in K The Kelmans-Seymour conjecture II: 2-Vertices Series B Journal of Combinatorial Theory,

Contents lists available atScienceDirect

Journal of Combinatorial Theory, Series B

www.elsevier.com/locate/jctb

The Kelmans-Seymour conjecture II: 2-Vertices in K

Dawei He¹, Yan Wang¹, Xingxing Yu²

SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta, GA30332-0160,USA

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received24February2016 Availableonline11December2019

Keywords:

Subdivisionofgraph Independentpaths Nonseparatingpath Planargraph

WeuseK⁻₄ todenotethegraphobtainedfromK4byremoving an edge, and useT K5 to denote a subdivision of K5. Let Gbe a 5-connected nonplanar graphand{x1,x2,y1,y2} ⊆ V(G) suchthatG[{x1,x2,y1,y2}]∼=K₄⁻withy1y2∈/E(G).

Letw1,w2,w3∈N(y2)− {x1,x2}bedistinct.Weshowthat G contains a T K5 in which y2 is not a branch vertex, or G−y2containsK⁻₄,orGhasaspecial5-separation,orG− {y2v:v /∈ {w1,w2,w3,x1,x2}}containsT K5.Thisresultwill beusedtoprovetheKelmans-Seymourconjecturethatevery 5-connectednonplanargraphcontainsT K5.

©2019ElsevierInc.Allrightsreserved.

1. Introduction

Weusenotation andterminology from [3]. Inparticular, foragraphK, weuseT K to denoteasubdivision ofK. Thevertices inaT K corresponding tothe verticesof K are itsbranch vertices. Kelmans [5] and, independently,Seymour [10] conjectured that

E-mailaddresses:dhe9@math.gatech.edu(D. He),yanwang@gatech.edu(Y. Wang), yu@math.gatech.edu(X. Yu).

1 PartiallysupportedbyNSFgrantthroughX.Yu.

2 PartiallysupportedbyNSFgrantsDMS-1265564andDMS-1600738.

https://doi.org/10.1016/j.jctb.2019.11.007 0095-8956/©2019ElsevierInc. Allrightsreserved.

every 5-connectednonplanar graphcontains T K5. In [6,7], this conjecture is shownto betrueforgraphscontainingK₄⁻.(WhenwesaythatagraphGcontainsanothergraph H,denotedasH ⊆G,wemeanthatGhasasubgraphisomorphictoH.)

This paper is the second in aseries of four papers inwhich we give aproof of the Kelmans-Seymourconjecture.Inthefirstpaper[3] oftheseries,weoutlinethestrategy of our proof, which we briefly sketchbelow. Let Gbe a 5-connectednonplanar graph, and letM beamaximalconnected subgraphofGsuchthatG/M (thegraphobtained from GbycontractingM)is 5-connectedand nonplanar.(NotethatweallowM to be asinglevertex graph.)Letz denote thevertex ofG/M representingthecontractionof M,andletH =G/M.Thenoneofthefollowingholds:

(a) H containsasubgraphK suchthatK∼=K₄⁻ andz hasdegree2inK.

(b) H containsasubgraphK suchthatK∼=K₄⁻ andz hasdegree3inK.

(c) H doesnotcontainK₄⁻ andthere existsT ⊆H,with z∈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.

Weremarkthatif(c)occursthenbyapplyingProposition 4.1in[3] wecanconclude thatH−V(T) containsK₄⁻,andhence,GcontainsK₄⁻;soGcontainsT K5bythemain result in[7]. Cases(c) and(d) willbe treatedintwo subsequentpapers.In thispaper, we deal with (a) bytaking advantage of the K₄⁻ containing z. Weprovethe following result,inwhichthevertex y₂ playstheroleofzabove.Notethatforpositiveintegerk, we usethenotation[k] for{1,. . . ,k}.

Theorem 1.1.LetG bea5-connectednonplanar graphand {x₁,x₂,y₁,y₂}⊆V(G) such that G[{x1,x2,y1,y2}]∼=K₄⁻ with y1y2∈/E(G).Thenoneof thefollowingholds:

(i) Gcontains aT K5 inwhichy2 isnota branch vertex.

(ii) G−y₂ containsK₄⁻.

(iii) G has a5-separation (G₁,G₂) suchthat V(G₁∩G₂)={y₂,a₁,a₂,a₃,a₄}, andG₂ is thegraph obtained from theedge-disjoint unionof the 8-cycle a1b1a2b2a3b3a4b4

a1 andthe4-cycle b1b2b3b4b1 byadding y2 andtheedgesy2bi fori∈[4].

(iv) For w1,w2,w3 ∈ N(y2)− {x1,x2}, G− {y2v : v /∈ {w1,w2,w3,x1,x2}} contains T K₅.

WenowbrieflyoutlinehowTheorem 1.1canbeused to resolvesituation(a);amore precise argument will be given inthefinal paper of theseries. Let Gbe a5-connected nonplanargraph,letMbeaconnectedsubgraphofG,andletzdenotethevertexinG/M representingthecontractionofM.SupposeG/M containsasubgraphK suchthatK∼= K₄⁻ andzhasdegree2inK.WeshowthatGcontainsT K₅.LetV(K)={x₁,x₂,y₁,y₂} with y1y2∈/E(G), andwemayassumez=y2.We nowapplyTheorem 1.1,with G/M

Fig. 1.The pathsX, A, B, C, P, Q.

playingtheroleof G. If(i) ofTheorem 1.1 occursthen G/M contains aT K5 inwhich y2isnotabranchvertex;soitiseasytoseethatGcontainsT K5.If(ii) ofTheorem1.1 occursthenG/M−y2containsK₄⁻;soGcontainsK₄⁻and,hence,GcontainsT K5bythe mainresultin[7].If(iii) ofTheorem1.1occursthenwemayapplyProposition 1.3in[3]

toreducetheproblemtoasituationsimilartothecasewhen(iv) ofTheorem1.1occurs.

Nowassume (iv) of Theorem 1.1occurs.Let M denote thesubgraph of Ginducedby M∪N(M), whereN(M) isthesetofallneighborsofM inG.SinceGis5-connected, thereexistx∈V(M) andw₁,w₂,w₃∈N(M)−{x₁,x₂}suchthatMhasfivepathsfrom wtox₁,x₂,w₁,w₂,w₃,withonlywincommon.Nowapply(iv) ofTheorem1.1withthis choiceofw1,w2,w3. Wecansee thataT K5 inG/M− {y2v :v /∈ {w1,w2,w3,x1,x2}}

canbe modified(ifnecessary)togiveaT K5 inG.

The arguments used inthis paper to proveTheorem 1.1 is similar to those used in [6,7]. Namely, we first find asubstructure in the graphwhich consists of eight special paths X,Y,Z,A,B,C,P,Q, see Fig. 1, where X is the path inbold and Y,Z are not shown.WewillthenattempttousethisintermediatestructuretofindthedesiredT K5

forTheorem 1.1.However,sincetheT K₅ weare lookingfor mustavoidy₂ asabranch vertexorusecertainspecialedgesaty₂,theargumentsherearemoreinvolvedandmake heavyuseoftheoption(ii).

Weorganize this paper as follows. InSection2, we collectafew knownresultsthat willbeusedintheproofofTheorem1.1.InSection3,wefindthepathX inGbetween x1 andx2whose deletionresultsinagraphsatisfyingcertainconnectivityrequirement.

In Section4, we find thepaths Y,Z,A,B,C,P,Qin Gand produce thedesired inter- mediatestructure inG. InSection5, we usethis structureto findthe desiredT K5 for Theorem1.1.

Weendthis sectionwith somenotation andterminology.LetGbe agraph.A sepa- ration inGconsists ofapairofsubgraphs G₁,G₂of G,denotedas(G₁,G₂),suchthat E(G₁)∪E(G₂)=E(G),E(G₁∩G₂)=∅, and neitherG₁ norG₂ is asubgraphof the other. The order of this separation is |V(G₁)∩V(G₂)|, and (G₁,G₂) is said to be a k-separation ifitsorder isk.WhenK⊆GandL⊆G,weletK−L=K−V(K∩L).

ForS⊆V(G),wemayviewS asasubgraphofGwithvertex setS andedgeset∅.For H ⊆G,NG(H) denotestheneighborhoodofH(notincludingtheverticesinV(H)).For

any x∈V(G),weuseNG(x) todenotetheneighborhood ofxinG.Whenunderstood, the reference to G maybe dropped. We mayview pathsas sequences ofvertices. The ends of apath P are the vertices of the minimum degree in P, and all other vertices of P (ifany) areitsinternal vertices. Acollectionof pathsis said tobeindependent if no vertex of any path in this collection is an internal vertex of any other path in the collection.

2. Previousresults

In thissection, we listafew knownresultsthatweneed. Webegin with atechnical notion.A3-planargraph(G,A) consistsofagraphGandacollectionA={A₁,. . . ,A_k} of pairwisedisjointsubsetsofV(G) (possiblyA=∅)suchthat

• fordistincti,j∈[k],N(Ai)∩Aj=∅,

• fori∈[k],|N(Ai)|≤3,and

• ifp(G,A) denotesthegraphobtainedfrom Gby(for eachi∈[k])deleting A_i and addingnewedges joiningeverypair ofdistinctvertices inN(Ai), thenp(G,A) can bedrawninacloseddiscwithnocrossingedges.

If, inaddition, b₁,. . . ,b_n are verticesinG suchthatb_i ∈/A_j foralli ∈[n] andj ∈[k], p(G,A) canbedrawninacloseddiscintheplanewithnocrossingedges,andb1,. . . ,bn

occurontheboundaryofthediscinthiscyclicorder,thenwesaythat(G,A,b1,. . . ,bn) is 3-planar. If there is no need to specify A, we will simply say that (G,b₁,. . . ,b_n) is 3-planar.

LetGbeagraphandA⊆V(G),andletkbeapositiveinteger.Recallfrom[3] that G is (k,A)-connected if, for any cutT of G with |T| < k, everycomponent of G−T contains avertex from A. SupposeGis(4,{b1,. . . ,bn})-connected.If (G,A,b1,. . . ,bn) is 3-planar then A=∅.Inthis case,we saythat(G,b₁,. . . ,b_n) is planar,and ifwe do notwishto specifytheorder ofb1,. . . ,bn thenwesaythat(G,{b1,. . . ,bn}) isplanar.

We can now state the following result of Seymour [11]; equivalent versions can be found in[1,13,12].

Lemma2.1. LetGbeagraphands₁,s₂,t₁,t₂bedistinctverticesofG.Thenexactlyone of thefollowingholds:

(i) Gcontains disjointpaths froms1,s2 tot1,t2,respectively.

(ii) (G,s₁,s₂,t₁,t₂)is 3-planar.

WealsostateageneralizationofLemma2.1,whichisaconsequenceofTheorems 2.3 and 2.4in[9].

Lemma 2.2. LetG be a graph, v1,. . . ,vn ∈ V(G) be distinct, and n≥4.Then exactly oneof thefollowingholds:

(i) Thereexist1≤i< j < k < l≤nsuchthat Gcontainsdisjointpaths fromv_i,v_j to v_k,v_l,respectively.

(ii) (G,v1,v2,. . . ,vn)is 3-planar.

Notethattheoutcomes(i),(ii) and(iii) inthenextthreelemmasareessentiallythe same.Thefirstofthese threelemmas isTheorem 1.1in[3].

Lemma2.3. LetGbe a5-connectednonplanar graph andlet (G1,G2)be a5-separation inG.Suppose|V(G_i)|≥7fori∈[2],a∈V(G₁∩G₂),and(G₂−a,V(G₁∩G₂)− {a}) isplanar.Then oneof thefollowingholds:

(i) GcontainsaT K5 inwhich aisnota branch vertex.

(ii) G−acontains K₄⁻.

(iii) G has a 5-separation (G₁,G₂) such that V(G₁∩G₂) = {a,a₁,a₂,a₃,a₄}, G₁ ⊆ G₁, and G₂ is the graph obtained from the edge-disjoint union of the 8-cycle a1b1a2b2a3b3a4b4a1 and the 4-cycle b1b2b3b4b1 by adding a and the edges abi for i∈[4].

ThenextresultweneedisTheorem 1.2from[3].

Lemma2.4. LetGbe a5-connectedgraph and(G₁,G₂)be a5-separationinG.Suppose that|V(Gi)|≥7fori∈[2] andG[V(G1∩G2)]containsatriangleaa1a2a.Thenoneof thefollowingholds:

(i) GcontainsaT K₅ inwhich aisnotabranch 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 addingaandtheedgesab_i fori∈[4].

(iv) For any distinct u₁,u₂,u₃ ∈ N(a)− {a₁,a₂}, G− {av : v /∈ {a₁,a₂,u₁,u₂,u₃}}

containsT K5.

WealsoneedProposition 4.2from[3].

Lemma 2.5.Let Gbe a 5-connected nonplanar graph and a∈V(G) suchthat G−a is planar.Thenone of thefollowingholds:

(i) GcontainsaT K₅ inwhich aisnota branch vertex.

(ii) G−acontains K₄⁻.

(iii) Ghas a5-separation(G1,G2)suchthatV(G1∩G2)={a,a1,a2,a3,a4}andG2 is the graph obtained from the edge-disjointunion of the 8-cycle a1b1a2b2a3b3a4b4a1

andthe4-cycle b1b2b3b4b1 byadding aandtheedgesabi fori∈[4].

We will make useof thefollowing result ofPerfect [8] on independent paths.A col- lection ofpathsinagrapharesaidto beindependent ifnointernalvertex ofapathin this collectionbelongsto anotherpathinthecollection.

Lemma 2.6. Let G be a graph, u ∈ V(G), and A ⊆ V(G−u). Suppose there exist k independent paths fromutodistinct a₁,. . . ,a_k∈A,respectively,andotherwise disjoint from A.Then forany n≥k,if there existnindependent pathsP1,. . . ,Pn in Gfrom u tondistinct verticesinAandotherwisedisjointfromA thenP1,. . . ,Pn may bechosen so thata_i∈V(P_i)fori∈[k].

WewillalsousearesultofWatkinsandMesner[14] oncyclesthroughthreevertices.

Lemma 2.7. Let G be a2-connected graph and let y1,y2,y3 be three distinct vertices of G.ThenthereisnocycleinGcontaining{y1,y2,y3}if,andonlyif,oneofthefollowing statements holds:

(i) Thereexistsa2-cutSinGandthereexistpairwisedisjointsubgraphsDyi ofG−S, i= 1,2,3,suchthatyi∈V(Dyi)andeach Dyi isaunion ofcomponentsofG−S.

(ii) There exist2-cuts S_yi of G,i= 1,2,3,z ∈S_y1∩S_y2∩S_y3,and pairwise disjoint subgraphs D_yi of G,such that y_i ∈V(D_yi),each D_yi isaunion of componentsof G−Syi,andSy1− {z},Sy2− {z},Sv− {z}are pairwisedisjoint.

(iii) There exist pairwise disjoint 2-cuts Syi in G, i = 1,2,3, and pairwise disjoint subgraphsD_yi ofR−S_yi suchthaty_i ∈V(D_yi),eachD_yi isaunionofcomponents ofG−S_yi,andG−V(D_y1∪D_y2∪D_y3)haspreciselytwocomponents,eachcontaining exactlyone vertexfrom Syi fori∈[3].

3. Nonseparatingpaths

OurstrategytoproveTheorem1.1istofindthepathsX,A,B,C,P,QinGthatform thestructure showninFig.1.Thegoalof thissectionistocomplete thefirststep:find thepathX whichisshowninbold-faceinFig.5.Aprecisestatementofthepropertiesof X isgiveninoutcome(iv) ofLemma3.2.ButfirstweneedtoproveLemma3.1,which willbe usedintheproofofLemma3.2tomodifycertainpathstoobtainthedesiredX.

Weneedtheconceptofchainofblocks.LetGbeagraphand{u,v}⊆V(G).Ablock in Gis eithera maximal 2-connectedsubgraph of Gor a subgraphof G inducedby a cutedgeofG.WesaythatasequenceofblocksB₁,. . . ,B_k inGisachainofblocksinG from uto v ifeitherk= 1 and u,v∈V(B₁) aredistinct,or k≥2,u∈V(B₁)−V(B₂), v ∈ V(Bk)−V(Bk−1), |V(Bi)∩V(Bi+1)|= 1 for i∈[k−1],and V(Bi)∩V(Bj)=∅

forany i,j ∈[k] with|i−j|≥2.For convenience,we alsoviewthis chainof blocksas k

i=1Bi, asubgraphofG.

The following result was implicit in [2,4]. Since it has not been stated and proved explicitlybefore,weincludeaproof.Weneedtheconceptofabridge.LetGbeagraph andH asubgraphofG.ThenanH-bridgeofGisasubgraphofGthatiseitherinduced by an edge of G−E(H) with both ends in V(H), or induced by the edges in some componentofG−H as wellas thoseedgesofGfromthatcomponenttoH.

Lemma 3.1.Let G be a graph and let x1,x2,y1,y2 ∈ V(G) be distinct such that G is (4,{x1,x2,y1,y2})-connected.Suppose thereexists apath X inG−x1x2 fromx1 tox2

suchthat G−X contains achain of blocksB fromy₁ toy₂.Then oneof thefollowing holds:

(i) Thereis a 4-separation (G1,G2)in Gsuch that B+{x1,x2}⊆G1, |V(G2)| ≥6, and(G2,V(G1∩G2))isplanar.

(ii) There exists aninduced path X in G−x₁x₂ from x₁ to x₂ suchthat G−X isa chainof blocksfromy₁ toy₂ andcontains B.

Proof. Without loss ofgenerality, we mayassume thatX isinduced inG−x1x2. We choosesuchX that

(1) B ismaximal(withrespectto subgraphcontainment),

(2) subjectto(1),thesmallestsizeofacomponentofG−X disjoint fromB (ifexists) isminimal, and

(3) subjectto (2),thenumberof componentsofG−X isminimal.

We claim that G−X is connected. For, suppose G−X is not connected and let D be a component of G−X other than B such that |V(D)| is minimal. Since G is (4,{x1,x2,y1,y2})-connected,thereexist distinctu,v∈N(D)∩V(X).Wechoosesuch u,vthatuXvismaximal.SinceGis(4,{x₁,x₂,y₁,y₂})-connected,uXv−{u,v}contains aneighbor of some component of G−X other than D. Let Q be an induced path in G[D+{u,v}] from uto v, and let X be obtainedfrom X by replacing uXv with Q.

ThenBiscontainedinB,thechainofblocksinG−Xfromy1toy2.Moreover,either thesmallestsizeofacomponentofG−X disjointfromB issmallerthanthesmallest sizeofa componentofG−X disjoint fromB,or thenumberofcomponentsofG−X issmallerthanthenumberofcomponentsofG−X.Thisgivesacontradictionto(1)or (2)or(3). Hence,G−X isconnected.

If G−X =B, then (ii) holdswith X := X. So assumeG−X = B. By(1), each B-bridgeof G−X hasexactly onevertex inB. Thus, foreach B-bridgeD of G−X, there exist unique b_D ∈ V(D)∩V(B). Since G is (4,{x₁,x₂,y₁,y₂})-connected, there existdistinctu_D,v_D∈N(D−b_D)∩V(X),andwechooseu_D andv_Dsuchthatu_DXv_D ismaximal.

We now define anew graphB suchthat V(B) isthe set of allB-bridges of G−X, and two B-bridges of G−X, C and D, are adjacent if uCXvC− {uC,vC}contains a neighbor of D−bD or uDXvD− {uD,vD}contains aneighborof C−bC.Let D be a connectedsubgraphofB.Then

D∈V(D)u_DXv_DisasubpathofX.LetS_Dbetheunion of {b_D :D∈V(D)}and theset ofthose vertices b∈V(B) withthe property thatbis containedinsome(B∪X)-bridgeofGthatcontainsinternal vertexof

D∈V(D)uDXvD. Wemayassumethat

(4) |S_D|≥3 for anycomponentDofB.

For,supposeBhasacomponentDsuchthat|S_D|≤2.Letu,v∈V(X) suchthatuXv=

D∈V(D)uDXvD.Then{u,v}∪S_DisacutinG.SinceGis(4,{x1,x2,y1,y2})-connected,

|S_D|= 2.Sothereis a4-separation(G1,G2) inGsuchthatV(G1∩G2)={u,v}∪S_D, B+{x₁,x₂}⊆G₁,andD⊆G₂forD∈V(D).Note that|V(G₂)|≥5.

Suppose|V(G₂)|= 5.ThenDconsistsofauniqueB-bridgeofG−X,sayD.Moreover,

|V(D)|= 2, and thevertex in V(D)−V(B) has allits neighborscontained{u,v,bD}, contradicting theassumptionthatGis(4,{x1,x2,y1,y2})-connected.

Therefore, |V(G2)|≥6.LetS_D ={b1,b2}.IfG2 hasdisjoint pathsS1,S2 from b1,u to b₂,v, respectively, then choose S₁ to be induced and let X = x₁Xu∪S₁∪vXx₂; now B∪S₂ iscontained inthe chainof blocks inG−X from y₁ to y₂, contradicting (1). Sono such two paths exist.Hence, byLemma2.1, (G2,u,b1,v,b2) is planar;thus (i) holds. 2

Now letDbe acomponentofB.Weclaimthat

(5) there exist D ∈ V(D), w₁,w₂ ∈ V(u_DXv_D)− {u_D,v_D}, and distinct b₁,b₂ ∈ S_D suchthatforeachi∈[2],{bi,wi}iscontainedina(B∪X)-bridgeofGdisjointfrom D−bD.

Toprove(5),letV(D)={D₁,. . . ,D_k}andassumethatu_D1Xv_D1 ismaximalamongall u_DiXv_Di,i∈[k].Moreover,sinceDisconnected,wemayassumethatforeachj∈[k], {D1,. . . ,Dj}inducesaconnectedsubgraphDj ofD.

Suppose (5) fails. Then for each j ∈ [k], all paths inG thatare from uDjXvDj − {uDj,vDj}to B andinternallydisjoint fromB∪X∪Dj mustendat acommonvertex inS_D.

We now apply induction on j to show that |S_Dj| ≤ 2 for j ∈ [k]. This is clear if j = 1; as otherwise we would have (5). Now assume that |S_Dj| ≤ 2 for some j ∈ [k−1].Bydefinition, Dj is connected.Therefore, sinceuD1XvD1 is maximaland Gis (4,{x1,x2,y1,y2})-connected, Dj+1 musthaveaneighborinuDiXvDi− {uDi,vDi}for some i ∈[j]. Hence,b_Dj+1 ∈ S_Dj. Toshow that |S_Dj+1|≤ 2,we also need to consider those vertices b ∈ V(B) with the property that there exists w ∈ V(u_Dj+1Xv_Dj+1)− {uDj+1,vDj+1}suchthat{b,w}iscontainedinsome(B∪X)-bridgeofGdisjoint from

Dj+1. Suppose b ∈/ S_Dj for some choice of b. Then w /∈ V(uDiXvDi)− {uDi,vDi} and, hence, Di has a neighbor in uDj+1XvDj+1 − {uDj+1,vDj+1}; which implies that b=bDi ∈S_Dj,acontradiction.Therefore,b∈S_Dj forallchoicesofsuchb;so|S_Dj+1|≤2.

Thus,|S_D|≤2,contradicting(4).So(5)holds. 2

LetP denote an inducedpath inG[D+{uD,vD}] betweenuD andvD, andlet X beobtainedfromX byreplacinguDXvD withP.Clearly,thechainofblocksinG−X fromy₁ toy₂ containsB as wellas apathfromb₁ tob₂ thatisinternallydisjointfrom D∪B. Thisisacontradiction to(1).

WenowshowthattheconclusionofTheorem1.1holdsorwecanfindapathX inG suchthaty1,y2 ∈/V(X) and (G−y2)−X is2-connected. Notethattheoutcomes (i), (ii) and(iii) in thenextlemma arethesameasthose inLemmas 2.3,2.4and2.5.

Lemma 3.2. Let G be a 5-connected nonplanar graph and let x1,x2,y1,y2 ∈ V(G) be distinctsuchthat G[{x1,x2,y1,y2}]∼=K₄⁻ with y1y2∈/E(G).Thenone ofthefollowing holds:

(i) GcontainsaT K₅ inwhich y₂ isnot abranchvertex.

(ii) G−y2 contains K₄⁻.

(iii) Ghasa5-separation(G1,G2)suchthat V(G1∩G2)={y2,a1,a2,a3,a4}andG2 is thegraph obtained from theedge-disjoint union of the8-cycle a₁b₁a₂b₂a₃b₃a₄b₄a₁ andthe4-cycle b₁b₂b₃b₄b₁ by addingy₂ andtheedgesy₂b_i fori∈[4].

(iv) For w1,w2,w3 ∈ N(y2)− {x1,x2}, G− {y2v : v /∈ {w1,w2,w3,x1,x2}} contains T K5,or G−x1x2 has aninduced pathX fromx1 to x2 such that y1,y2∈/V(X), w₁,w₂,w₃∈V(X),and(G−y₂)−X is2-connected.

Proof. First,wemayassumethat

(1) G−x1x2 has an induced path X from x1 to x2 such that y1,y2 ∈/ V(X) and (G−y₂)−X is2-connected.

To see this, wefirst show thatthere exists z ∈N(y1)− {x1,x2}suchthatK := (G− x1x2)−y2−y1z has disjoint paths from x1,y1 to x2,z, respectively. For, suppose z ∈ N(y1)− {x1,x2}andKhasnodisjointpathsfromx1,y1 tox2,z,respectively.Thenby Lemma2.1, (K,x₁,y₁,x₂,z) is planar. Let z₁,z₂ ∈N(y₁)− {x₁,x₂,z}. Then, since G is5-connectedand(K,x₁,y₁,x₂,z) isplanar,K−y₁z₁ hasdisjoint pathsfrom x₁,y₁to x2,z2, respective.Hence,z2 givesthedesiredchoiceforz.

Since Gis 5-connected,(G−x1x2)− {y1,y2,z}hasa pathX from x1 to x2. Thus, wemayapply Lemma3.1to G−y₂,X andB=y₁z.

Suppose(i) ofLemma3.1 holds.Then Ghasa5-separation(G₁,G₂) suchthaty₂∈ V(G1∩G2),{x1,x2,y1,z}⊆V(G1) andy1z∈E(G1),|V(G2)|≥7,and(G2−y2,V(G1∩

G2)− {y2}) is planar. If |V(G1)| ≥ 7 then, by Lemma 2.3, (i) or (ii) or (iii) of this lemmaholds.Nowassume|V(G1)|= 5.ThenV(G1∩G2)={x1,x2,y1,y2,z}.Moreover, (G2−y2,x1,x2,y1,z) isplanar,since(G−x1x2)−y2−y1zhasdisjointpathsfromx1,y1

tox₂,z,respectively.Hence,G₁−y₂hasaK₄⁻orG−y₂ isplanar;so(ii) ofthislemma holdsintheformercase,and(i) or(ii) or(iii) ofthislemmaholdsinthelattercaseby Lemma2.5.Thus wemayassumethat|V(G1)|= 6.Letv∈V(G1−G2).Thenv=y2. SinceGis5-connected,vmustbeadjacenttoallverticesinV(G1∩G2).Thus,v=y1as y1y2∈/E(G).Now |V(G1∩G2)∩ {x1,x2,z}|≥2.Therefore,G[{v,y1}∪(V(G1∩G2)∩ {x₁,x₂,z})] containsK₄⁻;so(ii) holds.

Sowemayassumethat(ii) ofLemma3.1holds.Then(G−y2)−x1x2hasaninduced path,alsodenotedbyX,fromx1tox2suchthat(G−y2)−X isachainofblocksfrom y1to z.Sincezy1∈E(G),(G−y2)−X isinfactablock.IfV((G−y2)−X)={y1,z} then,sinceGis5-connectedandX isinducedin(G−y₂)−x₁x₂,G[{x₁,x₂,z,y₁}]∼=K₄; so (ii) holds.Hence,V((G−y₂)−X)={y₁,z}and(G−y₂)−X is2-connected;thus (1) holds. 2

Wewish toprove(iv).Soletw1,w2,w3∈N(y2)− {x1,x2}andassumethat G:=G− {y₂v:v /∈ {w₁, w₂, w₃, x₁, x₂}}

doesnotcontainT K₅.Wemayassumethat (2) w1,w2,w3∈/V(X).

For, suppose not. Ifw₁,w₂,w₃ ∈V(X) then (iv) holds. So, without loss of generality, we may assume w1 ∈ V(X)− {x1,x2} and w2 ∈ V(G−X). Since X is induced in G−x1x2 and G is 5-connected, w1 has at least two neighbors in (G−y2)−X; so (G−y2)−(X−w1) is2-connectedand,hence,containsindependentpathsP1,P2fromy1

tow₁,w₂,respectively.Thenw₁Xx₁∪w₁Xx₂∪w₁y₂∪P₁∪(y₂w₂∪P₂)∪G[{x₁,x₂,y₁,y₂}] is aT K₅inG withbranchverticesw₁,x₁,x₂,y₁,y₂,acontradiction. 2

(3) Forany u∈V(x1Xx2)− {x1,x2}, {u,y1,y2}is notcontainedinany cycleinG− (X−u).

For, suppose thereexists u∈V(x₁Xx₂)− {x₁,x₂}suchthat{u,y₁,y₂}is containedin acycleC inG−(X−u).Then uXx1∪uXx2∪C∪G[{x1,x2,y1,y2}] is aT K5 inG with branchvertices u,x1,x2,y1,y2,acontradiction.Sowehave(3). 2

Lety3∈V(X) suchthaty3x2∈E(X),and letH :=G−(X−y3). By(1) and(2), G−X is2-connected.Hence,sinceX isinducedinG−x₁x₂ andGis5-connected,y₃ hasatleasttwoneighborsinG−X;soH is2-connected.By(3),nocycleinH contains {y1,y2,y3}.Thus,weapplyLemma2.7toH.Inordertotreatsimultaneouslythethree

outcomes of Lemma 2.7, we introduce some notation (based on those in Lemma 2.7).

Let Syi = {ai,bi}for i ∈ [3], such that: if Lemma2.7(i) occurs we let a1 = a2 = a3, b1 = b2 = b3, and Syi =S for i∈ [3]; ifLemma 2.7(ii) occurs then leta1 = a2 =a3; andifLemma2.7(iii) then{a1,a2,a3}and{b1,b2,b3}belongtodifferentcomponentsof H−V(D_y1∪D_y2∪D_y3).LetB_a,B_b denotethecomponentsofH−V(D_y1∪D_y2∪D_y3) suchthat for i∈ [3], a_i ∈V(B_a) and b_i ∈V(B_b). Note that B_a =B_b is possible,but onlyifLemma2.7(i) orLemma2.7(ii) occurs.

For convenience, let D_i := G[Dyi+{ai,bi}] for i ∈ [3]. We choose the cutsSyi so that

(4) D₁∪D₂∪D₃ismaximal.

By(4), ifLemma 2.7(i) occursthen we havea1 =a2 =a3 and V(Ba)={a1}, and b1=b2=b3 andV(Bb)={b1}.SinceH is2-connected,D_i,foreachi∈[3],containsa pathYi fromai tobi andthroughyi.Inaddition,since(G−y2)−X is2-connected,for anyv∈V(D₃)− {a₃,b₃,y₃},D₃ −y₃containsapathfroma₃ tob₃throughv.

(5) If B_a∩B_b = ∅ then |V(B_a)| = 1 or B_a is 2-connected, and |V(B_b)| = 1 or B_b is 2-connected.IfBa∩Bb=∅thenBa=Bb and Ba−a3 is2-connected.

First,supposeBa∩Bb=∅.Bysymmetry,weonlyprovetheclaimforBa.If|V(Ba)|= 1 orB_a is2-connectedthenwehave(5).Soassume|V(B_a)|>1 (hence|V(B_a)|≥3) and B_a isnot2-connected.Then B_ahasaseparation(B₁,B₂) suchthat|V(B₁∩B₂)|≤1.

Since Ba is connected and H is 2-connected, |V(B1 ∩B2)| = 1 and we may assume that, for some permutation ijk of [3], ai ∈ V(B1)−V(B2) and aj,ak ∈ V(B2) (or bi∈V(B1)−V(B2) andbj,bk∈V(B2) whenLemma2.7(ii) occurs).Replacing Syi,D_i byV(B1∩B2)∪ {bi},D_i∪B1, respectively,while keeping Syj,D_j,Syk,D_k unchanged, wederiveacontradictionto(4).

Now assume B_a∩B_b = ∅. Then B_a = B_b by definition, and a₁ = a₂ = a₃ by our assumptionaboveforthecasewhenLemma2.7(ii) occurs.NotethatBa−a3isconnected (sinceH is2-connected).SupposeBa−a3isnot2-connected.ThenBahasa2-separation (B1,B2) with a3 ∈ V(B1∩B2). First, suppose for some permutation ijk of [3], bi ∈ V(B₁)−V(B₂) and b_j,b_k ∈ V(B₂). Then replacing S_yi,D_i by V(B₁∩B₂),D_i∪B₁, respectively,whilekeepingS_yj,D_j,S_yk,D_k unchanged,wederiveacontradictionto (4).

Therefore, bythesymmetry betweenB₁ and B₂,we mayassume{b₁,b₂,b₃}⊆V(B₁).

Since G is 5-connected, there exists rr ∈ E(G) such that r ∈ V(X)− {y3,x2} and r ∈V(B2−B1). Let R be a pathin B2−(B1−a3) from a3 to r, and R apath in B1−a3fromb1tob2.Then(R∪rr∪rXx1)∪(a3Y3y3∪y3x2)∪a3Y1y1∪a3Y2y2∪(y1Y1b1∪ R∪b₂Y₂y₂)∪G[{x₁,x₂,y₁,y₂}] is aT K₅ inG withbranchverticesa₃,x₁,x₂,y₁,y₂,a contradiction. 2

(6) Dyi isconnectedfori∈[3].