presented at ICALP2008



Historical Notes

Why it is so important ?

[Jerry Spinrad’ book 03]

The new [linear time] algorithm [MS99] is currently too complex todescribe easily [...] The first O(n2) partitioning algorithms weresimilarly complex ; I hope and believe that in a number of years thelinear algorithm can be simplified as well.



Historical Notes

Applications of modular decomposition

◮ A very natural operation to define on discrete structures,searching regularities.

◮ A structure theory for comparability graphs

◮ A compact encoding using module contraction and if we keepat each prime node the structure of the prime graph.

◮ Divide and conquer paradigm can be applied to solveoptimization problems. For example to test isomorphism.



Historical Notes

◮ A very basic graph algorithmic problem (similar to graphisomorphism problem).

◮ A better understanding of graph algorithms and their datastructures.



Bottom up Techniques

Minimal Modules

Minimal module containing a set

For every A ⊆ V there exists a unique minimal module containingA

Proof :

Since the module family is partitive and therefore closed under ∩.




Splitters

Definition

A splitter for a A ⊆ V , is a vertex z /∈ As.t. ∃x , y ∈ A with zx ∈ E and zy /∈ E .

Modules

A ⊆ V is a module iff A does not have any splitter.

Usefull lemma

If z is a splitter for a A ⊆ V , then any module containing A mustalso contain z .




Submodularity

Let us denote by s(A) the number of splitters of a set A, then s isa submodular function.

Definition

A function is submodular if∀A,B ⊆ Ef (A ∪ B) + f (A ∩ B) ≤ f (A) + f (B)

This is the basis idea of Uno and Yagura’s algorithm for themodular decomposition of permutation graphs in O(n).




Bottom-up Techniques

Sketch of the algorithm

For each pair of vertices x , y ∈ VCompute the minimal module m(x , y) containing x and y .

Closure with splitters

While there exists a splitter add it to the set.

Complexity

O(n2.(n + m))




Primality testing

One can derive a primality test since if there exists a non trivialmodule, it contains at least two vertices.




For some problems Bottom-Up techniques are the best known.




Origins : Golumbic, Kaplan, Shamir 1995

Input : G1 = (V ,E1) and G2 = (V ,E2) 2 undirected graphs suchthat E1 ⊆ E2 and Π be a graph property.Results : a sandwich graph Gs = (V ,Es) satisfying property Π andsuch that E1 ⊆ Es ⊆ E2.Edges of E1 are forced, those of E2 are optional ones, but those ofE3 = E2 are forbidden.

Unfortunately most cases are NP-complete, as for example of Π

◮ Gs being comparability, chordal, strongly chordal, . . .




Only few polynomial cases

◮ cographs Golumbic, Kaplan, Shamir (1995)

◮ sandwich module Cerioli, Everett, de Figueiredo, Klein (1998). . .

Natural question

Find efficient algorithms for these polynomial cases.




Sandwich module problem

Input : G1 = (V ,E1) and G2 = (V ,E2) 2 undirected graphs suchthat E1 ⊆ E2.Result : a sandwich graph Gs = (V ,Es) having a non trivialmodule and such that E1 ⊆ Es ⊆ E2.




Minimal Sandwich Module

Splitter

For a subset A ⊆ V , a splitter is a vertex z /∈ As.t. ∃x , y ∈ A with zx ∈ E1 and zy /∈ E2 (or equivalently zy ∈ E3)A splitter is also called bias vertex.

Algorithm

The computation of a minimal sandwich module can be done inO(n2.(n + m1 + m3)).

Hard to do better with this idea, using a bottom up approach.



Top down techniques

An O(n + mlogn) algorithm

Input: A partition P of the vertex set V of a graph G

Output: The coarsest modular partition Q smaller than P



Top down techniques

begin

Let Z be the largest part of P ;Q ← P ; K ← {Z} ; L← {X | X 6= Z,X ∈ P};while L ∪ K 6= ∅ do

if there exists X ∈ L then S ← X and L← L \ {X};else

1 Let X be the first part K and x arbitrarily selected in X ;S ← {x} and K ← K \ {X};

foreach vertex x ∈ S do

foreach part Y 6= X such that N(x) ⊥ Y do

2 Replace in Q, Y by Y1 = Y ∩ N(x) and Y2 = Y \ N(x);Let Ymin (resp. Ymax) be the smallest part (resp. largest)among Y1 and Y2;if Y ∈ L then L← L ∪ {Ymin,Ymax} \ {Y};else

L← L ∪ {Ymin};if Y ∈ K then Replace Y by Ymax in K ;else Add Ymax at the end of K ;



Factoring Permutation


A factoring permutation of a graph G = (V ,E ) is a permutationof V in which any strong module of G is a factor. [CH 97]

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◮ From modular decomposition tree to factoring permutation.Just take a left-right ordering of the leaves of the tree.

◮ From a factoring permutation to the modular decompositiontree.Only O(n2) possible strong modules.




Splitter interpretation

Starting with the partition {N(x), {x},N(x)}, we maintain thefollowing invariant :It exists a factoring permutation smaller than the current partition.




Generic Refinement Algorithm

Input : P a partition and S a set of pivotsWhile S 6= ∅Choose S ∈ SP = Refine(S ,P)add all new generated parts to S




This technique is very powerfull not only for graph algorithms.First used by Corneil for Isomorphism Algorithms 1970Hopcroft Automata minimisation 1971Cardon and Crochemore string sorting 1981J. Spinrad Graph Partitionning (generic tool) 1986Paigue, Tarjan 1987 (generic tool presented on three problems). . .




Applications

◮ QUICKSORT

◮ Minimal deterministic automaton Hopcroft O(nlogn) 1971.O(n) ?

◮ Relational coarset partition Paige, Tarjan O(nlogn) 1987

◮ Coarsest functional partition Paigue, Tarjan O(nlogn) 1987improved to O(n) by Paigue, Tarjan, Bonic 1985 andChrochemore 1982.

◮ String sorting O(nlogn)

◮ Doubly Lexicographic ordering Paige Tarjan 1987 O(LlogL),where L = #ones in the matrix using a 2-dimensionalrefinement technique. O(L) ?




Many other applications on graphs

◮ Interval graph recognition O(n + m) using parttion refinementon maximal cliques, 1-consecutiveness property O(n + m),Habib, McConnell, Paul and Viennot 2000.

◮ Modular decomposition,

◮ Cograph recognition O(n + m), Habib, Paul 2000.

◮ Transitive orientation




Partition refinement techniques are kind of dual to Union-Finddata structures.

◮ x and y belong to the same part → Union-Find

◮ x and y cannot belong to the same part → Partitionrefinement




Vertex splitting

Also called vertex partitionningUse N(x) as a pivot set.




A linear algorithm ?

We have seen a simple algorithms in O(n + mlogn) in which theneighbourhood of a given vertex can be used to refine the partitionat most logn times.How to improve this to a constant number ?




Three explored directions

◮ Ehrenfeucht et al approach

◮ Using Factoring Permutation

◮ Using LexBFS (as for cographs)



Ehrenfeucht et al approach


M(G , v) is composed by {v} and the maximal modules of G notcontaining v .

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Principle of the Ehrenfeucht et al.’s algorithm

1. Compute M(G , v)

2. Compute MD(G/M(G ,v))

3. For each X ∈M(G , v) compute MD(G [X ])




ComputingM(G , v) via Partition Refinement

Splitter again

If z is a splitter of A ⊆ V then any strong module contained in Ais either contained in N(z) ∩ A or in A− N(z).




Computation ofM(G , v)

⇒ O(n + m log n) time using vertex partitioning algorihtm.

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How to reconstruct the modular decomposition tree from thepartiton M(G , v) ?The most difficult step in many algorithms.




Computation of MD(G/M(G ,v))

◮ The modules of G/M(G ,v) are linearly nested :any non-trivial module contains v

◮ The forcing graph F(G , v) has edge −→xy iff y separates x and v

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Complexity

◮ [Ehrenfeucht et al.’94] gives a O(n2) complexity.

◮ [MS00] gives a very simple O(n + m log n) algorithm based onvertex partitioning.

◮ [DGM’01] proposes a O(n + m.α(n,m)) and a morecomplicated O(n + m) implementation.

Other algorithms

◮ [CH94] and [MS94] present the first linear algorithms.

◮ [MS99] present a new linear time algorithm which extends totransitive orientation.


Cographs

Cographs

Recursive Definition

The class of cographs is the smallest class of graphs containing G0

and closed under series and parallel compositions.

Characterisation Theorem

A graph is a cograph iff it does not contain a P4.

Obvious

With the structure of prime graphs.


Cographs

An example

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Cographs

Comments

◮ In other words, the modular decomposition tree of a cographhas only series and parallel nodes.

◮ Despite its structural simplicity it is not so obvious to build alinear time algorithm to recognize cographs.


Cographs

Brute Force Algorithm

Using the decomposition theorem, we only have to compute atmost n times some connected components of G or its complement.O(n.(n + m)) complexity.


Cographs

Cotree

Definition

The modular decomposition tree of a cograph is called a cotree.

Properties of the cotree

◮ Vertices of the cotree can be labelled with 0 (parallel) and 1(series).

◮ From G to G just exchange 0’s and 1’s.

◮ xy ∈ E iff LCA(x , y) in the cotree is labelled with 1


Cographs

A simplicial scheme for cographs

G is a cograph iff it exists an ordering of the verticess.t. xi has a twin (false or true) in G{xi+1, . . . xn}


Cographs

3 approaches for a linear time algorithm

1. Corneil, Perl, Stewart 1981, O(n + m) cograph recognition.Incremental : add a vertex x at each step and update the treein O(d(x)).

2. Habib, Paul 2001 : Using partition refinement and a factoringpermutation

3. Brestcher, Corneil, Habib, Paul, 2006 , using 2 consecutiveLexBFS searches


Cographs

An algorithm based on factoring permutation

Factoring permutation (of cographs) via vertex partitionning

At each step, the partition can be refined into a factoringpermutation.

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Cographs

An algorithm based on factoring permutation

1. Best known complexity : O(n + m log n) and O(n + m) forcographs

2. Recent algorithmic results around this notion [UY00][BXHP05] [BCdMR05]

3. For some graph families, computing MD(G ) from a factoringpermutation costs O(n) time

4. Generalization for other decomposition like for example thesplit decomposition ?...


Cographs

An algorithm based on LexBFS

The best Algorithm Using LexBFS

A cograph recognition algorithm [BCHP03]

1. σ ← LexBFS(G )

2. σ ← LexBFS−(G , σ)

3. If σ and σ both have the NS-property then

3.1 Answer ”G is a cograph”3.2 Build MD(G)

4. Else Output a P4


Cographs


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Cographs


Computing a LexBFS ordering σ

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Computing σ=LexBFS−(G , σ)

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Lemma

M(G , x) is composed by the slices Si(x) of σ and Sj(x) of σ.

Theorem

If G is a cograph, then MD(G ) can be retrieved from the slicestructure of σ and σ.


Cographs


Data: a module M of G ;

Result: the cotree of G (M)

if M = {s} then T (M,G )← sTake a vertex s ∈ M and compute :LexBFS(s,G ) = [s,N(s),S1(s), . . . ,Sk(s)]LexBFS(s,G ) = [s,N(s),S1(s), . . . Sh(s)]if N(s) = ∅ (resp. N(s) = ∅ ) ; then

T (M,G )← s⊕

T (S1(s),G )(resp. T (M,G )← s

⊗T (S1(s),G ))

Else Let us consider two vertices : a ∈ S1(s) et b ∈ S1(s)if ab ∈ E then

T (M,G ) ← s⊕

T (S1(s),G )⊗

T (S1(s),G )⊕

T (S2(s),G ). . .

Else T (M,G )← s⊗

T (S1(s),G )⊕

T (S1(s),G )⊗

T (S2(s),G ) . . .


Cographs


Elements de preuve de l’algorithme

1. Soit S une couche lexicographique de LexBFS σ sur G , alorsla restriction de σ a S est un LexBFS de G (S).

2. Il n’existe pas d’arete entre les Si(x). Les sommets d’un Si(x)ont donc les memes voisins a l’exterieur, et les Si(x) sontdonc des modules de G .

3. Les Si (resp. Si) sont des modules maximaux de G − x .

4. Les Si verifient une condition d’inclusion des voisinages quijustifie les formules de la fonction T .

5. Le deuxieme parcours en − permet d’assurer la recursivite. Eneffet Si(x) est un module facteur de σ, d’apres le lemme celapermet de garantir que les parcours de G et G vontcommencer par le meme sommet de Si(x).


Cographs


Preprocessing

Data: a graph G ;

Result: A pair of LexBFS

Take x a vertex in G and computeσ = LexBFS(x ,G ) and σ− = LexBFS−(x ,G )and for each vertex of x of G , we contruct the lists Succ(x)containing the first vertices with respect to σ of the Si(x)′s (i.e.Succ(x) = minσS1(x), next.Succ(x) = minσS2(x) and so on ).Same definition for Succ(x) using σ.


Cographs


The optimised version

Data: s the first vertex of a module M of G wrt σ ;

Result: the cotree of G (M)

if Succ(s) = Nil and Succ(s) = Nil then

t(s,G )← s

if Succ(s) = Nil (resp. Succ(s) = Nil ) ; then

t(s,G )← s⊕

t(Succ(s),G )(resp. t(s,G )← s

⊗t(Succ(s),G ))

if Succ(s)Succ(s) ∈ E then

t(s,G )← s⊕

t(Succ(s),G )⊗

t(Succ(s),G )⊕

t(next.Succ(s),G ). . .

Else t(s,G )← s⊗

t(Succ(s),G )⊕

t(Succ(s),G )⊗

t(next.Succ(s),G )


Cographs


Couches lexicographiques

On appelle couche lexicographique d’un LexBFS un ensemble desommets qui a une etape du parcours sont equivalents. Seul unchoix arbitraire ou une regle de tie-break permet de les separer.

Initialement l’ensemble des sommets du graphe constitue unecouche lexicographique. A chaque etape de l’algorithme, lessommets non explores sont partitionnes en coucheslexicographiques. Un gestion de ces couches a l’aide d’unalgorithme de partitionnement permet une implementation simpleet efficace d’un LexBFS.


Cographs


Remarque

Tant qu’aucun sommet d’un module M n’a ete explore, lessommets de M appartiennent tous a une meme couchelexicographique du LexBFS.

Lemme

Soit M un module ”fort”qui est un facteur de σ, alors M est aussiun facteur de σ− et les deux facteurs commencent par le memesommet.

Definition

On notera lorsque le voisinage du sommet x de depart du parcoursa ete totalement explore, S1(x), . . . Sk(x) les coucheslexicographiques de son non-voisinage (i.e. les sommets ayant lesmemes voisins dans l’ensemble {x} ∪ N(x)).


Cographs


L’algorithme

◮ Initialisations :Calculs de σ = LexBFS(x ,G ) et σ − = LexBFS−(x ,G ) et despartitions associees.

◮ Pour obtenir le coarbre il suffit d’appeler la fonction T (M,G ),fonction recursive definie ci-apres pour un module M de G , etqui calcule le coarbre du sous-graphe G (M).


Cographs


Elements de preuve de l’algorithme

1. Soit S une couche lexicographique de LexBFS σ sur G , alorsla restriction de σ a S est un LexBFS de G (S).

2. Il n’existe pas d’arete entre les Si(x). Les sommets d’un Si(x)ont donc les memes voisins a l’exterieur, et les Si(x) sontdonc des modules de G .

3. Les Si (resp. Si) sont des modules maximaux de G − x .

4. Les Si verifient une condition d’inclusion des voisinages quijustifie les formules de la fonction T .

5. Le deuxieme parcours en − permet d’assurer la recursivite. Eneffet Si(x) est un module facteur de σ, d’apres le lemme celapermet de garantir que les parcours de G et G vontcommencer par le meme sommet de Si(x).


Cographs


Preprocessing

Data: a graph G ;

Result: A pair of LexBFS

Take x a vertex in G and computeσ = LexBFS(x ,G ) and σ− = LexBFS−(x ,G )and for each vertex of x of G , we contruct the lists Succ(x)containing the first vertices with respect to σ of the Si(x)′s (i.e.Succ(x) = minσS1(x), next.Succ(x) = minσS2(x) and so on ).Same definition for Succ(x) using σ.


Cographs


Interpretation de cet algorithme

Ce qu’il faut retenir de ce mecanisme a deux parcours

Pour chaque sommet x premier sommet explore par le LexBFSd’un module de G , le parcours sur G ordonne N(x), tandis que ledeuxieme parcours sur G ordonne le voisinage N(x).Les deux parcours LexbFS gerent sur les ensembles Si (resp. Si) dela meme facon leurs cas d’egalite (tie-break).

Remarque

L’arbre propose par l’algorithme n’est pas necessairementcanonique (i.e. il peut y avoir deux noeuds 0 (resp. 1) consecutifs).


Cographs


Generalisation to arbitrary graphs ?

1. There are many similarities between two-LexBFS-sweepalgorithm and the linear implementation of Ehrenfeucht etal.’s algorithm [DGM01]

2. LexBFS is useful for the transitive orientation problem. Couldit lead to a simple linear time algorithm for this problem ?


Cographs


Retour sur ces caracterisations

Caracterisation [Dirac 1961 ]

Un graphe est triangule ss’il possede un ordre d’eliminationsimplicial.A vertex is simplicial iff its neighbourhood is a clique (completegraph).

Caracterisation LexBFS [Rose, Tarjan et Lueker 1976]

Un graphe G est triangule ssi tout ordre ”inverse” LexBFS de Gest un ordre d’elimination simplicial.


Cographs


Umbrellas

When restricted to cographs, LexBFS orderings can be shown to beumbrella-free. An umbrella of an ordering σ is a triple of verticesx , y , z such that x <σ y <σ z where xz ∈ E , xy /∈ E and yz /∈ E .

Characterisation

G is a cograph iff Any LexBFS ordering σ of a cograph G isumbrella-free


Cographs


Jeu des differences

Characterisation

G is a cograph iff Any LexBFS σ of a cograph G is umbrella-free(there does not exist x <σ y <σ z such that xz ∈ E , xy /∈ E andyz /∈ E ).

Characterisation LexBFS [Rose, Tarjan et Lueker 1976]

A graph G is chordal iff Any LexBFS ordering σ of G is a simplicialelimination scheme


Cographs


Generalisation to arbitrary graphs ?

1. There are many similarities between two-LexBFS-sweepalgorithm and the linear implementation of Ehrenfeucht etal.’s algorithm [DGM01]

2. LexBFS is useful for the transitive orientation problem. Couldit lead to a simple linear time algorithm for this problem ?

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