K The Kelmans-Seymour conjecture III: 3-vertices in Series B Journal of Combinatorial Theory,

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Contents lists available atScienceDirect

Journal of Combinatorial Theory, Series B

www.elsevier.com/locate/jctb

The Kelmans-Seymour conjecture III: 3-vertices in K

₄⁻

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:

Received19September2016 Availableonline9December2019

Keywords:

Subdivisionofgraph Independentpaths Nonseparatingpath Planargraph

LetGbea5-connectednonplanargraphandletx1,x2,y1,y2∈ V(G) be distinct, such that G[{x1,x2,y1,y2}] ∼= K₄⁻ and y1y2 ∈/ E(G). We show that one of the following holds:

G−x1 contains K₄⁻, or G contains a K₄⁻ in which x1 is ofdegree2,orGcontainsaT K5inwhichx1isnotabranch vertex,or{x2,y1,y2}maybechosensothatforanydistinct z0,z1∈N(x1)−{x2,y1,y2},G−{x1v:v /∈ {z0,z1,x2,y1,y2}}

containsT K5.

1. Introduction

Weusenotationand terminologyfrom[2,3].ForagraphK,weuse T Kto denotea subdivisionofK.TheverticesofT KcorrespondingtotheverticesofKareitsbranchver- tices.Kelmans [6] and,independently,Seymour [11] conjecturedthatevery5-connected nonplanargraphcontains T K5. In[7,8], this conjecture is shownto betrue forgraphs containingK₄⁻.

E-mailaddresses:[email protected](D. He),[email protected](Y. Wang), [email protected](X. Yu).

1 PartiallysupportedbyNSFgrantsDMS-1265564andDMS-1600738throughX.Yu.

2 PartiallysupportedbyNSFgrantsDMS-1265564andDMS-1600738.

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In[2] weoutlineastrategytoprovetheKelmans-Seymourconjectureforgraphsnot containingK₄⁻.LetGbea5-connectednonplanargraphnotcontainingK₄⁻.Thenbya resultof Kawarabayashi [4], Gcontainsanedgee suchthatG/e is5-connected.IfG/e is planar,we canapply a discharging argument. So assume G/e is notplanar. Let M be amaximal connected subgraph of Gsuch thatG/M is 5-connectedand nonplanar (so |V(M)| ≥ 2). Let z denote the vertex representing the contraction of M, and let H =G/M.Thenoneofthefollowingholds:

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

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

(c) H doesnotcontainK₄⁻,andthere existsT ⊆H,withz∈V(T) andeitherT ∼=K₂ or T ∼=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.

Note that local structure around z (in particular, K₄⁻ containing z) will help us ﬁnd T K₅ inGfrom certainT K₅ inH.

In[2] wedealwithcertainspecialseparationsandtheresultscanbeusedtotakecare of(c).In[3] weproveresultsthatcanbeusedtotakecareof(a).Inthispaper,weprove thefollowing,whichcanbe usedtotakecareof(b).

Theorem 1.1. LetGbe a5-connectednonplanar graph and x1,x2,y1,y2∈V(G) be dis- tinct such that G[{x1,x2,y1,y2}] ∼= K₄⁻ and y1y2 ∈/ E(G). Then one of the following holds:

(i) Gcontains aT K5 inwhichx1 isnotabranch vertex.

(ii) G−x1 containsK₄⁻,orGcontainsasubgraphisomorphic toK₄⁻ inwhichx1isof degree 2.

(iii) x2,y1,y2 maybe chosensothat foranydistinct z0,z1∈N(x1)− {x2,y1,y2},G− {x1v:v /∈ {z0,z1,x2,y1,y2}}containsT K5.

This paper is organized as follows. InSection 2, we list anumber of known results thatwill beused intheproof ofTheorem1.1.ThestepswetaketoproveTheorem1.1 is quite similar to the arguments in[3]. First,we find apath inGfrom x1 to x2 such that the graph obtained from G by removing that path satisfies certain connectivity requirements. What isdifferent hereis thatwe need the pathto include x1z0 or x1z1. WefindthispathinSection3,seeFigs.1and2.InSection4,wederivefurtherstructural information ofthegraphG.InSection5,wefind asubstructure ofGconsisting offive additional paths, see Fig. 3. In Section 6, we use this substructure to find a T K5 in G− {x1v:v /∈ {z0,z1,x2,y1,y2}}.

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2. Lemmas

Foreachpositiveintegerm,let[m]={1,. . . ,m}.Forconvenience,werecallatechnical notionfrom[2] (originatedfrom[12]).A3-planargraph(G,A) consistsofagraphGand aset A={A1,. . . ,Ak}ofpairwisedisjoint subsetsofV(G) (possibly A=∅)suchthat

• fordistincti,j ∈[k], N(Ai)∩Aj =∅,

• fori∈[k], |N(A_i)|≤3,and

• ifp(G,A) denotes thegraphobtainedfrom Gby (foreachi∈[k]) deletingA_i and adding newedgesjoiningeverypairofdistinct verticesinN(Ai),thenp(G,A) can be drawninacloseddisc intheplanewithnoedgecrossing.

If,inaddition, b₁,. . . ,b_n arevertices inGsuchthatb_i∈/A_j for alli∈[n] andj ∈[k], p(G,A) canbedrawninacloseddiscintheplanewithnoedgecrossing,andb1,. . . ,bn

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

We can now state the following result of Seymour [12]; equivalent versions can be foundin[1,14,13].

Lemma2.1.LetGbeagraphands₁,s₂,t₁,t₂bedistinct verticesof G.Thenexactlyone ofthefollowingholds:

(i) Gcontains disjointpathsfrom s1,s2 tot1,t2,respectively.

(ii) (G,s1,s2,t1,t2)is3-planar.

WealsostateageneralizationofLemma2.1,whichisaconsequenceofTheorems2.3 and2.4in[10].

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 fromvi,vj to vk,vl,respectively.

(ii) (G,v₁,v₂,. . . ,v_n)is 3-planar.

WewillmakeuseofthefollowingresultofPerfect[9].Acollectionofpathsinagraph aresaidtobeindependent ifnointernalvertexofany pathinthiscollectionbelongsto anotherpathinthecollection.

Lemma 2.3.Let G be a graph, u ∈ V(G), and A ⊆ V(G−u). Suppose there exist k independentpaths fromutodistinct a1,. . . ,ak ∈A,respectively,and otherwisedisjoint

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from A.Then forany n≥k,if there existnindependent pathsP1,. . . ,Pn in Gfrom u tondistinct verticesinAandotherwisedisjointfromA thenP1,. . . ,Pn may bechosen so thatai∈V(Pi)fori∈[k].

WewillalsousearesultofWatkinsandMesner[15] oncyclesthroughthreevertices.

Lemma2.4. LetGbea2-connectedgraphandlety1,y2,y3bethreedistinctverticesofG.

There isnocycleinGcontaining {y1,y2,y3}if, andonlyif,oneof thefollowingholds:

(i) Thereexistsa2-cutSinGandthereexistpairwisedisjointsubgraphsDyiofG−S, i∈[3],suchthatyi∈V(Dyi)andeach Dyi isa unionof componentsofG−S.

(ii) There exist 2-cuts Syi of G, i ∈ [3], and pairwise disjoint subgraphs Dyi of G, suchthat yi∈V(Dyi),each Dyi is aunionof componentsof G−Syi,there exists z∈S_y₁∩S_y₂∩S_y₃,andS_y₁− {z},S_y₂− {z},S_y₃− {z}arepairwisedisjoint.

(iii) Thereexistpairwisedisjoint2-cutsS_y_i inG,i∈[3],andpairwisedisjointsubgraphs D_y_i of G−S_y_i such that y_i ∈ V(D_y_i), D_y_i isa union of components of G−S_y_i, andG−V(D_y₁∪D_y₂∪D_y₃)haspreciselytwocomponents,eachcontainingexactly one vertexfromS_y_i fori∈[3].

ThenextresultisTheorem3.2from [7].

Lemma 2.5. Let G be a 5-connected nonplanar graph and let x1,x2,y1,y2 ∈ V(G) be distinct suchthat G[{x1,x2,y1,y2}]∼=K₄⁻ andy1y2∈/E(G).SupposeG−x1x2contains a path X betweenx1 and x2 such that G−X is 2-connected, X−x2 isinduced in G, and y1,y2 ∈/ V(X). Letv ∈ V(X) such that x2v ∈ E(X). Then G contains a T K5 in which x2v isan edgeandx1,x2,y1,y2 arebranch vertices.

ItiseasytoseethatundertheconditionsofLemma2.5,G−{x2u:u∈ {/ v,x1,y1,y2}}

containsT K5.ThenextresultisCorollary2.11in[5].LetGbeagraphandA⊆V(G).

Foranypositiveintegerk,wesaythatGis(k,A)-connectedif,foreachS⊆V(G) with

|S|< k, everycomponentof G−S must containavertex from A.We saythat(G,A) is plane if G is drawn in the plane with no edge crossings, and the vertices in A are incidentwiththeouterfaceofG;andwesaythat(G,A) isplanar ifGadmitsaplanar drawing suchthat(G,A) isplane.

Lemma 2.6.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. Suppose there exists w ∈ N(a) such thatw isnotincident with theouterface ofG−a.Then

(i) theverticesofG−acofacialwith winduce acycleCw inG−a,and

(ii) G−a containspaths P1,P2,P3 fromw toA− {a}suchthat V(Pi∩Pj)={w}for 1≤i< j ≤3,and|V(Pi∩Cw)|=|V(Pi)∩A|= 1fori∈[3].

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Thenextfourresultsare(essentially)Theorem1.1,Theorem1.2,Proposition2.3and Proposition4.2,respectively,in[2].Notethatcondition(iii) inthreeofthesefourresults (Theorem1.1,Theorem 1.2andProposition4.2in[2])statesthatGhasa5-separation (G₁,G₂) such thatV(G₁∩G₂)={a,a₁,a₂,a₃,a₄}and G₂ isthe graphobtainedfrom theedge-disjointunionof the8-cyclea₁b₁a₂b₂a₃b₃a₄b₄a₁ and the4-cycleb₁b₂b₃b₄b₁ by addinga andthe edgesab_i fori∈[4].This conditionimplies thatGcontainsaK₄⁻ in whichaisofdegree2.Weonlyneedtheweakerversionsofthese results.

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

(i) Gcontains aT K5 in whichaisnot abranchvertex.

(ii) G−a contains K₄⁻,or G contains a subgraph isomorphic to K₄⁻ in which a isof degree 2.

Lemma2.8. 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 inwhicha isnotabranch vertex.

(ii) G−a contains K₄⁻,or G containsa subgraphisomorphic toK₄⁻ in whicha isof degree2.

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

containsT K5.

Lemma 2.9. Let G be a graph, A ⊆V(G), and a ∈ A such that |A| = 6, |V(G)| ≥ 8, (G−a,A− {a})isplanar, andGis(5,A)-connected.Thenone ofthefollowingholds:

(i) G−acontainsK₄⁻,orGcontainsasubgraphisomorphic toK₄⁻ inwhichthedegree of ais2.

(ii) Ghas a5-separation(G1,G2)suchthata∈V(G1∩G2),A⊆V(G1),|V(G2)|≥7, and(G2−a,V(G1∩G2)− {a})isplanar.

Lemma2.10.LetG bea5-connectednonplanar graph anda∈V(G)such thatG−ais planar.Thenone of thefollowingholds:

(i) Gcontains aT K5 in whichaisnot abranchvertex.

(ii) G−a contains K₄⁻,or G contains a subgraph isomorphic to K₄⁻ in which a isof degree 2.

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Wealso needLemma3.1 in[3].LetGbe agraphand{u,v}⊆V(G).Wesaythata sequenceofblocksB1,. . . ,BkinGisachainofblocksfromutovif|V(Bi)∩V(Bi+1)|= 1 for i ∈ [k−1], V(Bi)∩V(Bj) =∅ for any 1 ≤i < i+ 1 < j ≤ k, u,v ∈ V(B1) are distinct whenk= 1,andu∈V(B₁)−V(B₂) andv∈V(B_k)−V(B_k−1) whenk≥2.A blockisnontrivial ifitis2-connected.

Lemma 2.11. Let G be a graph and A = {x1,x2,y1,y2} ⊆ V(G) such that G is (4,A)-connected. Suppose there exists a path X in G−x1x2 from x1 to x2 such that G−X containsachain ofblocksB fromy1 toy2.Thenoneof thefollowingholds:

(i) Thereis a 4-separation (G₁,G₂) in G suchthat B+{x₁,x₂}⊆G₁,|V(G₂)|≥6, and(G2,V(G1∩G2))is planar.

(ii) There existsan induced path X in G−x1x2 from x1 tox2 such that G−X is a chainof blocksfromy1 toy2 andcontains B.

3. Nonseparatingpaths

LetGbea5-connectednonplanargraphandx1,x2,y1,y2∈V(G) bedistinctsuchthat G[{x1,x2,y1,y2}]∼=K₄⁻andy1y2∈/E(G).Totakecareofcase(b)describedinSection1, we needto ﬁndapathinGsatisfyingcertain properties(see (iv) ofLemma3.2).As a ﬁrststep,weprovethefollowing.

Lemma3.1. LetGbea5-connectednonplanargraphandx1,x2,y1,y2∈V(G)bedistinct suchthat G[{x1,x2,y1,y2}]∼=K₄⁻ andy1y2∈/E(G).Letz0,z1∈N(x1)− {x2,y1,y2}be distinct. Thenoneof thefollowingholds:

(i) Gcontains aT K₅ inwhichx₁ isnotabranch vertex.

(iii) There exist i ∈ {0,1} and an induced path X in G−x1 from zi to x2 such that (G−x₁)−X isachain of blocksfromy₁ toy₂,z_1−i∈/V(X),andone ofy₁,y₂ is containedin anontrivial blockof (G−x₁)−X.

Proof. We may assume G−x1 contains disjoint paths X,Y from z1,y1 to x2,y2, respectively. For, otherwise, since G is 5-connected, it follows from Lemma 2.1 that (G−x₁,z₁,y₁,x₂,y₂) isplanar;so (i) or(ii) holdsbyLemma2.10.

Hence(G−x1)−X containsachainofblocks fromy1 toy2,sayB.Wemayassume that (G−x1)−X is a chain of blocks from y1 to y2. For otherwise, we may apply Lemma2.11toconcludethatGhasa5-separation(G1,G2) suchthatx1∈V(G1∩G2), B +{x₁,x₂,z₁} ⊆ G₁, |V(G₂)| ≥ 7, and (G₂−x₁,V(G₁∩G₂)− {x₁}) is planar. If

|V(G₁)|≥7 then(i) or(ii) followsfromLemma2.7.Soassume|V(G₁)|≤6.Sincey₁y₂∈/ E(G),|V(G1)|= 6 and|V(B)|= 3.LetV(B)={y1,y2,v}.SinceGis 5-connectedand

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Fig. 1.Structure ofGin (iii) of Lemma3.2.

Fig. 2.Structure ofGin (iv) of Lemma3.2(withj= 1).

y1y2 ∈/E(G),{x1,x2,y1,y2,z1}=V(G1∩G2)=N(v).Hence,G[{v,x1,x2,y1}]−x1x2

isaK₄⁻ inwhichx₁ isofdegree2,and(ii) holds.

We may further assume that z0 ∈/ V(X). For, suppose z0 ∈ V(X). Since G is 5-connected and X is induced in G−x₁, every vertex of X has at least two neigh- borsin(G−x1)−X.Hence,(G−x1)−z0Xx2 isalsoachainof blocksfrom y1 toy2. Sowemayusez₀Xx₂ asX.

LetB1,B2betheblocksin(G−x1)−Xcontainingy1,y2,respectively.IfoneofB1,B2

isnontrivial,then(iii) holds.Sowemayassumethat|V(B₁)|=|V(B₂)|= 2.SinceX is inducedandGis5-connected,thereexistsz∈N(x2)−({x1,y1,y2}∪V(X)),andy1and y₂eachhaveatleasttwoneighborsonX−x₂.LetZbeapathin(G−x₁)−X−{y₁,y₂} from z0 to z. Then(G−x1)−Z containsachainof blocks,sayB, from y1 to y2,and the blocks in (G−x₁)−Z containing y₁ or y₂ are nontrivial. Thus, we may apply Lemma2.11to G, Z and B. If (ii) ofLemma2.11 holds, we have(iii). Soassume (i) ofLemma2.11holds. Then,as inthesecond paragraphofthis proof,(i) or(ii) follows fromLemma2.7. 2

Wehaveresultsfrom [2,3,8] that canbe used todeal with(i) or(ii) of Lemma3.1.

Inthis paper, wedeal with(iii) ofLemma3.1. Parts(iii) and(iv) of thenextlemma givemoredetailedstructure of Gwhen(iii) of Lemma3.1 occurs.Werefer thereader toFig.1for(iii) ofLemma3.2,and Fig.2for(iv) ofLemma3.2.

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For agraphH and asubgraph Lof H, anL-bridgeof H isasubgraphof H thatis induced byanedgeinE(H)−E(L) with bothincident verticesinV(L),or is induced bytheedgesinacomponentofH−Las wellasedges fromthatcomponentto L.

Lemma3.2. LetGbea5-connectednonplanargraphandx1,x2,y1,y2∈V(G)bedistinct such that G[{x₁,x₂,y₁,y₂}] ∼=K₄⁻ and y₁y₂ ∈/ E(G). Let z₀,z₁ ∈N(x₁)− {x₂,y₁,y₂} be distinct andlet G :=G− {x1x:x∈ {/ x2,y1,y2,z0,z1}}.Then one of the following holds:

(i) G containsT K5,or GcontainsaT K5 inwhich x1 isnot abranchvertex.

(iii) The notationofy1,y2,z0,z1 maybechosensothat(G−x1)−x2y2 has aninduced path X fromz₁ tox₂ suchthat z₀,y₁∈/V(X),and(G−x₁)−X is2-connected.

(iv) Thenotationofz0,z1maybechosensothatthereexistsaninducedpathX inG−x1

from z1 tox2 such that z0∈/V(X), (G−x1)−X isa chain of blocksB1,. . . ,Bk

from y₁ to y₂ with B₁ nontrivial, z₀ ∈V(B₁)when z₁ has at least two neighbors in B1, and (G−x1)−x2y2 has a 3-separation (Y1,Y2) such that V(Y1∩Y2) = {b,p₁,p₂},z₁,p₁,p₂,x₂ occuronX inthis order,Y₁=G[B₁∪z₁Xp₁∪p₂Xx₂+b], p1Xp2+y2⊆Y2,andp1,p2eachhaveatleasttwoneighborsinY2−B1.Moreover,if b∈/V(B1)thenV(B2)={b1,b}withb1∈V(B1),andthereexistssomej∈[2]such that p_3−j has auniqueneighborb₁ inB₁,b hasauniqueneighborv inX−p₁Xp₂ suchthat vp3−j∈E(X)−E(p1Xp2),vb1∈/E(G),andpjb∈/E(G).

Proof. We begin our proof by applying Lemma 3.1 to G,x₁,x₂,y₁,y₂. If (i) or (ii) of Lemma3.1holdsthen assertion(i) or (ii) ofthislemmaholds. Sowe mayassumethat (iii) ofLemma3.1holds.Thenwemayassume(G−x1)−x2y2 hasaninducedpathX fromz₁tox₂suchthatz₀,y₁∈/V(X),(G−x₁)−X hasanontrivialblockB₁containing y1, and y1 is not a cut vertex of (G−x1)−X. (Note that we are not requiring the stronger condition that (G−x₁)−X be achain of blocks from y₁ to y₂.) We choose suchapathX that

(1) B1ismaximal,

(2) subject to(1), wheneverpossible,(G−x₁)−X hasachainofblocksfrom y₁ toy₂ andcontainingB1,and

(3) subject to(2), thecomponentH of(G−x1)−X containingB1 ismaximal.

LetC betheset ofallcomponentsof(G−x₁)−X diﬀerent fromH.Then

(4) C=∅,H = (G−x₁)−X,andify₂∈/V(X) then H isachainofblocksfrom y₁ to y2andcontainingB1.

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First, suppose C = ∅. Then H = (G−x1)−X. Suppose y2 ∈/ V(X). Then H has a chainof blocks,say B, from y1 to y2 and containing B1. By applying Lemma 2.11to G−x₁,z₁,x₂,y₁,y₂,X canbechosensothat(G−x₁)−X isachainofblocksfromy₁to y₂,orGhasa5-separation(G₁,G₂) suchthatx₁∈V(G₁∩G₂),B+{x₁,x₂,z₁}⊆G₁,

|V(G2)|≥7 and(G2−x1,V(G1∩G2)− {x1}) isplanar.Wemayassumethelatteras otherwise(4)holds.SinceB1isnontrivialandy1y2∈/E(G),|V(B)|≥4.So|V(G1)|≥7;

and(i) or(ii) followsfrom Lemma2.7.

Now suppose C =∅. For each D ∈ C, letuD,vD ∈ V(X) be the neighborsof D in G−x2y2withuDXvDmaximal,andassumethatz1,uD,vD,x2occuronXinthisorder.

DeﬁneanewgraphG_CsuchthatV(G_C)=C,andtwocomponentsC,D∈ Careadjacent inG_C ifu_CXv_C− {u_C,v_C}containsaneighborofD oru_DXv_D− {u_D,v_D}containsa neighborofC.

Notethat,foranycomponentDofG_C,

D∈V(D)uDXvDisasubpathofX.SinceGis 5-connected,thereexisty∈V(H) andC∈V(D) withN(y)∩V(u_CXv_C−{u_C,v_C})=∅. Ify=y1thenletQbeaninducedpathinG[C+{uC,vC}]−x2y2fromuCtovC,and letX beobtainedfromXbyreplacinguCXvCwithQ.ThenB1iscontainedinablock of(G−x₁)−X,andy₁ isnotacutvertexof(G−x₁)−X.Moreover,if(G−x₁)−X hasachainofblocksfromy₁ toy₂thensodoes(G−x₁)−X.However,thecomponent of(G−x1)−X containingB1 islargerthanH,contradicting(3).

So we may assume that y = y1 for all choices of y and C. Let uXv :=

D∈V(D)u_DXv_D. Since G is 5-connected, y₂ ∈ V(

D∈V(D)D)∪V(uXv − {u,v}) and G has a separation (G1,G2) such that V(G1 ∩G2) = {u,v,x1,x2,y1}, G1 :=

G[

D∈V(D)D∪uXv+{x1,x2,y1}],andB1∪z1Xu∪vXx2⊆G2.Clearly,|V(Gi)|≥7 fori∈[2].SinceG[{x₁,x₂,y₁}]∼=K₃,(i) or(ii) followsfromLemma2.8.Thiscompletes theproofof(4).

LetBbethesetofallB₁-bridgesofH. ForeachD∈ B,letb_D∈V(D)∩V(B₁) and u_D,v_D ∈V(X) betheneighborsofD−b_D inG−x₂y₂ withu_DXv_D maximal. Deﬁne anew graphG_B such thatV(G_B)= B, and two B1-bridgesC,D ∈ B are adjacent in G_BifuCXvC− {uC,vC}containsaneighborofD−bDoruDXvD− {uD,vD}contains a neighbor of C−b_C. Note that, for any component D of G_B,

D∈V(D)u_DXv_D is a subpath of X, whose ends are denoted by u_D,v_D. We let S_D := {bD : D ∈ V(D)}∪ (N(u_DXv_D− {u_D,v_D})∩V(B1)).Wemayassumethat

(5) foranycomponentDofG_B,|S_D|≤2 andy2∈

D∈V(D)V(D)−S_D

∪V(u_DXv_D− {u_D,v_D}).

First, we may assume |S_D| ≤ 2. For, suppose |S_D| ≥ 3. Then there exist D ∈ V(D), r1,r2 ∈ V(uDXvD)− {uD,vD}, and distinct r₁,r₂ ∈ V(B1) such that for i ∈ [2], rir_i ∈E(G) or r_i =bDi forsomeDi ∈V(D)− {D}. (Toseethis, wechooseD∈V(D) such that there is a maximum number of vertices in B₁ from which G has a path to uDXvD− {uD,vD}andinternallydisjointfromB1∪D∪X.Ifthisnumberisatmost1,

(10)

wecanshowthat|S_D|≤2.)LetRi=rir_iifrir_i∈E(G);andotherwiseletRi beapath inG[Di+ri] fromri tor_i andinternallydisjointfromX.LetQdenoteaninducedpath inG[D+{u_D,v_D}]−b_D−x₂y₂betweenu_D andv_D,andletX beobtainedfromX by replacinguDXvDwithQ.Clearly,theblockof(G−x1)−X containingy1 containsB1

as wellasthepathR1∪r1Xr2∪R2.Notethaty1=bD(asy1 isnotacutvertex inH).

Moreover,ify₁=r_i forsomei∈[2] thenD_i isnotdeﬁnedandr_ir_i∈E(G).Soy₁isnot acutvertexof(G−x1)−X.Thus,X contradictsthechoiceof X,becauseof(1).

Now assume y₂ ∈/

D∈V(D)V(D)∪ V(u_DXv_D) −({u_D,v_D}∪S_D). Then S_D ∪ {u_D,v_D,x₁} is a cut in G; so |S_D| = 2 (as G is 5-connected). Let S_D = {p,q}. Then G has a 5-separation (G1,G2) such that V(G1 ∩ G2) = {p,q,u_D,v_D,x1}, B₁∪z₁Xu_D ∪v_DXx₂ ⊆ G₁, and G₂ contains u_DXv_D and the B₁-bridges of H contained inD.If (G2−x1,u_D,p,v_D,q) is planarthen, since|V(Gi)|≥7 for i∈[2],then (i) or(ii) followsfrom Lemma2.7.Sowemayassumethat(G2−x1,u_D,p,v_D,q) isnot planar. Then by Lemma2.1, G₂−x₁ contains disjoint paths S,T from u_D,pto v_D,q, respectively.

We apply Lemma 2.11 to G2 −x1 and {u_D,v_D,p,q}. If (i) of Lemma 2.11 holds then from theseparationinG₂−x₁, wederive a5-separation (G₁,G₂) inG suchthat x1 ∈V(G₁∩G₂),B1∪T+x1 ⊆G₁, |V(G₂)|≥7,and (G₂−x1,V(G₁∩G₂)− {x1}) is planar. So (i) or (ii) follows from Lemma 2.7. We may thus assume that (ii) of Lemma 2.11holds. Thus, there is an inducedpath S in G2−x1 from u_D to v_D such that(G2−x1)−S isachainofblocksfrom ptoq.NowletX beobtainedfromX by replacing u_DXv_D with S.Then y₁ isnotacutvertex of(G−x₁)−X,and theblock of (G−x1)−X containingy1 containsB1and (G2−x1)−S,contradicting (1).This completes theproofof(5).

Wemayalsoassumethat

(6) foranyB1-bridgeD ofH,y2∈/V(uDXvD)− {uD,vD}.

For,suppose y₂∈V(u_DXv_D)− {u_D,v_D}forsomeB₁-bridgeD ofH.ChooseX andD so that,subjectto (1)-(3),uDXvDismaximal.

We claim that {D} is a component of G_B. For, otherwise, by the maximality of u_DXv_D,thereexistsaB₁-bridgeCofHsuchthatN(C)∩V(u_DXv_D−{u_D,v_D})=∅.Let T beaninducedpathinG[D+{uD,vD}]−bD−x2y2fromuDtovD.ByreplacinguDXvD

with T weobtainapathX from X suchthaty₁ isnotacutvertex in(G−x₁)−X, B₁ is contained ina blockof (G−x₁)−X, and (G−x₁)−X hasa chainof blocks from y1 toy2and containingB1,contradictingthechoiceofX (in (2)asy2∈V(X)).

Hence,by(5),V(G_B)={D}.IfGhasanedgefromu_DXv_D−{u_D,v_D}toB₁−y₁orif y1hastwoneighbors,oneonuDXy2−uDandoneonvDXy2−vD,thenletXbeobtained fromX byreplacinguDXvD withaninducedpathinG[D+{uD,vD}]−bD−x2y2from u_D to v_D. Inthe former case, (G−x₁)−X has achain of blocksfrom y₁ to y₂ and containingB1,contradicting(2).Inthelattercase,(G−x1)−X hasacyclecontaining

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{y1,y2}.SobyLemmas2.11and2.7,(i) or(ii) holds,orthereisaninducedpathX^∗in G−x1 from z1 tox2 suchthaty1,y2∈/V(X^∗) and (G−x1)−X^∗ is 2-connected,and (iii) holds.

Therefore, we may assume N(uDXvD− {uD,vD})∩V(B1) = {y1}, and N(y1)∩ V(uDXvD− {uD,vD})⊆V(uDXy2) orN(y1)∩V(uDXvD− {uD,vD})⊆V(vDXy2).

LetL=G[D∪uDXvD] andletL=G[L+y1].

Suppose L has disjoint paths from uD,bD to vD,y2, respectively. We may apply Lemma 2.11 to L and {uD,vD,bD,y2}. If L has an induced path S from uD to vD

suchthatL−S isachainof blocksfrom bD to y2 then letX be obtainedfrom X by replacing uDXvD with S; now (G−x1)−X is a chain of blocks from y1 to y2 and containingB1,contradicting (2).Sowe mayassumethatLhasa4-separationas given in(i) ofLemma2.11. ThusGhasa5-separation (G1,G2) such thatx1 ∈V(G1∩G2),

|V(Gi)|≥7 for i∈[2], and (G2−x1,V(G1∩G2)− {x1}) isplanar. Hence,(i) or (ii) followsfrom Lemma2.7.

Thus,wemayassumethatsuchdisjointpathsdonotexistinL.ByLemma2.1,there existsacollectionAofsubsetsofV(L)−{bD,uD,vD,y2}suchthat(L,A,uD,bD,vD,y2) is3-planar.

Wenowshowthat(L−y1vD,uD,bD,vD,y2,y1) isplanar(whenN(y1)∩V(uDXvD− {uD,vD}) ⊆ V(uDXy2)), or (L −y1uD,uD,bD,vD,y1,y2) is planar (when N(y1)∩ V(uDXvD− {uD,vD})⊆V(vDXy2)).Since thearguments forthese two casesarethe same, we consider only the case when N(y1)∩V(uDXvD− {uD,vD}) ⊆ V(uDXy2).

Since Gis 5-connected, for eachA ∈ A, {x₁,y₁}⊆N(A) and |N_L(A)|= 3; and since N(y₁)∩V(L−b_D)⊆V(u_DXy₂) andGis 5-connected,|N_L(A)∩V(X)|= 2. Foreach suchA,leta₁,a₂∈N_L(A)∩V(X) andleta∈N_L(A)−V(X).IfforeachA∈ A,(G[A∪ {a,a₁,a₂,y₁}],a₁,a,a₂,y₁) isplanar,then(L−y₁v_D,u_D,b_D,v_D,y₂,y₁) isplanar.Sowe mayassumethat,forsomechoiceofA,(G[A∪ {a,a₁,a₂,y₁}],a₁,a,a₂,y₁) isnotplanar.

(NotethatG[A∪ {a,a₁,a₂,y₁}] is (4,{a,a₁,a₂,y₁})-connected.)Hence,byLemma2.1, G[A∪ {a,a₁,a₂,y₁}] containsdisjointpathsfroma₁,atoa₂,y₁,respectively.Sowecan applyLemma2.11toG[A∪{a,a₁,a₂,y₁}] and{a,a₁,a₂,y₁}.If(i) ofLemma2.11occurs thenGhasa5-separation(G₁,G₂) suchthatx₁∈V(G₁∩G₂),|V(G_i)|≥7 fori∈[2], and (G₂−x₁,V(G₁∩G₂)− {x₁}) is planar; so (i) or (ii) follows from Lemma 2.7.

Hence,we mayassumethat(ii) of Lemma2.11occurs.ThenG[A∪ {a,a₁,a₂,y₁}] has aninducedpathS froma₁to a₂ suchthatG[A∪ {a,a₁,a₂,y₁}]−S isachainofblocks fromy₁ toa.LetX beobtainedfrom X byreplacinga₁Xa₂withS.Thentheblockof (G−x₁)−X containingy₁ containsB₁ andG[A∪N_L(A)∪ {y₁}]−S,andy₁is nota cutvertexin(G−x₁)−X,contradicting(1).

Hence,Ghasa6-separation(G₁,G₂) withV(G₁∩G₂)={b_D,u_D,v_D,x₁,y₁,y₂}and G2−x1 = L−y1vD (or G2−x1 = L −y1uD). Since (L−y1vD,uD,bD,vD,y2,y1) (or (L −y1uD,uD,bD,vD,y1,y2)) is planar and |V(G2)| ≥ 8, (i) or(ii) follows from Lemma2.9 andthenLemma2.7.Thiscompletestheproofof(6).

Ify2∈V(X) thenby(4), (5)and(6),H is2-connected;so (iii) holds.Thuswemay assume y2 ∈/ V(X). Then by (4), H is achain of blocksfrom y1 to y2 and containing