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Journal of Combinatorial Theory, Series B
www.elsevier.com/locate/jctb
The Kelmans-Seymour conjecture I: Special separations
✩Dawei Hea,1,Yan Wangb,2, Xingxing Yub,3
aDepartmentofMathematics,ShanghaiKeyLaboratoryofPMMP,EastChina NormalUniversity,Shanghai200241,China
bSchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,GA30332, UnitedStatesofAmerica
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received11November2015 Availableonline11December2019
Keywords:
Subdivision Independentpaths Separation Planargraph Apexgraph
Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of K5. This conjecture was proved by Ma and Yu for graphs containing K4−, and an important step in theirproofisto dealwitha 5-separationinthegraphwith a planar side. In order to establish the Kelmans-Seymour conjecturefor all graphs, weneed to consider5-separations and6-separations with less restrictive structures. The goal of this paper is to deal with special 5-separations and 6-separations,includingthosewithanapexside.Resultswill beusedinsubsequentpaperstoprovetheKelmans-Seymour conjecture.
©2019ElsevierInc.Allrightsreserved.
✩ ThisworkstartedwhenDHwasastudentatECNUandXYwasvisitingECNU.
E-mailaddresses:dhe9@math.gatech.edu(D. He),yanwang@gatech.edu(Y. Wang), yu@math.gatech.edu(X. Yu).
1 PartiallysupportedbyScienceandTechnologyCommissionofShanghaiMunicipality(STCSM)grant No.13dz2260400.
2 PartiallysupportedbyNSFgrantsDMS-1265564andDMS-1600738throughX.Yu.
3 PartiallysupportedbyNSFgrantsDMS-1265564andDMS-1600738.
https://doi.org/10.1016/j.jctb.2019.11.008 0095-8956/©2019ElsevierInc. Allrightsreserved.
1. Introduction
Let K be agraph; weuse T K to denote asubdivisionof K and call thevertices of the T K corresponding to the vertices of K its branch vertices. Kuratowski’s theorem statesthatagraphisplanarif,andonlyif,itcontainsneitherT K3,3norT K5. Graphs containingnoT K3,3 aresubgraphsofthosegraphsconstructedfromplanargraphsand copies ofK5 bypastingthem alongcliquesofsize atmosttwo.Thestructureofgraphs containingnoT K5isnotwellunderstood.Kelmans [4] and,independently,Seymour [10]
conjecturedthat5-connectednonplanargraphsmustcontainT K5.Thus,iftheKelmans- Seymour conjectureistruethengraphscontainingnoT K5 isplanaroradmits acutof size at most4.Note thatthe requirement on connectivityis bestpossible (see,for ex- ample, K4,4).
Ma andYu [5,6] provedtheKelmans-Seymourconjecture forgraphscontaining K4−, and Kawarabayashi,MaandYu [3] provedtheKelmans-Seymourconjectureforgraphs containingK2,3.Wereferthereaderto [5,6,3] forproblemsandresults(aswell asrefer- ences) relatedtotheKelmans-Seymourconjecture.
It turns outthat K4− is theright intermediate structure for studying the Kelmans- Seymourstructure.ByaresultofKawarabayashi[2],any5-connectedgraphcontaining noK4−hasanedgeethatiscontractible(i.e.,G/eisalso5-connected).Thus,ourstrategy for provingtheKelmans-Seymourconjecture isto keepcontractingedgesincident with a special vertex to produce asmaller 5-connected graph. To avoid trivial components associated with 5-cuts or 6-cuts, we also contract triangles (but we give preference to edges).
We now give amoredetailed descriptionof our strategy.For agraph Gand acon- nectedsubgraphM ofG(respectively,anedgeeofG),weuseG/M (respectively,G/e) to denote the graph obtained from G by contracting M (respectively,e). Let G be a 5-connected nonplanargraphcontaining noK4−. Then Gcontains anedge esuch that G/e is 5-connected.If G/eis planar, wecanapply adischargingargument. So assume G/e is not planar. Let M be a maximal connected subgraph of G such that G/M is 5-connected and nonplanar.Let z denote thevertex ofG/M representingthecontrac- tionofM,andletH=G/M.Thenoneofthefollowingholds.
(a) H containsasubgraphK suchthatK∼=K4− andz hasdegree2inK.
(b) H containsasubgraphK suchthatK∼=K4− andz hasdegree3inK.
(c) H doesnotcontainK4−,andthereexistsT ⊆H,withz∈V(T) andeitherT ∼=K2 orT ∼=K3,suchthatH/T is5-connectedandplanar.
(d) H doesnot containK4−,and for anyT ⊆H withz ∈V(T) and eitherT ∼=K2 or T ∼=K3,H/T isnot5-connected.
NotethatbyT ⊆H wemeanthatT isasubgraphofH.Weplanaseriesoffourpapers to establish the Kelmans-Seymour conjecture. The purpose of this paper is to prove several resultsabout5-separations and6-separations, whichwill beused insubsequent
papers to reduce (c) and (d) to (a) or (b). Note that 5-separations and 6-separations arisenaturallywhen(d)occurs.Inthesecondpaper,wewillhandle(a).Thethirdpaper takes care of (b). In the final paper, we will deal with (c) and (d). In our arguments throughout this series, we frequently encounter the casewhen H −z contains K4−; so theexclusionofK4− (in [5,6])isveryuseful.
One of the main steps in the proofs in [5,6] is to deal with 5-connected nonplanar graphsthatadmita5-separationwithaplanarside.AseparationinagraphGconsists ofapairofsubgraphsG1,G2ofG,denotedas(G1,G2),suchthatE(G1)∪E(G2)=E(G), E(G1)∩E(G2) = ∅, and neither G1 nor G2 is asubgraph of the other. The order of thisseparationis|V(G1)∩V(G2)|,and(G1,G2) issaidtobeak-separationifitsorder is k. Let Gbe agraphand A ⊆V(G). We saythat (G,A) is planar ifG hasa plane representationinwhichthevertices inAare incidentwithacommon face.Ma andYu [5] provedthatifGhasa5-separation(G1,G2) suchthat|V(Gi)|≥7 (fori= 1,2) and (G2,V(G1)∩V(G2)) isplanarthenGcontainsaT K5withallbranchverticescontained inG2.In order to establish theKelmans-Seymour conjecture forgraphs containingno K4−,weneedtostudy 5-separationsand6-separationswithlessrestrictivestructures.
LetG be agraph.For two subgraphsG1,G2 ofG, we useG1∪G2 and G1∩G2 to denotetheirunionandintersection,respectively.Fora∈V(G),weuseG−atodenote thesubgraph obtained from Gby deletinga and alledges of Gincident with a. Fora positive integer n, we let [n] = {1,. . . ,n}. We often represent a path (or cycle) by a sequenceofvertices. Thefollowing resultdealswithonetypeof5-separations.
Theorem1.1. LetGbea5-connectednonplanargraphandlet(G1,G2)bea5-separation inG.Suppose|V(Gi)|≥7fori∈[2],a∈V(G1∩G2),and(G2−a,V(G1∩G2)− {a}) isplanar.Then oneof thefollowingholds.
(i) Forany a∗ ∈ V(G1−G2)∪ {a}, G contains a T K5 in which a∗ isnot a branch vertex.
(ii) G−acontains K4−.
(iii) Ghasa5-separation(G1,G2)suchthat V(G1∩G2)={a,a1,a2,a3,a4}andG2 is thegraph obtained from theedge-disjoint union of the8-cycle a1b1a2b2a3b3a4b4a1 andthe4-cycle b1b2b3b4b1 by addingaandtheedgesabi fori∈[4].
LetGbe agraph. ForS⊆V(G), weuseG[S] todenote thesubgraph ofGinduced byS,andletG−S =G[V(G)−S].ForH ⊆G,wewriteG[H] insteadofG[V(H)].For S⊆E(G),G−SdenotesthegraphobtainedfromGbydeletingtheedgesinS.Another typeof5-separationsconsideredinthispaperarethose(G1,G2) withthepropertythat G[V(G1∩G2)] containsatriangle.
Theorem1.2. LetGbea5-connectedgraphand(G1,G2)bea5-separationinG.Suppose that|V(Gi)|≥7fori∈[2] andG[V(G1∩G2)]containsatriangleaa1a2a.Thenoneof thefollowingholds.
(i) Gcontains aT K5 inwhichais notabranch vertex.
(ii) G−a containsK4−.
(iii) Ghas a5-separation(G1,G2)suchthatV(G1∩G2)={a,a1,a2,a3,a4}andG2 is the graph obtained from the edge-disjointunion of the8-cycle a1b1a2b2a3b3a4b4a1 andthe4-cycle b1b2b3b4b1 byadding aandtheedgesabi fori∈[4].
(iv) For any distinct u1,u2,u3 ∈ N(a)− {a1,a2}, G− {av : v /∈ {a1,a2,u1,u2,u3}}
contains T K5.
In the applications of Theorems 1.1 and 1.2, the vertex a will represent the special vertex resultingfrom thecontractionof aconnectedsubgraphformed byasequence of edges andtriangles. (Allowing thecontractionoftriangles willensurethat|V(Gi)|≥7 intheapplicationsofTheorems1.1and1.2.)So(i) ofTheorems1.1and1.2givesaT K5
inthe originalgraph,and (ii) of Theorems1.1 and1.2 allowsus toapply theresultof MaandYu [6] togetaT K5intheoriginalgraph.TheT K5givenin(iv) ofTheorem1.2 canbe usedto deriveaT K5 intheoriginalgraph.When(iii) ofTheorems 1.1and1.2 occurs,we will useProposition1.3 below,whoseproof isincluded inthissection (as it is short).
Let G be a graph. Recall thatwe useH ⊆G to mean that H is asubgraph of G.
WhenK⊆GandL⊆G,weletK−L=K−V(K∩L).ForS⊆V(G),wemayview S as a subgraph of G with vertex set S and edge set ∅. For H ⊆ G, NG(H) denotes theneighborhood ofH (notincludingtheverticesinV(H)).Forany x∈V(G),weuse NG(x) todenotetheneighborhoodofxinG.Whenunderstood,thereferencetoGmay be dropped.Wemayviewpathsas sequencesofvertices. Theends ofapath P are the vertices oftheminimumdegreeinP,andallotherverticesofP (ifany)areitsinternal vertices. Acollectionofpathsaresaidtobeindependent ifnovertexofanypathinthis collectionisaninternalvertexofanyother pathinthecollection.
Proposition 1.3.LetGbea5-connectednonplanargraph, (G1,G2)a5-separationinG, V(G1∩G2)={a,a1,a2,a3,a4}suchthatG2 isthegraphobtainedfromtheedge-disjoint union of the8-cyclea1b1a2b2a3b3a4b4a1 andthe4-cycleb1b2b3b4b1 by addingaand the edges abi fori ∈ [4]. Suppose |V(G1)| ≥7.Then, for any u1,u2 ∈N(a)− {b1,b2,b3}, G− {av:v /∈ {b1,b2,b3,u1,u2}}contains T K5.
Proof. Bysymmetrybetweenu1andu2,wemayassumethatu1=b4.Forconvenience, letG:=G− {av:v /∈ {b1,b2,b3,u1,u2}}.
First, suppose u1 ∈ V(G1−G2). Since G is 5-connected, G1−a has independent paths Q1,Q2,Q3,Q4 from u1 to a1,a2,a3,a4, respectively.Then G[{a,b1,b2,b3,u1}]∪ (Q1∪a1b1)∪(Q2∪a2b2)∪(Q3∪a3b3)∪b1b4b3 is a T K5 inG with branch vertices a,b1,b2,b3,u1.
Nowsuppose u1∈ {a2,a3}.Bysymmetry,wemayassumeu1=a2.NotethatG1−a contains apath R from a2 to a3; forotherwise, a2 and a3 are indifferent components of G1−aand,since|V(G1)|≥6,{a,a1,a2,a4}or{a,a1,a3,a4}wouldbeacutinG,a
contradiction.ThenG[{a,b1,b2,b3,u1}]∪(R∪a3b3)∪b1b4b3 isaT K5inGwithbranch verticesa,b1,b2,b3,u1.
Finally, assume u1 ∈ {a1,a4}. By symmetry, we may assume u1 = a1. If G1−a hasindependentpathsR1,R2 froma1to a2,a3,respectively,thenG[{a,b1,b2,b3,u1}]∪ (R1∪a2b2)∪(R2∪a3b3)∪b1b4b3 is aT K5 inG with branch vertices a,b1,b2,b3,u1. SowemayassumethatsuchR1,R2 donotexist.Thenthere existsaseparation(K,L) inG1−a suchthat |V(K∩L)| ≤1, a1 ∈V(K−L),and {a2,a3}⊆V(L). We claim thatV(K) ⊆ {a1,a4}∪V(K∩L); for, otherwise, {a,a1,a4}∪V(K∩L) woulda cut inG, contradicting the assumption that Gis 5-connected. If a4 ∈ V(L) then N(a) ⊆ {a,b1,b4}∪V(K∩L),contradiction.Soa4∈/V(L).ThenV(L)={a2,a3}∪V(K∩L);
asotherwise{a,a2,a3}∪V(K∩L) wouldbeacutinGofsizeatmost4,acontradiction.
Butthisimplies|V(G1)|≤6,acontradiction. 2
Thenextresultdealswithaspecialtypeof 6-separations,whose proofmakes useof Theorems1.1and1.2.
Theorem1.4. LetGbea5-connectedgraph,let(G1,G2)bea6-separationinG,andlet x,x1,x2∈V(G1∩G2)such that xx1x2xisatrianglein Gand |V(Gi)|≥7fori∈[2].
Moreover, assume that (G1,G2)is chosen sothat, subject to{x,x1,x2}⊆V(G1∩G2) and|V(Gi)|≥7fori∈[2],G1isminimal.LetV(G1∩G2)={x,x1,x2,v1,v2,v3}.Then N(x)∩V(G1−G2)=∅,orone ofthefollowingholds.
(i) GcontainsaT K5 inwhich xis notabranch vertex.
(ii) GcontainsK4−.
(iii) There exists x3 ∈ N(x) such that for any distinct y1,y2 ∈ N(x)− {x1,x2,x3}, G− {xv:v /∈ {x1,x2,x3,y1,y2}}containsT K5.
(iv) Thereexisti∈[2]andj ∈[3]suchthat N(xi)⊆V(G1−G2)∪ {x,x3−i},andany threeindependentpaths inG1−xfrom {x1,x2}tov1,v2,v3,respectively,withtwo fromxi andone fromx3−i,must containapath fromx3−i tovj.
Notethatin(ii) of Theorem 1.4, weask thatGcontainK4− instead ofthestronger statement“G−acontainsK4−”asin(ii)ofTheorems1.1and1.2.Thus,(iii) ofTheo- rems1.1and1.2doesnotoccurinTheorem1.4as itgives(ii) ofTheorem1.4.
Thispaper isorganizedas follows.InSection2, weusethedischarging techniqueto provetwolemmas aboutK4− inapexgraphs.(Agraphis apexifithasavertex whose removalresults inaplanar graph.)InSection3, wecollect anumberof knownresults andproveTheorem1.1. InSection4, weprovearesultaboutapexgraphs(fromwhich one cansee how (c) might be taken care of).In Section 5, we prove Theorem 1.2. In Section6,weproveTheorem1.4,usingTheorems 1.1and1.2.
Weend thissectionwithadditionalnotationandterminology.LetGbeagraph.Let K⊆G,S⊆V(G),andT acollectionof2-elementsubsetsofV(K)∪S;thenK+ (S∪T)
denotesthegraphwithvertexsetV(K)∪SandedgesetE(K)∪T.IfT ={{x,y}}and x,y∈V(K),wewriteK+xy insteadofK+{{x,y}}.
A set S ⊆V(G) is ak-cut (or a cut of size k) in agraph G, where k is a positive integer, if |S| = k and G has a separation (G1,G2) such that V(G1∩G2) = S and V(Gi−S)=∅fori∈[2].Ifv∈V(G) and{v}isacutofG, thenv issaid tobe acut vertex ofG.
Given apathP inagraphandx,y ∈V(P),xP ydenotes thesubpathof P between xandy(inclusive).ApathP withendsuandv(oranu-v path)isalsosaidtobefrom utov or betweenuand v.
A plane graphisagraphdrawnintheplanewith noedge-crossings.Theunbounded face ofaplanegraphisusuallycalled itsouterface.Theouterwalk ofaplanegraphG is thesubgraphof Gconsisting ofthoseverticesandedgesofGincidentwiththeouter face ofG.(An outercycleissimplyanouterwalkthatisacycle.)Itiswell knownthat if Gisa2-connectedplane graphtheneveryfacialboundaryof Gis acycle.Let D be acycleinaplanegraph.Givenx,y∈V(D),ifx=y then xDydenotes thesubpathof D betweenxandy (inclusive)inclockwise order; and ifx=y thenxDy issimplythe trivialpathwiththesinglevertexx=y.
2. DischargingandK4−
Inthissection,weproveresultsaboutK4−incertainapexgraphs,usingthedischarg- ing technique. First,we give asimple lemma on discharging. Fora plane graphG, let F(G) denotethesetofallfacesofGand,foreachf ∈F(G),letd(f) denotethenumber of edges of Gincident with f (with each cutedge counted twice).For eachv ∈ V(G), we usedG(v) (ord(v) whenGisunderstood) todenotethedegreeofv inG.
Lemma2.1. LetGbeaconnectedplane graph andletσ:V(G)∪F(G)−→Z,thesetof integers, suchthat σ(t)= 4−d(t)forallt∈V(G)∪F(G)(which iscalled thenumber of units of charge of t). Let τ be obtained from σ asfollows: For each f ∈ F(G) with d(f) = 3, choose two vertices incident with f and send charge 1/2 from f to each of thesetwovertices.Then
v∈V(G)
σ(v) +
f∈F(G)
σ(f) = 8,
and ifK4−Gthen,forv∈V(G),
τ(v)≤
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
4−3k, ifd(v) = 4k;
3−3k, ifd(v) = 4k+ 1;
5/2−3k, ifd(v) = 4k+ 2;
3/2−3k, ifd(v) = 4k+ 3.
Proof. ByEuler’sformula,wehave
v∈V(G)
σ(v) +
f∈F(G)
σ(f) = 8.
Nowsuppose K4− G.Then for eachv ∈V(G), v is containedinat mostd(v)/2 facialtriangles. So τ(v) ≤σ(v)+d(v)/2/2 = 4−d(v)+d(v)/2/2. Byconsidering d(v) modulo4,wegetthedesiredbounds onτ(v). 2
To state the remaining results in this section, we need a concept on connectivity.
Let G be a graph and A ⊆ V(G), and let k be a positive integer. We say that G is (k,A)-connected if,forany cutT ofGwith|T|< k,eachcomponentofG−T contains avertexinA. Thus,everyvertex ofGnotinAhasdegreeat leastk inG. Recallthat (G,A) is planarifG hasaplanerepresentation inwhich thevertices inAare incident withacommon face.
Lemma 2.2.Let Gbe aconnected graph and A⊆V(G) suchthat |A|= 5,|V(G)|≥7, Gis(5,A)-connected,and(G,A)isplanar. Then,forany a∈A,G−acontainsK4−. Proof. Since G is (5,A)-connected, each component of G−A must contain a neigh- bor of every vertex in A; thus G−E(G[A]) is connected. Clearly, G−E(G[A]) is (5,A)-connected, and (G−E(G[A]),A) is planar. Therefore, it suffices to prove the lemma for the case when A is an independent set in G. So we assume that A is an independentsetinG.
First,we show thatG−ais connected for any a∈ A. For, ifnot, then we maylet C1,C2 betwo componentsof G−asuch thatV(C1)A.Since Gis(5,A)-connected, V(Ci)∩A=∅for i∈[2].Now S := (A∩V(C1))∪ {a}isacutinGsuchthat|S|≤4 andG−S hasacomponentcontainedinC1−A, contradictingtheassumptionthatG is(5,A)-connected.
Take a plane representation of G such thatthe vertices in A are incident with the outerfaceofG.Letσ:V(G−a)∪F(G−a)−→Zsuchthatσ(t)= 4−dG−a(t) forall t∈V(G−a)∪F(G−a).ThenbyLemma2.1, thetotalchargeis
t∈V(G−a)∪F(G−a)
σ(t) = 8.
Notethatforany t∈V(G−a)∪F(G−a),ifσ(t)>0 thent∈A,or t∈F(G−a) and dG−a(t)= 3 (in whichcase, σ(t)= 1). For eachf ∈ F(G−a) withdG−a(f)= 3, choosetwo vertices of G−A incident with f (which exists as A is independent), and sendacharge1/2 fromf toeachofthesetwovertices.Letτdenotetheresultingcharge function.
Thenτ(f)≤0 forallf ∈F(G−a).SupposeK4−G−a.Then,foreachv∈V(G−a), τ(v) hastheupperboundinLemma2.1.Soτ(v)≤0 whenv /∈N(a)∪A, τ(v)≤1 for v∈N(a) (asA isindependent),andτ(v)=σ(v)≤3 for v∈A− {a}.
Let f∞ denote the outer face of G−a. Since A is independent in G, dG−a(f∞) ≥
|N(a)|+ 7;soτ(f∞)=σ(f∞)≤4−(|N(a)|+ 7)=−3− |N(a)|.Ifat leasttwovertices inA− {a}eachhavedegree2ormoreinG−a,then
v∈A−{a}
τ(v)≤2+ 2+ 3+ 3= 10;
so
τ(f∞) +
v∈A−{a}
τ(v)≤ −3− |N(a)|+ 10 = 7− |N(a)|.
Nowassumethatat mostonevertex inA− {a}hasdegree2ormoreinG−a.SinceG is(5,A)-connected,theverticesinA−awithdegree1inG−acannotshareaneighbor inG−a.Hence,sinceA isindependent,dG−a(f∞)≥ |N(a)|+ 9.Since
v∈A−{a}
τ(v)≤ 3+ 3+ 3+ 3= 12,
τ(f∞) +
v∈A−{a}
τ(v)≤4−(|N(a)|+ 9) + 12 = 7− |N(a)|.
Therefore, since
t∈V(G−a)∪F(G−a)
τ(t)≤τ(f∞) +
v∈A−{a}
τ(v) +
v∈N(a)
τ(v),
we have
t∈V(G−a)∪F(G−a)
τ(t)≤(7− |N(a)|) +|N(a)|= 7<8 =
t∈V(G−a)∪F(G−a)
σ(t),
acontradiction. 2
Thenextresultwillnotbeusedinthispaper,butwill beusedinsubsequentpapers intheseries;weincludeithereas itsproofalsousesdischarging.
Proposition 2.3.LetGbeagraph,A⊆V(G),anda∈Asuchthat |A|= 6,|V(G)|≥8, (G−a,A− {a})isplanar,and Gis(5,A)-connected.Thenoneof thefollowingholds.
(i) G−acontainsK4−,orGcontainsasubgraphK∼=K4− suchthata∈V(K)andthe degree of ainK is2.
(ii) Ghas a5-separation(G1,G2)suchthat a∈V(G1∩G2),A⊆V(G1),|V(G2)|≥7, and(G2−a,V(G1∩G2)− {a}) isplanar.
Proof. Wemayassumethat
(1) Ghasno5-separationor6-separation(G,G) suchthata∈V(G∩G),A⊆V(G),
|V(G)|≥8,and(G−a,V(G∩G)− {a}) isplanar.
For, suppose such a separation (G,G) does exist in G. Then G is (5,V(G ∩ G))-connected. If|V(G∩G)|= 5 then(ii) holds. If|V(G∩G)|= 6 wemaywork withGand V(G∩G) insteadofGandA.
NotethatAisindependentinG.For,ifthereexista1,a2∈Asuchthata1a2∈E(G) then let G = G−a1a2 and G consists of A and the edge a1a2. Now (G,G) is a separationinG,whichcontradicts(1).
(2) |V(G−a)|≥8 andeachvertex inA− {a}hasat leasttwoneighborsinG−A.
First,suppose |V(G−a)|= 7.Letb1,b2 denotethetwo vertices inV(G)−A.Since G is(5,A)-connected,|N(b1)∩N(b2)∩A|≥3.But thiscontradicts theassumption that (G−a,A− {a}) isplanar.
So|V(G−a)|≥8.Thenby(1), eachvertex inA− {a}hasat leasttwo neighborsin G−A,completingtheproofof(2).
Moreover,
(3) G−Aisconnected.
For, otherwise, let C1,C2 be two components of G−A. Since G is (5,A)-connected,
|N(Ci)∩(A−{a})|≥4 fori∈[2].Hence,|N(C1)∩N(C2)∩(A−{a})|≥3,contradicting theassumptionthat(G−a,A− {a}) isplanar.This proves(3).
We now apply a discharging argument to G−a by taking a plane representation of G−a in which the vertices in A − {a} are incident with its outer face. Let σ : V(G−a)∪F(G−a)−→Zsuchthatσ(t)= 4−dG−a(t) forallt∈V(G−a)∪F(G−a).
By(2) and(3),G−aisconnected.SobyLemma2.1,thetotalchargeis
t∈V(G−a)∪F(G−a)
σ(t) = 8.
Notethatforanyt∈V(G−a)∪F(G−a),ifσ(t)>0 thent∈A−{a},ort∈F(G−a) and dG−a(t)= 3 (in whichcase, σ(t)= 1). For eachf ∈ F(G−a) withdG−a(f)= 3, wemayassumethatf isincident withtwoverticesinV(G−a)−N(a);for,otherwise, (i) holds. So for each f ∈ F(G−a) with dG−a(f) = 3, we choose two vertices from V(G−a)−N(a) incidentwith f, and send acharge 1/2 from f to eachof these two vertices.Letτ denotetheresultingchargefunction.Thenτ(f)≤0 forallf ∈F(G−a).
WemayassumethatK4−G−a,as otherwise(i) holds.Thenforeachv∈V(G−a), τ(v) hastheupperboundinLemma2.1.Thus,τ(v)≤0 ifv /∈A(sinceverticesinN(a) donotreceive charge),andτ(v)≤5/2 ifv∈A− {a}(by(2)).
Denotebyf∞theouterfaceofG−a.SinceAisindependentinG,τ(f∞)=σ(f∞)≤
−6.Therefore,τ(f∞)+
v∈A−{a}τ(v)≤ −6+ 25/2= 13/2.Hence,thetotalchargeis
t∈V(G−a)∪F(G−a)
τ(t)≤τ(f∞) +
v∈A−{a}
τ(v)≤13/2<8 =
t∈V(G−a)∪F(G−a)
σ(t).
This isacontradiction. 2 3. Apexseparations
In this section, we prove Theorem 1.1. For convenience, we introduce the following terminology.LetGbeagraphandA⊆V(G).Wesaythat(G,A) isplane ifGisdrawn in the plane with no edge crossings such that the vertices in A are incident with the outerface.Moreover,fora1,. . . ,ak∈V(G),wesay(G,a1,. . . ,ak) isplane (respectively, planar)ifGisdrawn(respectively,hasadrawing)inacloseddiscintheplanewithno edgecrossingssuchthata1,. . . ,ak occurontheboundaryofthediscinthiscyclicorder (clockwise orcounterclockwise).
Wealsoneedafewknownresults.Thefirstresultisaconsequenceofamoregeneral resultofSeymour [9] (withequivalent versionsprovedin[1,12,11]).
Lemma 3.1.Let G be a graph and let s1,s2,t1,t2 ∈ V(G) be distinct such that G is (4,{s1,s2,t1,t2})-connected. Then either G contains disjoint paths from s1 to t1 and from s2 tot2,or (G,s1,s2,t1,t2)isplanar.
Thenextlemma isTheorem4.3in[5],whereitisusedtoprovethatifa5-connected nonplanargraphhasa5-separationwithonesideplanarandnontrivialthenitcontains T K5.
Lemma 3.2.Let G be a graph and let a1,a2,a3,a4,a5 ∈ V(G) be distinct such that (G,a1,a2,a3,a4,a5) is plane. Let A = {a1,a2,a3,a4,a5} and suppose G is (5,A)-connected and |V(G)| ≥ 7. Then there exist w ∈ V(G)−A, a cycle Cw in (G−A)−w,andfourpaths P1,P2,P3,P4 fromw toAsuchthat
(i) V(Pi∩Pj)={w}for1≤i< j≤4,and|V(Pi∩Cw)|= 1 fori∈[4],and (ii) there exist1≤i=j≤4 suchthat a1 isanend ofPi and a5 isanend ofPj.
We needtwo results from [3], which maybe viewed asapex versionsof Lemma3.2.
Theyareusedin[3] todealwith5-separationswithanapexside.Lemma3.3isCorollary 2.11in[3], andLemma3.4 isCorollary2.12in[3].
Lemma 3.3.Let G be a connected graph with |V(G)| ≥ 7, A ⊆ V(G) with |A| = 5, and a ∈ A, such that G is (5,A)-connected, (G−a,A− {a}) is plane, and G has no 5-separation (G1,G2) with A⊆G1 and |V(G2)|≥7.Let w∈N(a) such that w isnot incident with theouterface of G−a.Then
(i) theverticesof G−acofacial withw induceacycleCw inG−a,and
(ii) G−acontains pathsP1,P2,P3 from wtoA− {a}such thatV(Pi∩Pj)={w}for 1≤i< j≤3,and|V(Pi∩Cw)|=|V(Pi)∩A|= 1 fori∈[3].
Lemma 3.4.Let Gbe aconnected graph with |V(G)|≥7,A⊆V(G) with |A|= 5,and a∈A,suchthatK4−G−a,Gis(5,A)-connected,(G−a,(A− {a})∪N(a))isplane, and Ghas no 5-separation(G1,G2)with A⊆V(G1) and |V(G2)| ≥7.Then G−a is 2-connected.Moreover,eitherGisthegraphobtainedfromtheedge-disjointunionofthe 8-cycle a1b1a2b2a3b3a4b4a1 andthe4-cycleb1b2b3b4b1 by addingaandtheedgesabi for i∈[4],with A={a,a1,a2,a3,a4},or thereexistsw∈V(G)−A suchthat
(i) theverticesofG−acofacialwithwinduceacycleCwinG−asuchthatCw∩D=∅, whereD denotestheouter cycleofG−a,
(ii) there existpaths P1,P2,P3,P4 inG from w to A suchthat V(Pi∩Pj)={w}for 1≤i< j ≤4,and|V(Pi∩Cw)|=|V(Pi)∩A|= 1fori∈[4],and
(iii) either a ∈/ 4
i=1V(Pi), or a ∈ 4
i=1V(Pi) and we may write A − {a} = {a1,a2,a3,a4}suchthata∈V(P1),ai ∈V(Pi)for2≤i≤4,anda1,a2,a3,V(P1∩ D),a4 occur onD in acyclic order.
Proof of Theorem1.1. Leta∗ ∈V(G1−G2)∪ {a}. Wechoosetheseparation(G1,G2) sothatG2 isminimal.ThenwemayassumethatG2 hasno5-separation(G2,G2) such that |V(G2∩G2)| ≤ 5, V(G1 ∩G2) ⊆ V(G2), and |V(G2)| ≥ 7. For, suppose such (G2,G2) does exist. Then a ∈/ V(G2∩G2); otherwise, (G1∪G2,G2) contradicts the choiceof(G1,G2) (thatG2isminimal).Hence,(G2,V(G2∩G2)) isplanar;so(ii) holds byLemma2.2.
LetA:=V(G1∩G2)={a,a1,a2,a3,a4}suchthat(G2−a,a1,a2,a3,a4) isplane.Let D denote theouter walk ofG2−a;so a1,a2,a3,a4 occur on D inclockwise order. We mayassumethatneither(G1,A) nor(G2,A) isplanar;else (ii) holdsbyLemma2.2.
Supposethereexists somew∈N(a)∩V(G2−D).Then byLemma3.3, thevertices of G2−a cofacial with w induce a cycle Cw in G2−a, and G2 −a contains paths P1,P2,P3 from w to A− {a} such that V(Pi∩Pj) = {w} for 1 ≤ i < j ≤ 3, and
|V(Pi∩Cw)|=|V(Pi)∩A|= 1 fori∈[3].Withoutlossofgenerality,wemayassumethat V(Pi)∩A={ai}fori∈[3].Lety∈V(G1−A) withy=a∗.SinceGis5-connected,there exist independent paths Q1,Q2,Q3,Q4 inG1−a4 from y to a1,a2,a3,a, respectively.
ThenCw∪P1∪P2∪P3∪Q1∪Q2∪Q3∪(Q4∪wa) isaT K5 inGinwhicha∗ isnota branchvertex.So(i) holds.
Thus, we mayassume that N(a)∩V(G2) ⊆ V(D). By the minimality of G2, A is independent inG2. Hence(G2−a,(A−a)∪(N(a)∩V(G2))) isplanar. Moreover, we mayassume|N(a)∩V(G2−A)|≥2;for,otherwise, (G2,A) is planarand, hence, (ii) holdsbyLemma2.2.
Suppose (ii) and (iii) of Theorem 1.1 both fail. Then by Lemma 3.4, G2 −a is 2-connected(so Disacycle)andthereexists w∈V(G2)−Asuchthat
(1) theverticesofG2−acofacialwithwinduceacycleCwinG2−asuchthatCw∩D=∅, (2) there exist pathsP1,P2,P3,P4 inG2 from w to Asuch thatV(Pi∩Pj)={w}for
1≤i< j≤4,and|V(Pi∩Cw)|=|V(Pi)∩A|= 1 for i∈[4],and (3) eithera∈/ 4
i=1
V(Pi),ora∈ 4
i=1
V(Pi) andwemayrelabela1,a2,a3,a4(ifnecessary) such thata∈V(P1), ai ∈V(Pi) for 2≤i ≤4,and a1,a2,a3,V(P1∩D),a4 occur onD inacyclicorder.
Forconvenience, letL=Cw∪P1∪P2∪P3∪P4andassume,withoutlossofgenerality, that a1,a2,a3,a4 occur on D inclockwise order. Bythe planarity of G2−a, we may assumethatPi∩D isapathfori∈[4].
Case 1.a∈/ 4
i=1
V(Pi).
Without loss ofgenerality, wemayassumethatai∈V(Pi) fori∈[4]. IfG1−a has disjoint paths S1,S2 from a1,a2 to a3,a4, respectively, then L∪S1∪S2 is a T K5 in G in which a∗ is not a branch vertex; so (i) holds. Hence, we may assume that such S1,S2 donotexist.Then, sinceG1is (5,A)-connected,it followsfrom Lemma3.1 that (G1−a,a4,a3,a2,a1) isplaneand, hence,G1−A isconnected.
Subcase 1.1. G1−Aisnot2-connected.
If|V(G1−A)|= 2 thenletC1,C2denotethetwodisjoint1-vertexsubgraphsofG1−A, and otherwiselet (C1,C2) denotea1-separation inG1−A withV(Ci−C3−i)=∅for i∈[2].SinceGis5-connectedand(G1−a,a4,a3,a2,a1) isplane,N(a)∩V(Ci−C3−i)=∅ and Ci−C3−i isconnected, fori∈[2]. Withoutloss ofgenerality,wemayassumethat {a,a1,a2,a3}⊆N(C1) and{a,a1,a3,a4}⊆N(C2).
Suppose N(a)∩V(G2−A)⊆ V(P1∪P3). If N(a)∩V(G2−A)⊆ V(Pi) for some i ∈ {1,3} then (G2,A) is planar; so (ii) holds by Lemma 2.2. Thus, we may assume that there exist a ∈ N(a)∩V(P1)−V(P3) and a ∈ N(a)∩V(P3)−V(P1). Let L be obtainedfrom LbyreplacingP1,P3withwP1a,wP3a,respectively,and letR bea path inG1− {a,a1,a3}froma2 to a4. ThenL∪aaa∪R isaT K5 inGinwhich a∗ is notabranchvertex.So(i) holds.
Hence, we may assume that there exists a ∈ N(a)∩V(G2 −A) such that a ∈/ V(P1∪P3).RecallthatN(a)∩V(G2−A)⊆V(D).ThenD−V(P1∪P3) hasapathQ which iseitherfroma toa2 (anddisjoint from P4)or froma toa4 (anddisjoint from P2).
First,assumethatQisfromatoa2.LetP2 bethepathinP2∪Qbetweenwanda, andL beobtainedfromLbyreplacingP2withP2.LetR1beapathinG1[(C1−C2)+ {a1,a3}] froma1 toa3,andR2 beapathinG1[(C2−C1)+{a,a4}] fromatoa4.Now L∪R1∪(R2∪aa) isaT K5 inGinwhicha∗ isnotabranchvertex,and(i) holds.
SowemayassumethatQisfromatoa4.ThenletP4 bethepathinP4∪Qbetween w and a, and L be obtained from L by replacing P4 with P4. Let R1 be a path in G1[(C1−C2)+{a,a2}] fromatoa2,andR2beapathinG1[(C2−C1)+{a1,a3}] from
