数学外文+中文翻译.docx

数学外文+中文翻译.docx

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数学外文+中文翻译

SIAMJ.DISCRETEMATH.

Vol.26,No.1,pp.193–205

ROMANDOMINATIONON2-CONNECTEDGRAPHS∗

CHUN-HUNGLIU†

ANDGERARDJ.CHANG‡

Abstract.ARomandominatingfunctionofagraphGisafunctionf:

V（G）→{0,1,2}suchthatwheneverf（v）=0,thereexistsavertexuadjacenttovsuchthatf（u）=2.Theweightoffisw（f）=.TheRomandominationnumberofGistheminimumweightofaRomandominatingfunctionofGChambers,Kinnersley,Prince,andWest[SIAMJ.DiscreteMath.,23（2009）,pp.1575–1586]conjecturedthat≤[2n/3]forany2-connectedgraphGofnvertices.Thispapergivescounterexamplestotheconjectureandprovesthat≤max{[2n/3],23n/34}forany2-connectedgraphGofnvertices.Wealsocharacterize2-connectedgraphsGforwhich=23n/34when23n/34>[2n/3].

Keywords.domination,Romandomination,2-connectedgraph

AMS.subjectclassifications.05C69,05C35

DOI.10.1137/080733085

1.Introduction.ArticlesbyReVelle[14,15]intheJohnsHopkinsMagazinesuggestedanewvariationofdominationcalledRomandomination;seealso[16]foranintegerprogrammingformulationoftheproblem.Sincethen,therehavebeenseveralarticlesonRomandominationanditsvariations[1,2,3,4,5,7,8,9,10,11,13,17,18,19].EmperorConstantineimposedtherequirementthatanarmyorlegioncouldbesentfromitshometodefendaneighboringlocationonlyiftherewasasecondarmywhichwouldstayandprotectthehome.Thus,therearetwotypesofarmies,stationaryandtraveling.Eachvertex（city）thathasnoarmymusthaveaneighboringvertexwithatravelingarmy.Stationaryarmiesthendominatetheirownvertices;avertexwithtwoarmiesisdominatedbyitsstationaryarmy,anditsopenneighborhoodisdominatedbythetravelingarmy.

Inthispaper,weconsider（simple）graphsandlooplessmultigraphsGwithvertexsetV（G）andedgesetE（G）.Thedegreeofavertexv∈V（G）isthenumberofedgesincidenttov.NotethatthenumberofneighborsofvmaybelessthandegGvinalooplessmultigraph.ARomandominatingfunctionofagraphGisafunctionf:

V（G）→{0,1,2}suchthatwheneverf（v）=0,thereexistsavertexuadjacenttovsuchthatf（u）=2.Theweightoff,denotedbyw（f）,isdefinedas.ForanysubgraphHofG,letw（f,H）=.TheRomandominationnumberofGistheminimumweightofaRomandominatingfunction.

Amongthepapersmentionedabove,wearemostinterestedintheonebyChambersetal.[2]inwhichextremalproblemsofRomandominationarediscussed.Inparticular,theygavesharpboundsforgraphswithminimumdegree1or2andboundsof+and.Aftersettlingsomespecialcases,theygavethefollowingconjectureinanearlierversionofthepaper[2].Conjecture（Chambersetal.[2]）.Forany2-connectedgraphGofnvertices,≤[2n/3]。

Thispaperprovesthat≤max{[2n/3],23n/34}forany2-connectedgraphGofnvertices.Noticethat23n/34islargerthan2n/3byn/102.Wealsocharacterize2-connectedgraphsGwith=23n/34when23n/34>[2n/3].ThiswasinfactsuspectedbyWestthroughaprivatecommunicationandprovedaftersomediscussionswithhim.

2.Counterexamplestotheconjecture.Inthissection,wegivecounterexamplestotheconjecturebyChambersetal.[2].

TheexplosiongraphofalooplessmultigraphGisthegraphwithvertexsetV（）=V（G）∪{,,,,:

e=xy∈E（G）}andedgesetE）={x,y,,,,,e:

e=xy∈E（G）};seeFigure1.Noticethat{,,,}inducesa5-cyclein,denotedbyCe.Wecall,,theinnerverticesofCeandof.NotethatevenifGhasparalleledges,itsexplosiongraph,isasimplegrap

Theorem1.Thereareinfinitelymany2-connectedgraphswithRomandominationnumberatleast23n/34,wherenisthenumberofverticesinthegraph.

Proof.Considerkgraphs,,...,,eachisomorphicto,andtheirexplosiongraphs,,...,.LetGbea2-connectedgraphobtainedfromthedisjointunionoftheseexplosiongraphs’sbyaddingsuitableedgesbetweenverticesoftheoriginalgraphss;i.e.,theseaddededgesandthesforma2-connectedgraph.Then,Ghasn=34kvertices.

Weclaimthat≥23n/34=23k.Supposetothecontrarythat<23k.ChooseanoptimalRomandominatingfunctionfofG.Since=w（f）<23k,thereissomewithw（f,）<23.

Noticethatforanyedgexyin,nomatterwhatthevaluesoff（x）andf（y）are,itisalwaysthecasethatw（f,）≥3.Furthermore,iff（x）≤1andf（y）≤1,theninfactw（f,）≥4.Supposehasrverticesvwithf（v）≤1,where0≤r≤4.Therearethen（）edgesxyinwithw（f,）≥4.Thus

23>w（f,）≥r·0+（4−r）2+6·3+（）or,equivalently,2r>3+（）,whichisimpossibleas0≤r≤4.

Thelowerbound23n/34inthetheoremaboveisinfacttheexactvalueforthegivengraphG.Thiswillbeseenfromthefollowingtheorem,whoseproofemploysamethodthatisusefulintheentirepaper.

Fortechnicalreasons,weoftenconsiderthreeRomandominatingfunctions,,and.Weusetodenotethe3-tuple（,,）,and（v）for（（v）,（v）,（v））.Theweightofisw（）=.Notethatw（）≤w（）/3forsomej.Avertexvis-strongif（v）=2forsomej.

Theorem2.IfistheexplosiongraphofalooplessmultigraphGwithoutisolatedverticesandhasvertices,thenhasa3-tupleofRomandominatingfunctionssuchthatw（）≤69/34andeverynoninnervertexis-strong.Furthermore,ifsuchsatisfiesw（）=69/34,thenGisadisjointunionof’s.

Proof.IfGhasnverticesandmedges,then=n+5m.Weshallconstructbyfirstassigningvalues0,1,or2totheverticesinG.Forthispurpose,weordertheverticesofGinto,,...,andletbethesubgraphofGinducedby{,...,}asfollows.

Noticethat=G.Startingfromi=n,dothefollowingloop:

ifGihasavertexofdegreenotequalto3oravertexofdegree3thathasatmosttwoneighbors,thenchooseitasvi;otherwisechooseavertexofdegree3asvianditsthreeneighborsas,,.Fortheformercase,let=andthenreplaceibyi−1torepeattheloop;forthelattercase,let=fork=i,i−1,i−2,i–3andthenreplaceibyi−4torepeattheloop

EveryedgeofGservesasa“backedge”ofsome,i.e.,withi>k.Hence,m=。

Once（）hasbeendefinedfork

Case1.≥4.Inthiscase,let（）=（2,2,2）.Then,noneofthebackedgesofiscountedasatype-1edgeforsomes.Also,=1asdesired.

Case2.≤2or=3,buthasatmosttwoneighbors.Inthiscase,thereissomesuchthatforanybackedgeof.Define（）=2and（）=0forallj.Thebackedgesofarecountedastype-1edgesforsomes.foratmosttimes.

Case3.=3fori≤≤i+3,whereisabackedgeoffori≤≤i+2.Inthiscase,≤2fori≤≤i+2and=3.Fori≤≤i+3,similartoCase2,wemaychoosesome.foratleastmin{2,}backedgesof.Define（）=2and（）=0forallj.Thebackedgesofarecountedastype-1edgesforsomesforatmosttimesifi≤i’≤i+2andforatmost4=1+timesifi’=i+3.So,for,+1,+2,vi+3,thebackedgesarecountedastype-1edgesforsomeforatmost1+times,where≥6.

Now,wehaveassignedvalues（v）suchthatvis-strongforallverticesvinG.Wethenassignvaluesforverticesinall,wheree=xy,as

SinceGhasnoisolatedvertex,,,andareRomandominatingfunctionsof.Foredgesxywhicharetype1forsomeinG,（）=（）=2;foredgesxywhicharetype2forallinG,sincetherearesuchthat（x）=（y）=2,wehave（）=（）=2andsoandare-strong.

Next,wecalculatew（）.SupposethattherearecindicesisuchthatGiissimpleand3-regular,asinCase3.Then,4c≤nand6c≤:

isinCase3}≤m.Thesegivethat34c≤n+5m=n’andsoc≤n’/34.Thus,wehave

Furthermore,whenw（）=69n’/34,cmustbeequalton’/34andhence6c=:

isinCase3}=m,whichmeansthatCases1and2hadneverhappened.Bythealgorithmofordering{,,...,}describedatthebeginningoftheproof,Gmustbeadisjointunionof’s.Thiscompletestheproofofthetheorem.

3.Romandominationon2-connectedgraphs.Inthissection,weestablishourmainresultfortheRomandominationon2-connectedgraphs.

Theorem3.Forany2-connectedgraphGofnvertices,≤max{[2n/3],23n/34}.Furthermore,if23n/34>[2n/3]andGcontainsnospanningsubgraphisomorphictotheexplosiongraphofadisjointunionofK4’s,then<23n/34.

Proof.SupposetothecontrarythatthetheoremisfalseandGisaminimum

counterexampletothetheorem.Inotherwords,Gisa2-connectedgraphofordernwith>max{[2n/3],23n/34},andany2-connectedgraphoforderwith+|E（）|

Claim1.Gdoesnothaveacyclewithachord.

Lemma4（Dirac[6],Plummer[12]）.A2-connectedgraphisminimally2-connectedifandonlyifeverycycleisaninducedcycle.

Acycleispendentifithasnochordsandallofitsverticesexceptexactlytwononadjacentverticesareofdegree2.

Claim2.Anytwopendent5-cyclesinGarevertex-disjoint

Proof.SupposetothecontrarythatGhastwopendent5-cyclesandthatarenotvertex-disjoint.Let=and,bethetwoverticesofdegreeatleast3fori=1,2,where=andpossibly=.WenowconsiderthegraphobtainedfromthesubgraphofGinducedbyV（G）−{:

i=1,2,j=2,4,5}byaddingedgesand.Itisclearthatisa2-connectedgraphwith=n−6verticesand|E（）|<|E（G）|.BytheminimalityofG,wehaveγR（）≤max{[2/3],23/34}.ChooseaRomandominatingfunctionof