Graph Extremities Defined by Search Algorithms

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Graph Extremities Defined by Search Algorithms

Anne Berry, Jean Blair, Jean-Paul Bordat, Geneviève Simonet

To cite this version:

Anne Berry, Jean Blair, Jean-Paul Bordat, Geneviève Simonet. Graph Extremities Defined by Search

Algorithms. [Research Report] 05055, LIRMM. 2005, 17 p. �lirmm-00106696�

A. Berry  J. Blair y J.-P. Bordat z G. Simonet z Abstract

Graph search algorithms have exploited and in some cases identied graph extremities, such as the leaves of atree and the simplicial vertices of a chordal graph. Recently, several well-knowngraph search algorithms have beencollectivelyexpressed astwogeneric algorithmscalledMLSand MLSM,eachofwhichinstantiateswithparametersdeningthesetoflabels to be used, apartial orderon labels usedto choose at each stepa vertex ofmaximallabelandthewayalabelisinitializedandincremented. Inthis paper, we investigate the properties of the vertex that is numbered 1 by MLS ona chordal graphand by MLSM onanarbitrary graph. We show thattheminimalseparatorsincludedintheneighborhoodofthisvertexare totally orderedby inclusionandthat thisvertexis indeedanextremity,as itbelongstoamoplexofthegraph. Whentheorderonlabels istotal,this extremitypropertyis stillstronger sinceall thevertices ofthis moplexare numberedconsecutively. WhenMLSisrunonanon-chordalgraphwitha totalorderonlabels,vertexnumber1isaweakerkindextremity,calledan OCF-vertex.

1 Introduction

Variouspropertiesthatidentifyavertexasoutermostorasanextremityofagraph havelongbeen exploited in both graphtheory and the designof eÆcient graph algorithms. Theendpointsof apath, forexample,are itstwoextremities;leaves arethe extremities of atree. Because this simplenotion hasproved veryuseful indealingwithtrees,graphtheoristshaveendeavoredtoextendittowidergraph classes.

For chordalgraphs, extremities were dened as thesimplicial vertices (a vertex is simplicial if its neighborhood is a clique), concurrently by Dirac [9] and by LekkerkerkerandBoland[19]. ThisconceptledtoeÆcientrecognitionalgorithms forchordalgraphs,basedonthecharacterizationofFulkersonandGross[14],who showed that a graph is chordal if and only if it has a peo (perfect elimination ordering). Apeoisanorderingoftheverticesgivenbythesimplicial elimination scheme,whichrepeatedlyndsasimplicialvertexandremovesitfromthegraph. This was eÆciently implemented by Algorithm LexBFS, introduced by Rose, TarjanandLueker[21],whichndsapeoin asinglelinear-timepassiftheinput graph is chordal, numbering the vertices from n to 1. Thus LexBFS ends on a simplicial vertex. Tarjan and Yannakakis [22] later simplied Lex BFS into AlgorithmMCS.



LIMOS,EnsemblescientiquedesCezeaux,F-63177Aubiere,France. berry@isima.fr y

Dept. of EE & CS, United States Military Academy, West Point, NY 10996, USA. jean.blair@usma.edu

z

andGoss[16]denedanedge-eliminationforthesubclassofweaklychordalgraphs calledchordalbipartitegraphs. ThiswaslaterextendedbyHayward,Spinradand Sritharan [17] to an elimination scheme involving extremities of weakly chordal graphscalledco-pairs. Dahlhaus,Hammer,MarayandOlariu[13]usedMCSto ndadominationeliminationordering onHHD-free graphs. OnAT-freegraphs, Corneil, Olariu and Stewart [11] dened dominating pairs of vertices, and used LexBFSto ndsuchapaireÆciently[12], asthevertexnumbered1byLexBFS belongstoadominatingpair,andasecondpassofLexBFSwillndasecondsuch vertex.

Onarbitrary graphs,workwasdoneon computinga minimal triangulation of a graph(a chordalgraphobtainedfrom thisgraphbyaddinganinclusion-minimal setofedges). Ohtsuki,CheungandFujisawa[20]usedwhatiscalledthesimplicial elimination game,which simulates asimplicial eliminationscheme byrepeatedly choosingavertex,addingeveryedgewhoseabsenceviolatesthesimpliciality condi-tionandremovingitfromthecurrentgraph. Thegraphobtainedfromtheoriginal graphGbyaddingalledgesaddedinthisprocessischordal,and[20]showedthat itisaminimal triangulationof Gifand onlyifnoother orderingof thevertices canproducebythesimplicialeliminationgame astrictly inclusion-smallerset of addededges. Theycalledsuchanorderinganmeo(minimaleliminationordering) ofG,andshowedthatanorderingisameoifandonlyifateach stepofthe sim-plicialeliminationgame,aspecialvertex(whichwecallanOCF-vertex)ischosen. ThusOCF-verticescanbeviewedasextremitiesofanarbitrarygraph.

Algorithm LEX M, introduced in the same seminal paper as LexBFS [21] (LexBFS is astreamlinedversionof LEXM), nds anmeoeÆciently, sovertex number1of LEX M is always anOCF-vertex. LEXM was extendedto Algo-rithmMCS-M byBerry, Blair,Heggernesand Peyton[1]using thesamelabel simplicationasMCStocomputeanmeoeÆciently.

BerryandBordat[2]dened anewkindofextremitybyreningthenotionof a simplicialvertex into that of asimplicial moplex. This strengthened the notion of simplicial vertex in a chordal graph, as any vertex of a simplicial moplex is simplicial,but somesimplicialverticesmaynotbelong to anysimplicialmoplex. Inthesameway,thenotionofmoplexstrengthensthenotionofOCF-vertexinan arbitrarygraph,asanyvertexofamoplexisanOCF-vertex,whereassome OCF-vertices may notbelong to anymoplex. Thus emerged the notionsof weak and strong extremities ofa graph: the weak extremities areOCF-vertices, which are exactlythesimplicialverticeswhenthegraphischordal,andthestrongextremities arethe verticesbelonging to amoplex, which are exactlythe verticesbelonging to a simplicialmoplex when the graph is chordal. Blair and Peyton [8] studied AlgorithmMCSonachordalgraphinrelationwiththecliquetreerepresentation ofchordalgraphs. Thoughnotexplicitlystatedin [8], itfollowsfromtheir work that MCSrunonachordalgraphendson amoplex, i.e. numbertheverticesof amoplexlastwiththesmallestconsecutivenumbers. Inanarbitrarygraph,LEX MwasshownbyBerryandBordattoendonamoplex[3],whereasMCSendson anOCFvertexthatdoesnotnecessarilybelongtoamoplex[1]. Anotherrelevant resultisthat LexBFSalwaysendsonamoplex,evenonanon-chordalgraph[2]. Corneiland Krueger [10] introduced MNS (Maximal Neighborhood Search), as analgorithm that encompassesboth LexBFSandMCS, and showed that italso computesapeoifthegraphischordal. Recently,Berry,KruegerandSimonet[7] extended the family of search algorithms by dening ageneric algorithm MLS (MaximalLabelSearch). AlgorithmMLShastwoinputvariables: agraphanda labelling structure containing in particular aset of labelsand apartial order on

aspeciclabellingstructure. [7]showedthatthesetoforderingsoftheverticesof agivengraphcomputablebyMLS(withallpossiblelabellingstructures)isequal totheset oforderingscomputablebyMNS, which ensuresthat MLS ndsapeo if the graph is chordal. Its corresponding extension MLSM nds an meo and generalizesbothLEXMandMCS-M.

Inthispaper,weinvestigatethepropertiesofthevertexnumbered1byMLSand MLSM,inordertodetermineinwhat measurethesemoregeneralgraphsearches preservesomeof thestrong extremalproperties exhibitedby MCS,LexBFSand LEXM.Weshowthat indeedtheydo. Figure1summarizestheextremity prop-ertiesofAlgorithmsMLSandMLSM.Itpresentsknownresultsdescribedin this Sectionaswellasnewonesgiveninthis paper.

MLS with total order

LexBFS

MCS

whole moplex

MLS chordal graph

MLS=MNS

MLSM arbitrary graph

MLSM=MNSM

MLSM with total order

LEX M

MCS−M

MLS non−chordal graph

MLS=MNS

MLS with total order

whole moplex

LexBFS

MCS

OCF−vertex

not always an extremity *

OCF−vertex *

vertex of a moplex *

whole moplex *

whole moplex

whole moplex *

vertex of a moplex *

whole moplex *

whole moplex *

Figure1: Extremitypropertiesof AlgorithmsMLS andMLSM.These properties concern the vertex numbered 1 by the algorithms: it is an OCF-vertex (weak extremity),avertexofamoplex(strongextremity)oravertexofamoplexwhose vertices are numbered consecutively, denoted by 'whole moplex'. The equality MLS=MNS (resp. MLSM=MNSM)meansthat bothalgorithms computethe sameorderingson anygraph, andtherefore havethesameextremity properties. 'withtotalorder'means'withtotalorderonlabels'. Newresultspresentedinthis paperareindicatedby*.

Thepaperis organizedas follows: in Section2wegivesomenotationsand de-nitions,aswellassomeusefulpreviousresults. Section3describestheproperties oftheneighborhoodofvertex1. Section4describesthekindofgraphextremities whichvertex1belongstoforMLSandMLSM,anddiscussesthespecialproperties oftherelatedordering.

2 Preliminaries

2.1 Basic denitions and notations

All graphs in this work are undirected and nite. The readeris referredto the classicalworkof[15]foranydenitionsnotincludedinthissection. Thesubgraph ofGinducedbythesubsetAof verticesisdenotedG(A). Theneighborhood ofa vertexxinGisdenotedN

G

(x). TheclosedneighborhoodisN G

[x]=fxg[N G

(x). TheneighborhoodofasetofverticesAisN

G (A)=[ x2A N G (x)nA,andN G [A]= A[N G

(A). Wewill omitsubscript G whenthe graphweworkon isclear from thecontext.

simplicial ifitsneighborhoodisaclique. Amodule isasubsetX ofverticeswhich sharethesameexternalneighborhood: 8x;y2X;N(x)nX=N(y)nX,sothata cliquemoduleisasubsetX ofverticeswhichsharethesameclosedneighborhood: 8x;y2X;N[x]=N[y].

AnorderingonthesetV ofverticesisaone-to-onemappingfromf1;2;:::;ngto V. In everygureof thispapershowinganorderingonV, everyvertexx will be named by itsnumber

1

(x). Be carefulthat all algorithms considered here numbertheverticesfrom ndown to1, sotherst numberedvertex is(n) and thelast numberedoneis(1).

A chordal (or triangulated) graph is a graph with no chordless cycle of length greater or equal to 4. The notions of simplicial elimination scheme and game, minimal triangulation, peo and meohave been dened in Section 1. The graph obtainedbyrunningsimplicialeliminationgameongraphG,choosingthevertices accordingtoordering,isdenotedG

+  .

Itremainsto denethegraphextremitieswewill workon, namelyOCF-vertices andmoplexes.

Denition2.1 A vertex x in G = (V;E) is an OCF-vertex of G if, for any non-adjacent distinct neighbors y;z of x, there is a connected component C of G(V nN[x]) suchthaty andz belong toN(C).

AsubsetS ofverticesisaminimal separator ithere areat leasttwoconnected components of G(V nS) whose neighborhood in G is S (see [5] for extensive denitionsonminimalseparation).

Denition2.2 (Berry, Bordat [2])A moplexofagraphGisacliquemodule of Gwhoseneighborhoodisaminimal separator.

2.2 Algorithms MLS and MLSM

ThedenitionsofAlgorithmsMLSandMLSMarebasedonthenotionoflabelling structure.

Denition2.3 A labelling structureisastructure(L;;l 0

;Inc),where:  Lis aset(thesetof labels),

  is apartial order on L (which may be total or not,  denoting the cor-responding strict order), which will be usedto choose a vertex of maximal label,

 l 0

isanelementof Lwhich will beusedtoinitialize thelabels,

 IncisamappingfromLf2;3;:::;ngtoL,whichwillbeusedtoincrement alabel, andsuchthatforany integerifrom2tonandany labelslandl

0 in L

i

,the followingpropertieshold: (ls1)lInc(l;i) (ls2)if ll 0 thenInc(l;i)Inc(l 0 ;i) whereL i

isthe subsetofL denedby inductionby: - L

n =fl

0

g,andfor anyifrom ndown to2 - L i 1 =L i [fl2L j9l 0 2L i j l=Inc(l 0 ;i)g.

input :AgraphG=(V;E)andalabellingstructure L=(L;;l 0

;Inc). output:AnmeoonV andaminimaltriangulationH=G

+ 

ofG. Initializealllabelsasl

0 ;E 0 E;G 0 G; fori=ndownto 1do ChooseavertexxofG 0 ofmaximallabel; (i) x; foreachvertexy of G 0 dierent fromxdo if there isa path from x to y in G

0

suchthat every internalvertex on the pathhas alabelstrictlysmaller thanlabel(y)then

E 0

E 0

[fxyg;

foreachy inV suchthatxy2E 0

do label(y) Inc(label(y);i); Removex fromG 0 ; H (V;E 0 ); Figure2: AlgorithmMLSM.

Algorithm,MLSM,isgiveninFigure2.

AlgorithmMLScanbestreamlinedfromMLSM,exactlyas[21]streamlinedLEX M into LexBFS, by removing the rst foreach loop. Thus variables E

0

and H become useless (E

0

= E and H = G) and the second foreach loop becomes: foreach neighbory ofx inG

0

do. MLScomputesapeoofanychordalgraph. Whenusing labelling structure Lwith MLS (resp. MLSM),wewill referto the algorithmas L-MLS (resp. L-MLSM). LexBFS,MCSandMNS are instancesof algorithmL-MLSandLEXM,MCS-MandMNSM(denedin[7]fromMNS)are instancesofalgorithmL-MLSM,withthefollowingrespectivelabellingstructure L=(L;;l

0 ;Inc):

LexBFS and LEX M:Listheset oflists ofelementsof f2;3;:::;ngin strictly decreasingorder, is lexicographical order (a total order), l

0

is theempty list, Inc(l;i) is obtainedfrom l by adding i to theend of thelist, provided that i is strictlysmallerthanthelastintegerinl.

MCS and MCS-M: L = f0;1;2;:::;n 1g,  is  (a total order), l 0

= 0, Inc(l;i)=l+1.

MNSand MNSM:Listhepowersetoff2;3;:::;ng,is(notatotalorder), l

0

=;,Inc(l;i)=l[fig.

Wecall MLS (resp. MLSM, L-MLS, L-MLSM) ordering of a graphG=(V;E) anyorderingonV that canbecomputedbythegivenalgorithm. Thereadercan easilyconvincehimself ofthefollowingPropertyprovedin[7].

Property 2.4 ([7 ] ) Forany graphG andany labelling structureL, Algorithms L-MLSM computing ordering  on G has the same behavior as L-MLS on G

+  (theygivethesamelabelsandnumberstovertices,providedthattheybreak tiesin the sameway).

Property2.4hastwoconsequencesthatwillbelargelyused inthispaper.

1. AnyL-MLSMorderingofGisaL-MLSorderingofG + 

. Thusaproperty of MLS on chordalgraphscan in somecasesbeextended to apropertyof

MLSM on arbitrary graphssince G 

is chordal (Corollaries 3.10and 3.11 andTheorem4.5).

2. MLS andMLSM havethesamebehavioron achordalgraph,sincein that case, G

+ 

=G. Any execution of MLS on achordal graphcan be seen as an execution of MLSM, and conversely. Thus any property of MLSM on arbitrarygraphsalsoholdsforMLSonchordalgraphs(Corollary4.3).

3 The neighborhood of vertex 1

3.1 Super-components and slices

Inthis Section, we will studysomepropertiesof theneighborhood ofthe vertex numbered1byMLSorMLSM,whichwillleadtoextremitypropertiesinSection4.

Notations3.1 Forany graphG=(V;E)andany orderingonV, -acomponent of(G;)isaconnectedcomponentof G(V nN((1))),

-asuper-componentof (G;) istheunion ofallcomponents of(G;)having the sameneighborhood assomecomponentof (G;),

-pdenotesthenumberofsuper-componentsof(G;)and(D 1 ;D 2 ;:::;D p )denotes the sequenceof the super-components of (G;) orderedas follows: 8i2[1;p 1],  1 (d i ) > 1 (d i+1 ), where d i

denotes the vertex of D i

with maximum number ( 1 (d i )=maxf 1 (d);d2D i g), -theslicesof(G;)arethesubsetsT

1 ,T 2 ,...,T p ofV partitionningV asfollows: 8i2[1;p],T i =D i [S i ,whereS i =N(D i )n([ 1j<i N(D j ))=N(D i )n([ 1j<i S j ), andt i

denotes the vertexof T i

withmaximum number.

We say that  numbers the slices in increasing order if 8i 2 [1;p 1], 8t2 T i ,  1 (t)> 1 (t i+1 ).

The partition of V in slices can be rened into itspartition in thin slices, where thethinslicesof(G;)aredenedfromthecomponentsof(G;)inthesameway asthe slices aredenedfrom thesuper-components of(G;).

Example3.2 Figure 3 shows the slices dened by an execution of MCS on a chordal graph. N((1))=f8;3;2gand the components of (G;) are f7;5g,f6g, f4g and f1g, with N(f7;5g) = N(f6g) = f8g, N(f4g) = f8;3g and N(f1g) = f8;3;2g,sop=3and -D 1 =f7;5g[f6g=f7;6;5g,d 1 =7,S 1 =f8g,T 1 =f8;7;6;5gandt 1 =8, -D 2 =f4g, d 2 =4,S 2 =f3g, T 2 =f4;3gandt 2 =4, -D 3 =f1g, d 3 =1,S 3 =f2g, T 3 =f2;1gandt 3 =2. Weobserve that: -N(D 1 )N(D 2 )N(D 3 ),

-numberstheslicesinincreasing order(butnotthethinslices, sincesliceT 1

is partitionnedintothe thinslices T

0 1 =f8;7;5gandT 0 2 =f6g).

Wewillseehowthese propertiescanbegeneralizedtoMLS andMLSM orderings.

We will use the following notations and Lemmas, which are proved in the Ap-pendix.

Notations3.3 ForanyexecutionofMLS onanygraphGcomputingordering andanyverticesx andy of Gsuchthat 

1 (x) 1 (y), l x

(y)denotes the labelofy justbefore numbering x, NN

x

4

1

8

2

3

6

5

7

T1

T2

T3

S

Figure3: Theslices denedbyanexecutionofMCSonachordalgraph.

NN x (y)=fz2N(y)j  1 (z)> 1 (x)g.

Lemma3.4 ForanyMLS orderingofachordalgraph, 8i2[1::p], NN d i (d i )= NN di ((1))  N(D i ), NN ti (t i ) = NN ti ((1)), and if i < p then  1 (t i )   1 (d i )> 1 (t i+1 ).

Lemma3.5 ForanyMLS orderingofachordal graph, 8i2[1::p 1];8s2S i ,  1 (s)> 1 (t i+1 ).

Lemma3.6 ForanyMLSorderingofachordalgraph,ifthe orderonlabelsis totalthen8i2[1::p 1],8d2D i , 1 (d)> 1 (t i+1 ).

WehavethefollowingProperty.

Property 3.7 For any chordal graph G and any MLS ordering  of G 8i 2 [1;p 1],N(D

i

)N(D i+1

).

Proof: LetGbeachordalgraph,letbeaMLSorderingofGandi2[1;p 1]. Letus showthatN(D

i )N(D i+1 ). Lets2N(D i

),andletji suchthats2S j . ByLemmas3.4and3.5, 1 (s)>  1 (t j+1 )   1 (t i+1 )   1 (d i+1 ). So s 2 NN di+1 ((1)) and by Lemma 3.4 s2N(D i+1 ). ThusN(D i )N(D i+1 )and,asN(D i )6=N(D i+1 )bydenitionof thesequence(D i ),N(D i )N(D i+1 ). 2

ForanygraphGandanyorderingonV,theminimalseparatorsofGincluded in N((1)) are exactly the neighborhoods of the componentsof (G;) dierent from f(1)g([4]), which are also theneighborhoods of the super-componentsof (G;)dierentfrom f(1)g. Sowehavethefollowingresult.

Theorem3.8 Foranychordal graphGandany MLSorderingofG, the min-imalseparatorsof Gincludedin N((1)) aretotally orderedbyinclusion.

onlabelsis total thenany L-MLS ordering of Gnumbers the slices in increasing order.

ByProperty2.4 any L-MLSM ordering  of any graphG is a L-MLS ordering ofG

+ 

,whichveriestheprecedingneighborhoodpropertiessinceG + 

ischordal. Moreover, the neighborhood of (1)is the same in G and G

+  (by denition of G + 

)and sinceisameoofG,the componentsof(G;)and of(G + 

;) arethe samewith thesameneighborhoods(this followsforinstancefrom Theorem5.12, Invariant4.9andLemma6.2in[5]). Thus thesuper-components,their neighbor-hoodsandtheslicesof(G;)andof(G

+ 

;)arethesame,andTheorems3.8and 3.9extendto MLSMonanarbitrarygraph.

Corollary3.10 Forany graphG andany MLSM ordering of G, the minimal separatorsofGincludedinN((1))aretotallyorderedby inclusion.

Corollary3.11 For any graph Gand any labelling structure L, if the order on labels is total then any L-MLSM ordering of G numbers the slices in increasing order.

If the order on labels is not total, an L-MLS ordering  does not necessarily numberslicesin increasingorder. Figure4givesacounterexampleforthis.

1

2

3

5

6

4

T1

T2

Figure 4: This MNS ordering doesnot numberslices in increasing order onthis chordalgraph.

3.2 MLS on a non-chordal graph

WewillendthissectionbyafewremarksonrunningMLSonanon-chordalgraph. [2]showedTheorems3.8and3.9foranexecutionofLexBFSonanarbitrarygraph, withtheevenstrongerpropertythatitnumbersthethinslicesinincreasingorder insteadoftheslices.

Property 3.12 ([2 ])ForanygraphG(chordalornot)andanyLexBFSordering  of G, the minimal separators of G included in N((1)) are totally ordered by inclusion,andnumbersthe thinslicesinincreasing order.

Upon investigation, it turns out that in a non-chordal graph, MLS does not in generalexhibit theseproperties,evenwhentheorderistotal. Figure5givesa counter-exampleofthis.

1

3

4

2

8

5

7

6

Figure5: MCS onanon-chordalgraph. Thesuper-componentsof(G;)(which are also itscomponents), are: D

1 = f5;7g, D 2 = f6gand D 3 =f1g; N(D 1 ) = f2;8g, N(D 2 ) = f3;8g and N(D 3

) = f2;3;4;8g. The neighborhoods are not pairwise inclusive, does notnumberthe slices in increasing order,vertex(1) doesnotbelongtoamoplexofG. However,(1)isanOCF-vertexofG.

4 Extremities

4.1 The extremities dened by MLS and MLSM

ForanyMLSMorderingofagraphG,sinceisanmeoofG,by[20](1)isan OCF-vertexof G(weakextremity). Toshowthat (1)isastrongextremity,i.e. avertexofamoplexofG,weneedonlyusethepropertiesoftheneighborhoodof (1)describedin Section3andthefollowingLemmaprovedin theAppendix.

Lemma4.1 For any graph G andany vertex x of G, x isa vertexof amoplex ofGix isanOCF-vertexof Gandtheset ofminimal separatorsofGincluded inN(x)has agreatestelement S

max

for inclusion. Inthat case,the moplex containing xisN[x]nS

max .

We will also show that if moreover the order on labels is total then the ver-tices of the moplex X containing (1) are numbered consecutively, i.e. X = f(1);:::::(jXj)g. Wesaythat thealgorithmendsonamoplex.

Theorem4.2 For any non-clique graph G and any labelling structure L, L-MLSM on input graph G ends on a vertex of a moplex. If moreover the order onlabelsistotalthenitends onamoplex.

Proof: Since  is a meo of G (1) is an OCF-vertex of G [20]. Moreover, by Corollary 3.10 the set of minimal separators of G included in N((1)) has a greatest element S max for inclusion. If D p 6= f(1)g then S max = N((1)) elseS max =N(D p 1

)(p>1becauseotherwiseGwouldbereducedtotheclosed neighborhoodofOCF-vertex(1)andwouldbeaclique). ItfollowsbyLemma4.1 that(1)isavertexofamoplexX ofG,withX =N[(1)]nS

max

. SoXiseither equaltof(1)gortoD

p

. ThusiftheorderonlabelsistotalthenbyCorollary3.11 L-MLSMendsonX. 2

AsbyProperty2.4,anychordalgraphhasthesameL-MLSandL-MLSM order-ings,wehavethefollowingCorollary.

Corollary4.3 Foranynon-cliquechordalgraphGandanylabellingstructureL, L-MLSon inputgraphGends onavertexof amoplex. Ifmoreoverthe orderon labelsistotal thenitendson amoplex.

MLSdenesapeoofanychordalgraph,andMLSMdenesanmeoofanygraph. Wecanstrengthenthesepropertiestomoplexorderings,introducedbyBerryand Bordat[3]. Wedene amoplexeliminationschemein achordalgraphby repeat-edlychoosingamoplexof thecurrentgraph, which issimplicialsincethis graph is chordal, and removing it from the current graph, until this graph is aclique. Ifthe graphfails to be chordal, wedene moplex elimination game in the same wayaswedenedsimplicialeliminationgame: wesimulate amoplexelimination schemebyrepeatedlychoosingamoplexofthecurrentgraph,addingeveryedge whoseabsenceviolatesthesimplicialityof thismoplex,andremovingitfromthe currentgraph, until this graphis aclique. Thusweobtainan ordered partition (X 1 ;X 2 ;:::;X k )where X i

is amoplexof thecurrentgraphat stepiifi<k and X

k

isthenalclique. Amoplexordering isanorderingonV thatiscompatible withsuchanorderedpartition,i.e. suchthat8i2[1;k 1],8x2X

i ,8x 0 2X i+1 ,  1 (x)< 1 (x 0 ). Then G + 

isexactlythegraphobtainedfrom Gbyaddingall edgesaddedinthemoplexeliminationgame.

Berry and Bordat [3] showed that any moplex ordering of any graph is a meo of this graph. They also showed that any LexBFSordering of a chordal graph, respectivelyanyLEXMorderingofanygraph,isamoplexordering. Weextend theseresultstoMLSandMLSM.FollowingProperty4.4easilyfollowsfrom Corol-lary4.3,andTheorem4.5followsfromProperty2.4and4.4andfromsomeknown resultsaboutminimaltriangulation. Theirproofsaregivenin theAppendix.

Property 4.4 ForanychordalgraphGandanylabellingstructureL,iftheorder onlabelsistotalthenany L-MLS orderingofGis amoplexorderingof G.

Theorem4.5 For any graph G and any labelling structure L, if the order on labelsistotal thenany L-MLSM orderingofGisamoplex orderingof G.

4.3 MLS on a non-chordal graph

Again,wewill discuss the properties of MLS when runon anon-chordal graph. [2] showedthat even onanon-chordalgraph, LexBFSalwaysends onamoplex. Ingeneral,anMLSexecutiondoesnotendonamoplexanddoesnotevenendon avertexofamoplex,eveniftheorderonlabelsistotal. However,weshowthat iftheorderonlabelsistotalthenMLSendsonaweakextremity.

Theorem4.6 Forany graphG,anylabellingstructureLandanyL-MLS order-ingofG, if theorder onlabelsistotalthen(1)is anOCF-vertexof G.

ThisisthecaseinparticularforMCS(asshownin[1]). TheproofofTheorem4.6 isgivenintheAppendix.

Iftheorderonlabelsisnot totalthenMLSdoesnotnecessarilyendonan OCF-vertex. For instance, on the graphshown in Figure 6,  =(1;2;3;4;5;6) is an MNSorderingofGand1isnotanOCF-vertexofGsince2and4arenon-adjacent and 4does notbelong to the neighborhood of the uniqueconnected component f3;5gofG(V nN[1]).

4.4 The MLS-Terminal Vertex Problem on chordal graphs

Wedene theMLS-Terminal VertexProblem asfollows: given agraphGand a vertexxofG,isxMLS-terminal,i.e. isthereaMLSexecutionwhichgivesxthe

1

3

2

4

6

5

Figure6: MNSonanon-chordalgraphdoesnotendonanOCF-vertex

number1? Wedene theL-MLS-Terminal Vertex Problem in thesamewayfor anylabellingstructureL.

Lanlignel[18]showedthatLexBFS-TerminalVertexProblemisNP-completeeven intherestrictiveclassesofweaklychordalgraphsandHDP(Heriditary Dominat-ingPath) graphs,andleftopenitscomplexityonachordalgraph.

WewillgiveasimplecharacterizationofMNS-terminalverticesinachordalgraph.

Characterization 4.7 ForanychordalgraphGandanyvertexxofG,xis MNS-terminalix issimplicial andthe neighborhoods of the connectedcomponents of G(V nN[x]) aretotally orderedbyinclusion.

The MLS orderings of a graph are exactly its MNS orderings [7], so its MLS-terminalverticesareexactlyitsMNS-terminalverticesandwehavethefollowing Corollary.

Corollary4.8 MNS-Terminal (resp. MLS-Terminal)Vertex Problem is polyno-mialin theclass of chordal graphs.

5 Conclusion

Wehave shown that Algorithms MLS on achordalgraph and MLSM onan ar-bitrarygraphdeneasnumber1verticeswhicharewell-denedasextremitiesof thegraph,withspecial properties.

Whenrunonanon-chordalgraph,thepropertiesof MLSare weaker,exceptfor AlgorithmLexBFS,which standsoutashavingproperties in manywayssimilar to the meo-computingalgorithms such as LEXM, MCS-M, and more generally MLSM.Inviewofthis,perhapsitwouldbeinterestingto investigate experimen-tallywhethertheorderingproducedbyanexecutionofLexBFSonanon-chordal graphdenesatriangulation G

+ 

which is closeto minimal. MLS run ona non-chordal graph with a non-total order on labels does not always end on a weak extremityofthe graph. It is anopen questionwhether it endsin that caseon a stillweakerkindofextremity.

WealsoleaveopenthecomplexityofLexBFS-TerminalVertexProblemonchordal graphs,aswellas the complexityof MLS-TerminalVertex Problemon arbitrary graphs.

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Thesubscriptinnotationsl x

(y)andNN x

(y)willbeomittedwhentheconsidered stepisclearfromthecontext.

 WegivetheproofsoftheLemmasofSection3andofsomeotherusefulones.

Lemma5.1 At any step of an execution of MLS on a graph G, for any unnumberedvertices u;v of G,

(i)if NN(u)=NN(v)thenl(u)=l(v), (ii) ifNN(u)NN(v)thenl(u)l(v).

Proof: ThisLemmaisprovedin[7]. 2

Lemma 3.4 For any MLS ordering  of a chordal graph, 8i 2 [1::p], NN d i (d i ) = NN d i ((1))  N(D i ), NN t i (t i ) = NN t i ((1)), and if i < p then  1 (t i ) 1 (d i )> 1 (t i+1 ). Proof: NN d i (d i )NN d i ((1)) by denition of d i and NN d i (d i ) 6 NN d i ((1)) (otherwise by Lemma 5.1 l d i (d i )l d i

((1)), whichcontradictsthechoiceof d i

at thatstep). Hence NN d i (d i )=NN d i ((1))N[D i ]\N((1))=N(D i ). Wesupposethati<p. Bydenitionofd

i andt i , 1 (t i ) 1 (d i ). Letus showthat  1 (d i )> 1 (t i+1 ). Ift i+1 62N((1))thent i+1 =d i+1 andwe aredonesincebydenition ofd

i , 1 (d i )> 1 (d i+1 ). Otherwise,assume forcontradiction that

1 (t i+1 )> 1 (d i ). Then t i+1 2NN di ((1)), with NN di ((1))N(D i )V nT i+1 ,so t i+1 62T i+1 ,acontradiction. Letus showthat NN

ti (t i )=NN ti ((1)). Since8ji, 8t2T j , 1 (t i )  1 (t j ) 1 (t),NN ti (t i )N(t i )\([ j<i T j ). Since8j<i,N(t i )\D j =; (because T i \N(D j )= ;) NN ti (t i ) [ j<i S j  N((1)). So NN ti (t i )  NN ti

((1)),andweshowtheequalityasaboveford i

. 2

Lemma5.2 IfGisachordalgraph,achordlesspathinGandzavertex of G not belonging to and seeing both extremities of , thenz sees every vertexof .

Proof: It is a well-known property of chordal graphs which follows for instancefrom resultsfrom[6]. 2

Lemma5.3 Let G be achordal graph, let  be a MLS ordering of G, let i2[1::p], s2S i such that  1 (d i )> 1

(s), and let be achordless d i

s-path suchthat every internal vertexof  belongsto D

i

. Then the following property P holds justafter processing d

i

and remains true untilnumbering s.

P : thereisa vertexv of suchthat every vertexof [v;s] isunnumbered, NN((1))NN(v) and8t2[v;s]nfvg,NN(t)=NN((1)).

Proof: Let us show that P holds just after processing d i . NN d i (d i ) = NN d i ((1)) by Lemma 3.4, so NN d i ((1)) N(d i ). Also NN d i ((1))  N(s) since, as  is a peo of G, N((1)) is a clique of G. It follows by

di di NN

di

(t), and as the reverse inclusion holds by denition of d i and S i , NN di (t) = NN di ((1)). After processing d i , NN((1)) is unchangedand P istrue, taking as vertex v the vertexof  next to d

i

(which is theonly vertexofseeingd

i

sinceischordless).

Letusshowthat P ispreservedwhenprocessingsomevertexw, 1 (d i )>  1 (w)> 1 (s).

Firstcase: w62N((1)). Ifnovertexof[v;s]seesw thenw62[v;s]and P remainstruewiththesamevertexv,otherwiseP remainstrue,takingas newvertexv thelast vertexof[v;s]seeingw.

Secondcase: w2N((1)). ws isanedgeof G,asN((1)) isaclique. We getfromedgewsandpath[v;s]awv-pathsuchthateveryinternalvertext isunnumberedandveriesNN(t)=NN((1))NN(v),sobyLemma5.1 l(t)l(v). ByProperty 2.4,thisexecutionofMLSonchordalgraphGcan beseen asan execution of MLSM, accordingto which wv will bean edge of H with H =G

+ 

=G. It followsthat w sees v in G. ByLemma 5.2 w seeseveryvertexof[v;s]. Itfollowsthatwhenprocessingw,wisaddedto NN((1)) aswell as to NN(t)for everyvertex tof [v;s], andP remains truewiththesamevertexv. 2

Lemma3.5ForanyMLSorderingofachordalgraph,8i2[1::p 1];8s2 S i , 1 (s)> 1 (t i+1 ).

Proof: Suppose there is some s 2 S i such that  1 (t i+1 ) >  1 (s). Then by Lemma 3.4  1 (d i ) >  1 (t i+1 ) >  1 (s). Lemma 5.3 ensures the existence of some vertex v such that NN

t i+1 ((1))  NN t i+1 (v), so by Lemma 3.4 NN t i+1 (t i+1 ) NN t i+1 (v)and by Lemma 5.1 l t i+1 (t i+1 )  l t i+1

(v),which contradictsthechoiceoft i+1

atthat step. 2

Lemma 3.6 For any MLS ordering  of a chordal graph, if the order on labelsistotalthen8i2[1::p 1],8d2D

i , 1 (d)> 1 (t i+1 ).

Proof: Suppose there is some d 2 D i such that  1 (t i+1 ) >  1 (d). We can choose d as the rst vertex of D

i numbered after t i+1 . Just be-fore processing t i+1 , l(t i+1

) is maximum (since the order  is total) with l t i+1 (t i+1 )=l t i+1 ((1))byLemmas3.4and5.1,sol t i+1 (d)l t i+1 ((1)). As byLemma3.5,everyvertexofN(D

i )isnumberedbeforet i+1 ,l(d)remains equaltol ti+1

(d)untildisnumbered. Wethusonlyhavetoprovethatl((1)) isincrementedbefore numberingd. This willleadto acontradiction, since (1)isnumberedlast.

If t i+1

2 N((1)), l((1)) is incremented when numbering t i+1 and we aredone. If t i+1 = 2 N((1)), t i+1 = d i+1 . ByProperty 3.7, S i+1 is notempty, so lets 2 S i+1 .  1 (d i+1 )>  1

(s), and by Lemma 5.3, until numberings, there is someunnumberedvertex v such that NN((1))  NN(v), so by Lemma 5.1 l((1)) l(v), hence l(d) l(v). This implies that 

1 (s) > 

1

(d) and asl((1)) is incremented when processing s, it is incremented beforenumberingd. 2

moplexof Gix isanOCF-vertexofGand theset ofminimalseparators ofGincludedinN(x)hasagreatestelementS

max

forinclusion. Inthatcase,themoplexcontainingx isN[x]nS

max .

Proof: Wesupposethat x isavertexofamoplex X ofG. ThenN[x]= N[X]andthereisaconnectedcomponentCofG(Vn N[x])suchthatN(C)= N(X). Since X is a clique module of G, any non-adjacent neighbors of x must belong to N(X) i.e. to N(C), so x is anOCF-vertex of G, and the neighborhood of any connected component of G(V nN[x]) is a subset of N(X)i.e. ofN(C),soN(C)isthegreatestminimalseparatorofGincluded inN(x)forinclusion.

Conversely,wesupposethatxisanOCF-vertexofGandthesetofminimal separatorsofGincludedinN(x)hasagreatestelementS

max

forinclusion. LetX =N[x]nS

max

. Weimmediatelyverifythat X isacliquemodule of G,withN(X)=S

max

. LetC betheconnectedcomponentofG(V nN[x]) such that S

max

=N(C). X and C are distinct connected components of G(V nN(X))such that N(X)=N(C),soN(X)isaminimalseparatorof GandthereforeX isamoplexof G. 2

 WegivetheproofsofProperty4.4andTheorem4.5:

Property4.4ForanychordalgraphGandanylabellingstructureL,ifthe orderonlabelsistotalthenanyL-MLSordering ofGis amoplexordering ofG.

Proof: WehaveonlytopointoutthatbyCorollary4.3anyL-MLSordering of anon-clique chordal graphGends on amoplex X of Gand that the restrictionoftothechordalsubgraphG(V nX)isitselfaL-MLSordering ofthissubgraph,andtheprooffollowsbyinductiononjVj. 2

Theorem4.5ForanygraphGandanylabelling structureL, iftheorder onlabelsis totalthen any L-MLSMordering of Gis amoplexordering of G.

Proof: Let  be a L-MLSM ordering of G. Foranyk from 1to n, let G

k 

denotethecurrentgraphofthesimplicialelimination gamejust before processing(k),letV

k 

beitssetofvertices,i.e. V k  =f(k);:::;(n)g,and  k be the restrictionof to V k 

. To provethat is amoplex ordering of G,itis suÆcienttoprovethatforanyk from1ton suchthat G

k  isnota clique, k endsonamoplexofG k 

. Letk,1kn,suchthat G k 

isnota cliqueandletus showthat

k endsonamoplexofG k  . ByProperty 2.4is aL-MLS orderingof G +  ,so  k is aL-MLS ordering of G +  (V k  ), which is equalto (G k  ) +  k

. As  is ameo of G, its restriction 

k

is ameoofG k 

(otherwise there would existanother ordering k

onV k  thatproducessomebetterll-inthan

k

,andthereforeanordering onV that produces somebetter ll-in than, acontradiction). So,(G

k  ) + k is a minimaltriangulationofG k  andtherefore,asG k  isnotaclique,(G k  ) +  k is notacliqueeither. ItfollowsfromCorollary4.3that 

k endsonamoplex of (G k  ) +  k

, which is also a moplex of G k 

since any moplex of a minimal triangulationofagraphisamoplexofthisgraph[2]. 2

Lemma5.4 We consider aMLS executiononagraph Gcomputing and vertices u;v of G such that 

1 (u) >  1 (v). If l u (v)  l u ((1)) and every neighbor t of v such that 

1 (u) 1 (t)> 1 (v) isa neighbor of (1)thenl v (v)=l v

((1)) andevery neighbor t of(1) suchthat 1 (u)  1 (t)> 1 (v)isaneighbor ofv.

Proof: Ateverystepbetweennumberinguand numberingv, ifthelabel ofvisincrementedthenthelabelof(1)isincrementedtoo,soby(ls1)and (ls2)theinequalityl(v)l((1))ispreserved,andsincel(v)ismaximaljust beforenumberingv,l

v (v)=l

v

((1)). Nowifsomevertextnumberedbefore uandvwasaneighborof(1)butnotofvthentheinequalitywouldbecome l(v)l((1)),whichwouldbepreserveduntilnumberingv,acontradiction. 2

Theorem4.6For anygraphG, anylabellingstructureLand anyL-MLS orderingofG,iftheorderonlabelsistotalthen(1)isanOCF-vertexof G.

Proof: WedenethepartitionofV intosubsetsT 0 i

,1ip 0

,whichisa variantofthepartition ofV intothin slices. Foranyiin [1::p

0 ], -t 0 i =(n)ifi=1,otherwiset 0 i

istherst vertexnumberedafter c 0 i 1 not belongingtoN[C 0 i 1 ], - c 0 i =t 0 i ift 0 i 62 N((1)), otherwisec 0 i

is therst vertexnumbered after t 0 i notbelongingtoN((1))(c 0 p 0 =(1)), -C 0 i

isthecomponentof(G;) containingc 0 i (C 0 p 0 =f(1)g), - T 0 i

is the set of vertices t such that  1 (t 0 i )   1 (t) and (if i < p 0 )  1 (t) >  1 (t 0 i+1

), so by denition, the subsets T 0 i

are numbered in in-creasingorder.

Notethatforanyiin[1::p 0 ]T 0 i N((1))[C 0 i

andthattheremaybedistinct i,j suchthat C 0 i =C 0 j ,sop 0

canbegreaterthanthenumberofcomponents of(G;).

Werst showthat for anyi in [1::p 0

], l((1)) is maximumamong labelsof unnumbered vertices just before numbering t

0 i

. The proof goes by induc-tionon i. It obviously holds fori =1. We suppose that itholds for some i < p 0 . l t 0 i (t 0 i+1 )  l t 0 i ((1)). By Lemma 5.4 with u = t 0 i and v = t 0 i+1 l t 0 i+1 (t 0 i+1 )=l t 0 i+1 ((1)),andsincel(t 0 i+1

)ismaximumatthatstep(because theorderistotal)l((1))ismaximumtoo.

Nowlety;z2N((1))suchthat  1

(y)> 1

(z). Letusshowthat either y2N(z)orthereissomej<p 0 suchthaty;z2N(C 0 j ). Leti2[1::p 0 ]such thaty2T 0 i .

Firstcase: 9j2[i::p 0 1]j z2N(C 0 j ) Letj 0

bethe smallestintegersuchthat ij 0 andC 0 j 0 =C 0 j . If  1 (c 0 j 0 )  1 (y) then  1 (c 0 j 0 )   1 (y) >  1 (t 0 i+1 )   1 (t 0 j 0 +1 ), so by deni-tion oft 0 j 0 +1 , y 2N(C 0 j 0 ). Otherwise 1 (t 0 i )  1 (y)>  1 (c 0 j 0 ) andby Lemma 5.4 with u=t 0 i and v =c 0 j 0 , y 2 N(c 0 j 0 ), soy 2 N(C 0 j 0 ). Thus in bothcasesy;z2N(C 0 j ). Secondcase: 8j2[i::p

0 1],z62N(C 0 j )  1 (t 0 i )   1 (y) >  1 (z) so by Lemma 5.4 with u = t 0 i and v = z,

Hence(1)isanOCF-vertexoftheinputgraph. 2

 WenowpresenttheproofofCharacterization4.7:

Characterization4.7 ForanygraphGandanyvertexx ofG,x is MNS-terminalix issimplicialand theneighborhoods ofconnectedcomponents ofG(V nN[x]) aretotallyorderedbyinclusion.

Wewillusethefollowingwell-knownpropertyofchordalgraphs:

Property 5.5 (con uence vertex property) Let G = (V;E) be a chordal graph, let C be a connected subset such that N(C) isa clique. Then 9z 2 C j N(C)N(z).

Proof: (ofCharacterization4.7)If xisMNS-terminalthen xis simplicial since MNS computes a peo of G and by Theorem 3.8 neighborhoods are totallyorderedbyinclusion.

Conversely,wesupposethatxissimplicialandneighborhoodsofconnected componentsofG(VnN[x])aretotallyorderedbyinclusion.Let(C

1 ;C 2 ;:::;C q ) beanorderingonthecomponentsofG(V nN(x))suchthat 8j2[1::q 1], N(C j )  N(C j+1 ), with C q = fxg. . For any j 2 [1::q], let S 0 j = N(C j )nN(C j 1 ),letT 0 j =S 0 j [C j andletc j 2C j jN(C j )N(c j )(c j exists byProperty5.5becauseasxissimplicial,C

j

issimplicialtoo). Letusshow thatthereisaMNSexecutiononGnumberingthesetsT

0 j

inincreasingorder. Weprovebyinduction onj thatthere isaMNSexecutiononGnumbering successivelyT 0 1 , T 0 2 , ...,T 0 j

rst. It obviouslyholds for j =0. We suppose that it holds for j 1for some j  q. After numbering T

0 1 , T 0 2 , ..., T 0 j 1 , foreveryunnumberedvertext,l(t)=NN(t)N(C

j 1 )N(C j )N(c j ). Soc j

canbechosenat thatstep. It remainstoshowthat at anystepafter numberingc

j

andsome, but notall, vertices ofT 0 j

, it is possible to choose thenextvertexinT

0 j

. Lety beanunnumberedvertexinT 0 j

,beac j

y-path whoseinternalverticesarein C

j ,lety

0

betherstunnumberedvertexof from c

j andz

0

bethe numberedvertexpreceding y 0 on. Thenz 0 2l(y 0 ). Let y 00

be an unnumbered vertex whose label is maximal among those of unnumbered vertices t such that z

0

2 l(t). So the label of y 00

is maximal amongthoseofallunnumberedverticesandy

00 2T 0 j (sincez 0 2C j \N(y 00 )). Thusy 00

canbechosenatthatstep. HenceT 0 1 ,T 0 2 ,...,T 0 q canbenumberedin increasingorder. AstheverticesofT

0 q

form acliquemodule(sincex is sim-plicial)theycanbenumberedinanarbitraryorderandxcan benumbered last. 2