2ème Cours Cours MPRI 2010 2011habib/Documents/Cours_2_2010.pdfthe memory word GRAF[x,y] Therefore...

2eme Cours Cours MPRI 2010–2011

2eme CoursCours MPRI 2010–2011

Michel [email protected]

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

Chevaleret, septembre 2010

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


Schedule

Comments on last course

Chordal graphsLexicographic Breadth First Search LexBFSSimplicial elimination scheme

Exercices



Fagin’s Theorems again

Fagin’s theorems in structural complexity

Characterizations without any notion of machines or algorithms !

NP

The class of all graph-theoretic properties expressible in existentialsecond-order logic is precisely NP.

P

The class of all graph-theoretic properties expressible in Hornexistential second-order logic with successor is precisely P.



Quadratic space in linear time

◮ Select a 2-dimensional array GRAF of size n2

construct an auxillary unidimensional array of size m EDGE :For j=1 to mxy being the j th edge of GGRAF [x , y ] = jEDGE [j] = a pointer to the memory word GRAF [x , y ]

◮ The construction of the EDGE array requires O(m) time

◮ Memory used n2 +m ∈ O(n2)



◮ xy ∈ E iff EDGE [GRAF [x , y ]] contains a pointer pointing tothe memory word GRAF [x , y ]

◮ Therefore the query : xy ∈ E ?Can be done in 2 tests O(1).



Any graph solution for the rectangle problem ?


Chordal graphs

A nice graph

Start with the graph of the planar tiling and keep exactly 2integers edges by rectangle.This yields a graph (possibly with parallel edges) in whichall vertices have even degrees except the corners.Take a maximal path starting in one corner ....


Chordal graphs

Lexicographic Breadth First Search LexBFS

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 a 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

2The ordering of the LexBFS search is 7,6,5,4,3,2,1. Note that thereverse ordering is not simplicial, since G is not chordal


Chordal graphs


It is just a breadth first search with a tie break rule.We are now considering a characterization of the

order in which a LexBFS explores the vertices.


Chordal graphs


Property (LexB)

an order σ on V , if a < b < c and ac ∈ E but ab /∈ E , then itexists a vertex d such that d < a and db ∈ E and dc /∈ E .

d cba

Theorem

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


Chordal graphs


4 points condition

Questions◮ Under which condition an order σ on V correspond to some

graph search ?

◮ What are the properties of these orderings ?

Main reference :

D.G. Corneil et R. M. Krueger, A unified view of graph searching,SIAM J. Discrete Math, 22, N 4 (2008) 1259-1276


Chordal graphs


A characterisation theorem for chordal graphs

Theorem

Dirac 1961, Fulkerson, Gross 1965, Gavril 1974, Rose, Tarjan,Lueker 1976.

(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

Simplicial elimination scheme

Simplicial

5

1 4 38

6 7 2

A vertex is simplicial if its neighbourhood is a clique.


σ = [x1 . . . xi . . . xn] is a simplicial elimination scheme if xi issimplicial in the subgraph Gi = G [{xi . . . xn}]

ca b


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


Chordal graphs are hereditary


Chordal graphs


Theorem [Tarjan et Yannakakis, 1984]

G is a chordal graph iff every LexBFS ordering provides a simplicialelimination scheme.

1

1 8

7

6

5

4

32


Chordal graphs


How can we prove such an algorithmic theorem ?

1. A direct proof, finding the invariants ?

2. Find some structure of chordal graphs

3. Understand how LexBFS explores a chordal graph


Chordal graphs


A direct proof

Theorem [Tarjan et Yannakakis, 1984]

G is a chordal graph iff every LexBFS ordering provides a simplicialelimination scheme.


Chordal graphs


Demonstration.

Let c be the leftmost non simplicial vertex.Therefore it exists a < b ∈ N(c) with ab /∈ E . Using LexBproperty, it necessarily exists d < a with db ∈ E and dc /∈ E .Since G is chordal, we have ad /∈ E (else we would have the cycle[a, c , b, d ] without a chord).But then considering the triple d , a, b, it exists d ′ < d such thatd ′a ∈ E and d ′b /∈ E .If dd ′ ∈ E , using the cycle [d , d ′, a, c , b] we must have the chordd ′c ∈ E which provides the cycle [d , d ′c , b] which has no chord.Therefore dd ′ /∈ E .And we construct an infinite sequence of such d and d’.


Chordal graphs


Consequences

G has a linear number of maximal cliques.Computing a maximum clique ω(G ) is polynomial.Computing χ(G ) also


Chordal graphs


Let C(x , y) be the set of maximal cliques that contain x and y.

Clique Consecutivity property

In a lexBFS ordering τ if y is the first vertex after x s.t.(C (x , y)) = {C}, then the elements of C visited after x areconsecutive in τ .


Exercices

Helly Property

Definition

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

Exercise

Subtrees in a tree satisfy Helly property.


Exercices

Classes of twin vertices

Definition

x and y are called false twins, (resp. true twins) ifN(x) = N(y) (resp. N(x) ∪ {x} = N(y) ∪ {y}))

2ème Cours Cours MPRI 2010 2011habib/Documents/Cours_2_2010.pdfthe memory word GRAF[x,y] Therefore...

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