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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()hasbeendefinedforkCase1.≥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