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2eme Cours Cours MPRI 2010–2011
2eme CoursCours MPRI 2010–2011
Michel [email protected]
http://www.liafa.jussieu.fr/~habib
Chevaleret, septembre 2010
2eme Cours Cours MPRI 2010–2011
Schedule
Comments on last course
Chordal graphsLexicographic Breadth First Search LexBFSSimplicial elimination scheme
Exercices
2eme Cours Cours MPRI 2010–2011
Comments on last course
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.
2eme Cours Cours MPRI 2010–2011
Comments on last course
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)
2eme Cours Cours MPRI 2010–2011
Comments on last course
◮ 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).
2eme Cours Cours MPRI 2010–2011
Comments on last course
Any graph solution for the rectangle problem ?
2eme Cours Cours MPRI 2010–2011
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 ....
2eme Cours Cours MPRI 2010–2011
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
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Lexicographic Breadth First Search LexBFS
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
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Lexicographic Breadth First Search LexBFS
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.
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Lexicographic Breadth First Search LexBFS
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).
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Lexicographic Breadth First Search LexBFS
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
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Lexicographic Breadth First Search LexBFS
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
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Simplicial elimination scheme
Simplicial
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
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Simplicial elimination scheme
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 .
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Simplicial elimination scheme
Chordal graphs are hereditary
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Simplicial elimination scheme
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
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Simplicial elimination scheme
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
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Simplicial elimination scheme
A direct proof
Theorem [Tarjan et Yannakakis, 1984]
G is a chordal graph iff every LexBFS ordering provides a simplicialelimination scheme.
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Simplicial elimination scheme
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’.
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Simplicial elimination scheme
Consequences
G has a linear number of maximal cliques.Computing a maximum clique ω(G ) is polynomial.Computing χ(G ) also
2eme Cours Cours MPRI 2010–2011
Chordal graphs
Simplicial elimination scheme
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 .
2eme Cours Cours MPRI 2010–2011
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.
2eme Cours Cours MPRI 2010–2011
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}))