On some simplicial elimination schemes for chordal graphs

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On some simplicial elimination schemes for chordal

graphs

Michel Habib, Vincent Limouzy

To cite this version:

Michel Habib, Vincent Limouzy. On some simplicial elimination schemes for chordal graphs. 2008.

�hal-00353959v2�

hordal graphs

Mi hel Habib

1

LIAFA, CNRSand Université Paris Diderot - Paris7, Fran e

Vin ent Limouzy

2

Dept. of Computer S ien e, University of Toronto,Canada

Abstra t

We introdu ed here an interesting tool for the stru tural study of hordal graphs, namely the Redu ed Clique Graph. Using some of its ombinatorial properties we show that for any hordal graph we an onstru t in linear time a simpli ial elimination s heme starting withapendingmaximal lique atta hed via aminimal separator maximalunderin lusion among allminimal separators.

Keywords: Chordal graphs, minimalseparators, simpli ialelimination s heme, redu ed liquegraph.

1 Introdu tion

Inthe followingtext, agraphisalwaysnite,simple,loopless,undire tedand onne ted. A graph is hordal i it has no hordless y le of length

≥ 4

. The lass of hordal graphs is one of the rst lass to have been studied at

1

Email: habibliafa.jussieu.fr

been intensively studied, as an be seen inthe following books [9,2℄.

Let us re all the main notions dened for hordal graphs. A maximal lique of

G

is a omplete subgraph maximal under in lusion. A minimal separator isasubset ofverti es

S

forwhi h itexist

a, b

∈ G

su h that

a

and

b

arenot onne tedin

G

−S

,and

S

isminimalunderin lusionwiththis prop-erty. A vertex is simpli ialif its neighborhoodis a lique ( omplete graph). An ordering

x

1

, . . . , x

n

of the verti es is a simpli ial elimination s heme, ifforevery

i

∈ [1, n − 1] x

i

isasimpli ialvertexin

G[x

i+1

, . . . x

n

]

. Amaximal lique treeis a tree

T

that satises the following three onditions: Verti es of

T

are asso iated with the maximal liques of

G

. Edges of

T

orrespond to minimal separators. For any vertex

x

∈ G

, the liques ontaining

x

yield a subtree of

T

.

Using results of Dira [5℄, Fulkerson, Gross [6℄, Buneman [3℄, Gavril [8℄ and Rose, Tarjanand Lueker [12℄,we have:

Theorem 1.1 The following 5 statements are equivalent and hara terize hordalgraphs.

(i) G has a simpli ial elimination s heme (ii) Every minimal separator is a lique (iii)

G

admits a maximal liquetree.

(iv)

G

isthe interse tion graph of subtrees in a tree. (v) Any LexBFS provides a simpli ial elimination s heme.

2 The Redu ed Clique Graph

Denition 2.1 For a hordal graph

G

, we denote by

C

the set of maximal liques of

G

and by

C

r

(G)

the redu ed lique graph, i.e. the graph whose verti es are the maximal liques of

G

, and two liques are joined by an edge i their interse tion separates them (i.e. if for every

x

∈ C − (C ∩ C

′

₎

and every

y

∈ C

′

_{− (C ∩ C}

′

₎

,

C

∩ C

′

isa minimalseparators for

x

and

y

in

G

). Clearly

C

r

(G)

isasubgraphoftheinterse tiongraphofthemaximal liques of

G

. Ea h edge

CC

′

of

C

r

(G)

an be labelled with the minimal separator

S

= C ∩ C

′

.

Lemma 2.2 [7℄ Let us onsider three maximal liques

C

1

, C

2

, C

3

in

G

, su h that

S

= C

1

∩ C

2

and

U

= C

2

∩ C

3

are minimal separators in

G

, then

S

⊂ U

implies that

C

1

∩ C

3

is a minimal separator of

G

.

(d)

Fig. 1. An example of a hordal graph (a), its redu ed lique-graph (b), notethat although themaximal liques

{b, d, e}

and

{c, e, f }

interse tthe orrespondingedge is missing. Two maximal lique-treesareshown ( )-(d).

Lemma 2.3 [7℄Letus onsideratrianglein

C

r

(G)

togetherwithits3minimal separators labelling its edges. Then two of these minimal separators must be equal and in luded in the third.

Withthese two lemmas itis easyto prove the followingresult:

Proposition 2.4 [1,7℄ Fora hordalgraph

G

maximal liquetrees orrespond to maximum spanning trees of

C

r

(G)

when the edges are labelled with the size of the minimal separator they are asso iated with. Furthermore

C

r

(G)

is the union of all maximal liquetrees of

G

.

As a onsequen e, all maximal lique trees dene the same multiset of minimal separators, and from one maximal lique tree to another we an pro eed by ex hanging edges (with same label) on triangles. But the graph

C

r

(G)

has still more ombinatorial properties, that we now onsider. Let us now study the limit ase of the two previous lemmas, when

S

= U

. First we need a basi separating lemma (whi h an also be found in a more general setting of tree de ompositions,see lemma12.3.1 in [4℄).

Lemma 2.5 Separating lemma

two onne ted omponentsof

T

− C

1

C

2

. If wedene

V

i

fori=1,2 the unionof allmaximal liquesin

T

i

. Then

S

= C

1

∩ C

2

separates every

x

∈ V

1

− S

from any

y

∈ V

2

− S

.

Lemma 2.6 Let us onsiderthree maximal liques

C

1

, C

2

, C

3

in

G

, su h that

S

= C

1

∩ C

2

= U = C

2

∩ C

3

are minimal separators in

G

, then either the edge

C

1

C

3

∈ C

r

(G)

or the two edges

C

1

C

2

, C

2

C

3

annotbelong both to a same maximal lique tree.

Proof. Suppose that the edge

C

1

C

3

does not belong to

C

r

(G)

, i.e. that

S

=

C

1

∩ C

3

does not separate

C

1

− S

from

C

3

− S

. Therefore if it exists some maximal liquetree

T

ontainingbothedges

C

1

C

2

, C

2

C

3

,thiswould ontradi t the aboveseparating lemma 2.5.

2

Lemma 2.7 Let us onsiderthree maximal liques

C

1

, C

2

, C

3

in

G

, su h that

S

= C

1

∩ C

2

= U = C

2

∩ C

3

are minimal separators in

G

, if the edges

C

1

C

2

, C

2

C

3

belongboth to a samemaximal liquetree

T

. Then

C

1

C

3

∈ C

r

(G)

and

C

1

∩ C

3

= U

Proof. Using the previous lemma ne essarily

C

1

C

3

∈ C

r

(G)

, but lemma 1 just states that

C

1

∩ C

3

⊆ U = S

. If this is a stri t in lusion then one an build a new maximal lique tree

T

′

by ex hanging the edges

C

1

C

2

by

C

1

C

3

. But then

T

′

would be a better spanning tree than

T

whi h ontradi ts the optimalityof

T

and therefore

C

1

∩ C

3

= U = S

.

2

3 Min-max separators

For anite hordalgraph

G

,letus alla min-max(resp. min-min)separator

S

,aminimalseparatorthatismaximal(resp. minimal)underin lusionamong all minimalseparators of

G

.

Theorem 3.1 [10℄ Let

G

be a hordalgraph, thenit existsa maximal lique-tree

T

that admits a pending edge labelled with a min-maxseparator.

Proof. The proof will pro eed by transforming a maximal lique tree using the above lemmas. Let us onsider

T

a maximal lique tree of

G

and some edge

ab

∈ T

labelledwith a min-maxseparator

S

. First weneed to denean operationon liques trees, namelythe hain-redu tion. Suppose

ab

is nota pendingedge in

T

, therefore

T

− {ab}

isthe disjoint unionof twonon empty trees

T

a

, T

b

. Ifone ofthese trees,say

T

a

admitsalead edge

xy

labelledwitha minimal separator

S

′

_{⊂ S}

(

y

being the pending lique in

T

). Then the whole hain in

T

a

joining

ab

to

xy

is labelled with minimal separators ontaining

S

′

. Using this fa t and su essive appli ations of the above lemmas, we an inter hangein

T

a

the edges

xy

and

ay

(orequivalentlyin

T

ex hanging

xy

by

by

). Let usgo ba k to the proof of the theorem. Ifone of the subtrees

T

a

, T

b

, say

T

a

is made up with edges labelled with minimal separators in luded in

S

, thenusingthe hain-redu tion operationwe an produ eanothermaximal lique tree

T

′

in whi h allthe edges of

T

a

are leaves atta hed to

b

and

ab

isa leaf and we have nished. Else it exists in one of the subtrees

T

a

, T

b

, say

T

a

, some edge

zt

labelled with

S

′

whi h is not omparable with

S

. We re urse on the maximal minimal separator that ontains

S

′

and whi h ne essarily belongsto

T

a

. Thispro essne essarilyendsby ndingaleafinthe treewhi h islabelledwithamax-minseparator, be auseea htimewere urseonastri t

subtree.

2

Su h maximal lique trees seem to play an important role for the study of path graphs [10℄. The above proof alsosuggests a dual result for min-min separators. But as it was noti ed by M. Preissmann [11℄, su h a maximal lique tree does not always exist. The graph depi ted in gure 2 does not admit a min-mineliminations heme.

Fig. 2. Preissmann's ounter example [11℄, A graph, its redu ed lique graph and one maximal liquetree

Usingthe above onstru tiveproof, apolynomials heme an be obtained to ompute a min-max eliminations hemes. As shown in Figure 3, lassi al graph sear hes donot provide su h eliminations heme.

Fig. 3. An exemple of graph on whi h

M CS

,

LexBF S

fail to nd a max-min simpli ial vertex. For anystarting vertex, both sear heswill end on

e

of

f

.

Proof. Weprovetheresultinthemin-max ase. Toobtainsu hatreewe an rst ompute a maximal lique tree

T

of

G

as explainedin[7℄, with itsedges being labelled with the minimal separators of

G

. We an sort the minimal separators with respe t to their size in linear time, and therefore start with anedge

ab

labelledwithamax-minseparator

S

andthen explore

T

a

and stop eitherbe ausethewholesubtreeislabelledwithminimalseparators ontained in

S

, then it su es to modify the tree, or be ause we have found an edge labelledwithsomeedge

xy

labelledwithaminimalseparator

S

′

in omparable with

S

. In this ase, amongalledgesin

T

a

, onsider theedge

zt

labelledwith a min-maxseparator

S

′′

in omparablewith

S

, and re urse on

zt

. During this algorithm an edge of

T

is at most traversed twi e, whi h yields the linearity

of the wholepro ess.

2

Corollary 3.3 For any hordal graph there exist an elimination s heme that follows a linear extension of the ontainment ordering of the minimal separa-tors. It an be omputed in

O(n.m)

.

Proof. It is well-known, that one an produ e elimination s heme on the following way. Take any maximal lique tree

T

of a hordal graph

G

, and let

C

be a leaf of this tree, atta h to the tree via the minimal separator

S

. Su essively prune all verti es in

C

− S

and re urse on

T

− C

the maximal lique tree of

G

− {C − S}

. To nish the proof it su es toapply the above theorem. Ea h time the above algorithm is applied requires

O(n + m)

, this

yields the omplexity.

2

It should be noti ed that not every linear extension of the ontainment ordering an be obtained with aneliminations heme.

4 Reversible elimination s hemes

A reversible elimination s heme is just an ordering of the verti es whi h is simpli ialin both dire tions. As shown by the graph alled 3-sun, thereexist graphsforwhi hone anprovethatthere isnoreversibleeliminations heme. A vertex is said to be bisimpli ial if its neighbourhood an be partionned intotwo liques. Furthermore, if a graph

G

admits su h a reversible elimina-tion s heme, this implies that ea h vertex is either simpli ial or bisimpli ial. Therefore su ha graph annot ontain any law (

K

1,3

)as subgraph.

Proof. Letus onsider aunitintervalgraph

G

and oneofitsunitary interval representation. Therefore to ea h vertex

x

∈ G

we an asso iate an interval

I

(x) = [lef t(x), right(x)]

of length one of the real line, su h that

xy

is an edge i

I(x) ∩ I(y) 6= ∅

. Let us onsider the total ordering

τ

of the verti es of

G

dened as follows:

x

≤

τ

y

i

(right(x) < right(y))

. Let

x

be the rst vertex of this ordering, learly its neighborhood is a lique. Thus

τ

is an eliminations heme. Reversibility is straightforward. Conversely let us pro eedby ontradi tion. Letusassumethat

G

admitsareversibleelimination ordering and that

G

is not a proper intervalgraph. Asproper intervalgraph admita hara terizationby forbiddenindu ed subgraphs,we anassumethat our graph ontains one of the graph as a subgraph. The forbidden sugraphs for proper interval graphs are the net, the law and the sun of size

3

. These graphsaredepi tedingure4. Sotoproveour laimitissu ienttoseethat noneof thesegraphsadmitareversibleeliminationordering. Forthe law, we already noti edit. Consideringthe 3-sun, itis easyto he k thatea h vertex is bisimpli ial. Ifwe onsider the sugraph indu ed by

{a, b, c, d, e}

,this graph formsthe bull. Andthisgraphadmit onlyone reversible eliminationordering whi h is

a, b, d, c, e

. To onvin e ourself

a

and

e

has to be the extremities of the ordering(

d

is not a good andidate sin e it is not simpli ialin the whole graph). Then tosatisfy

b

,sin e

a

isalreadypositionned

c

and

d

have tobeon the right. In the same way tosatisfy

c

,sin e

e

is already positionned

b

and

d

have to be on the left. Finally the only ordering tofullll all the onstraints is

a, b, d, c, e

. Butnow, when wewantto add

f

, ea h positionin the previous order will violate the onstraint for at least one vertex. A ontradi tion. For the net, the proof issimilar.

2

(a) Claw:

K

1,3

(b)3-Sun ( )

net

Fig.4. Forbidden indu edsubgraphs for properintervalgraphs.s

[1℄J.R.S.BlairandB.Peyton. Anintrodu tionto hordalgraphsand liquetrees. GraphTheory andSparse MatrixMultipli ation,pages 129,1993.

[2℄A. Brandstädt, V.B. Le, and J.P. Spinrad. Graph Classes: A Survey. SIAM Monographs onDis . Math.and Appli ., 1999.

[3℄P.Buneman. A hara terization ofrigid ir uit graphs. Dis rete Math., 9:205 212,1974.

[4℄R.Diestel. Graph Theory. Springer Verlag,1997.

[5℄G.A.Dira . Onrigid ir uitgraphs. Abh.Math.Sem.Univ.Hamburg,25:7176, 1961.

[6℄D.R.FulkersonandO.A.Gross. In iden ematri esandintervalgraphs. Pa i J.of Math., 15:835855, 1965.

[7℄P. Galinier, M. Habib, and C. Paul. Chordal graphs and their lique graphs. In Springer-Verlag, editor, 21th Workshop on Graph-Theoreti Con epts in Computer S ien e, Aa hen, Le ture Notes in Computer S ien e 1017, pages 358371, 1995.

[8℄F. Gavril. The interse tion graphs of a path in a tree are exa tly the hordal graphs. J.Combinatorial Theory,16:4756,1974.

[9℄M. C. Golumbi . Algorithmi Graph Theory and Perfe t Graphs. A ademi Press, NewYork,1980.

[10℄B. Lévêque, F. Maray, and M. Preissmann. Chara terizing path graphs by forbidden indu edsubgraphs. Journal of Graph Theory,Toappear, 2008. [11℄M. Preissmann, 2009. private ommuni ation.

[12℄D.J. Rose, R.E. Tarjan, and G.S. Lueker. Algorithmi aspe ts of vertex elimination on graphs. SIAMJ. of Computing,5:266283,1976.