A characterization of graphs with no $K_{3,4}$ minor

A CHARA CTERIZA TION OF GRAPHS WITH NO K 3 , 4 MINOR ON-HEI SOLOMON LO Abstract. A complete structural characterization of graphs with no K 3 , 4 minor is obtained, and the follo wing consequences are established. Ev ery 4 -connected non-planar graph with at least seven v ertices and minimum degree at least five contains b oth K 3 , 4 and K − 6 as minors, thereby proving a conjecture of Kaw araba yashi and Maharry in a strengthened form. Moreov er, ev ery 4 -connected graph with no K 3 , 4 minor is hamiltonian-connected, extending a theorem of Thomassen, and admits an em b edding on the torus. 1. Introduction The theory of graph minors has long o ccupied a cen tral p osition in structural graph theory . A classical p oint of departure is the Kurato wski–W agner theorem [ 16 , 39 ], which c haracterizes planar graphs as those con taining neither K 5 nor K 3 , 3 as a minor. This theorem initiated the systematic study of excluded-minor structural characterizations and their consequences. A decisiv e adv ance w as ac hiev ed b y Rob ertson and Seymour in their Graph Minors series. They pro v ed that finite graph s are well-quasi-ordered under the minor relation, thereb y extending the Kurato wski–W agner theorem and settling W agner’s conjecture that ev ery minor-closed class is determined by a finite set of excluded minors [ 33 ]. Bey ond this fundamental finiteness result, they developed a far-reaching structural theory for graphs excluding a fixed minor. Their Minor Structure Theorem [ 32 , 30 ] shows that such graphs admit a tree-lik e decomp osition in to parts that are almost embeddable in surfaces of b ounded gen us. This yields a p ow erful global framew ork for minor-closed classes, but b y its nature pro vides only an approximate structure rather than an exact c haracterization. An imp ortan t direction is therefore to obtain precise structural descriptions for particular ex- cluded minors. This problem w as explicitly emphasized by Lo v ász [ 19 ] as one of the fundamental researc h directions in graph minor theory . Exact structural c haracterizations are known only in a limited n um b er of cases and are t ypically difficult to obtain. Among the most prominent op en problems are the structural characterizations of graphs with no K 6 minor and of graphs with no P etersen minor. These questions are closely tied to deep conjectures suc h as Hadwiger’s conjecture and T utte’s 4 -flow conjecture. A t presen t, a complete description of graphs excluding K 6 or the P etersen graph is still b ey ond reac h. Nonetheless, the sub ject has con tin ued to dra w considerable interest, and a num b er of substantial partial results ha v e b een obtained; see, for example, [ 14 , 34 ]. Not withstanding these difficulties, exact structural characterizations are av ailable for a num b er of smaller excluded minors. W agner [ 39 , 40 ] pro vided the classical characterizations for K 5 and K 3 , 3 . Maharry [ 21 ] obtained a corresp onding characterization for the cub e, Ding [ 4 ] for the o cta- hedron, Maharry and Rob ertson [ 22 ] for the W agner graph, and Ellingham, Marshall, Ozeki, and T suchiy a [ 7 ] for K 2 , 4 . These structural results hav e led to a wide range of applications, including v erifications of Hadwiger’s conjecture for graphs with no K 5 minor [ 39 ], inv estigations into the flexibilit y of embeddings [ 23 ], and results on hamiltonicity [ 8 , 18 ]. Graphs excluding K 3 ,t minors play a central role in graph minor theory . It is well known that ev ery graph embeddable in a fixed surface excludes K 3 ,t for some t . Moreo v er, Robertson and This researc h was partially supported by the F undamental Researc h F unds for the Central Univ ersities. 1 2 O.-H.S. L O Seymour [ 29 ] exhibited a finite family of obstructions, basically obtained by summing copies of K 5 and K 3 , 3 , that c haracterizes graphs of b ounded gen us in terms of excluded minors. These facts motiv ate a deep er inv estigation into the structure of graphs with no K 3 ,t minor. Along this line, Op orowski, Oxley , and Thomas [ 28 ] prov ed that every sufficien tly large 3 - connected graph with no K 3 ,t minor con tains a large wheel. Böhme, Maharry , and Mohar [ 2 ] sho w ed that every sufficien tly large 7 -connected graph of b ounded tree-width necessarily con tains a K 3 ,t minor. More recently , Kaw arabay ash i and Maharry [ 13 ] established that every sufficien tly large almost 5 -connected non-planar graph con tains b oth a K 3 , 4 minor and a K − 6 minor, and they prop osed the following strengthening. Conjecture 1.1 ([ 13 ]) . Every 5 -connected non-planar graph with at least sev en vertices contains b oth a K 3 , 4 minor and a K − 6 minor. Figure 1. The graphs K 3 , 4 (left) and K − 6 (righ t). It has been remarked in [ 3 ] that a complete c haracterization of graphs with no K 3 ,t minor “seems extremely hard to obtain. ” Ev en for the first op en case K 3 , 4 , the problem is describ ed as “hard” and “very c hallenging” in [ 13 , 6 ]. Note that the graph K 3 , 4 lies b etw een K 3 , 3 and the Petersen graph in the minor order. The main ob jectiv e of this pap er is to pro vide a complete structural characterization of graphs with no K 3 , 4 minor. W e begin b y analyzing the 4 -connected mem b ers of this class and then sho w ho w all graphs excluding K 3 , 4 can b e constructed from these fundamen tal building blo cks. Since Maharry and Slilat y [ 24 , 25 ] ha v e already characterized the pro jective-planar graphs with no K 3 , 4 minor, the cen tral fo cus of our w ork is on the non-projective-planar case, for whic h preliminary progress w as rep orted in our earlier pap er [ 17 ]. The main structural theorem of this pap er is as follows. Here, patc h graphs, in tro duced in [ 24 ], are recalled in Section 4.1 , and the family of oloidal graphs is defined in Section 4.2 . Theorem 1.2. A gr aph G is 4 -c onne cte d and has no K 3 , 4 minor if and only if one of the fol lowing holds: • G is a 4 -c onne cte d planar gr aph. • G is a 4 -c onne cte d non-planar sub gr aph of a r e duc e d p atch gr aph. • G is isomorphic to K 6 . • G is an oloidal gr aph. Sev eral consequences follow from Theorem 1.2 . First, w e resolv e Conjecture 1.1 in a strengthened form. Theorem 1.3. Every 4 -c onne cte d non-planar gr aph G with minimum de gr e e at le ast five c ontains K − 6 as a minor, and also c ontains K 3 , 4 as a minor if G is not isomorphic to K 6 . Our second application concerns hamiltonian properties under forbidden minors. T utte [ 38 ] pro v ed that ev ery 4 -connected graph with no K 3 , 3 minor is hamiltonian, and Thomassen [ 37 ] strengthened this result by showing th at such graphs are hamiltonian-connected; that is, any t w o v ertices can serve as the end-vertices of a hamiltonian path. W e extend Thomassen’s theorem by relaxing the excluded minor. Theorem 1.4. Every 4 -c onne cte d gr aph with no K 3 , 4 minor is hamiltonian-c onne cte d. Finally , we deriv e a consequence regarding the gen us of graphs excluding a K 3 , 4 minor. Theorem 1.5. Every 4 -c onne cte d gr aph with no K 3 , 4 minor admits an emb e dding on the torus. GRAPHS WITH NO K 3 , 4 MINOR 3 Organization of the pap er. The pap er is structured as follows. In Section 2 , we fix the basic definitions and notation that will b e used throughout. Section 3 introduces Seymour’s splitter theorem along with its v arian ts, which form essen tial to ols for our subsequen t argumen ts. In Section 4 , we establish the principal structural result, Theorem 1.2 , pro viding a characterization of 4 -connected graphs with no K 3 , 4 minor. Building up on this, Section 5 extends the characterization to all graphs excluding a K 3 , 4 minor, sho wing how they can b e constructed from the 4 -connected building blo cks. Finally , Section 6 is dev oted to the deriv ation of the consequences of Theorem 1.2 , sp ecifically Theorems 1.3 , 1.4 , and 1.5 . 2. Definitions and not a tion Throughout this pap er, we consider only finite simple graphs unless otherwise sp ecified. A k -cut in a connected graph G is a subset S ⊆ V ( G ) with | S | = k such that G − S is disconnected. The graph G is k -c onne cte d if | V ( G ) | > k and G has no k ′ -cut for any k ′ < k . A 3 -connected graph G is almost 4 -c onne cte d if either | V ( G ) | ≤ 6 , or every 3 -cut of G separates G into a single vertex and a remaining comp onen t. A 3 -connected graph G on at least five v ertices is internal ly 4 -c onne cte d if either G is isomorphic to K 3 , 3 , or ev ery 3 -cut of G is an indep endent set that separates G into a single vertex and a remaining comp onent. In particular, ev ery 4 -connected graph is b oth almost 4 -connected and internally 4 -connected. The de gr e e of a v ertex in a graph G is the num b er of its neighbors in G . A graph G is k -de gener ate if every subgraph of G contains a vertex of degree at most k . Equiv alently , a graph on n vertices is k -degenerate if and only if its vertices can be ordered as v 1 , v 2 , . . . , v n suc h that each vertex v i has at most k neighbors among v 1 , . . . , v i − 1 . A graph H is a minor of a graph G if it can b e obtained from G by deleting vertices, deleting edges, and contracting edges. This is denoted by G ⪰ H , and, when no ambiguit y arises, we also sa y that G contains H as a minor, or G has a minor of H . It is w ell kno wn that G ⪰ H if and only if there exists a mapping µ that assigns to eac h v ertex of H a pairwise disjoin t subset of V ( G ) and to each edge of H a path in G , such that for every v ∈ V ( H ) , the induced subgraph G [ µ ( v )] is connected, the paths corresp onding to distinct edges are pairwise in ternally disjoint, and for each uv ∈ E ( H ) , the path µ ( uv ) joins a vertex in µ ( u ) to a vertex in µ ( v ) with no in ternal vertex contained in S w ∈ V ( H ) µ ( w ) . W e call suc h a mapping µ a mo del of H , witnessing that H is a minor of G . Moreov er, if G is connected, we may assume that S w ∈ V ( H ) µ ( w ) = V ( G ) ; in this case, we call µ a sp anning mo del . Assume that G con tains a mo del µ of H . If a v ertex v ∈ V ( H ) has degree at most four, we may assume that the subgraph G [ µ ( v )] con tains a spanning path P such that, for every edge uv ∈ E ( H ) , the in tersection of µ ( v ) with the path µ ( uv ) is an end-vertex of P , and eac h end-v ertex of P is inciden t with at least tw o paths corresponding to edges inciden t with v . If µ ( v ) consists of a single v ertex for ev ery v ∈ V ( H ) , then w e sa y that G con tains a sub division of H . Note that if ev ery v ertex of H has degree three, then the existence of an H -minor is equiv alent to the existence of a sub division of H . Let A b e a set of edges not contained in G . W e write G + A for the graph obtained from G by adding the edges in A . Let D b e a set of vertices or edges con tained in G , and write G − D for the graph obtained from G by deleting the elemen ts of D . When A = { a } (resp ectively , D = { d } ), we ma y write G + a (resp ectively , G − d ) for brevit y . F or any path P and vertices u, v ∈ V ( P ) , we denote by P [ u, v ] the subpath of P with end-vertices u and v . A vertex of P is internal if it is not an end-vertex, and we write P ( u, v ) for the (p ossibly empt y) path obtained from P [ u, v ] by deleting u and v . The length of a w alk, path, or cycle is the num b er of edge o ccurrences it contains. A cycle in a graph em b edded on a surface that b ounds a face is called a facial cycle . More generally , the closed w alk that traces the b oundary of a face is called a facial walk . The length of a face is the length of 4 O.-H.S. L O its facial walk. T w o vertices of an embedded graph are c ofacial if they are incident with a common face. T wo edges are indep endent if they hav e no common end-v ertex. F or any p ositiv e in teger k , w e write [ k ] for the set of p ositive integers at most k . 3. Splitter theorems In many situations the graph G is not given explicitly; instead, w e are given a kno wn graph H  = G such that G ⪰ H , and we wish to exploit this information to understand the structure of G . By definition, H is obtained from G by a non-trivial sequence of deletions and con tractions. Equiv alen tly , one may view G as arising from H by a sequence of minor extensions, eac h pro ducing a larger graph that still o ccurs as a minor of G . In this wa y , starting from the fixed minor H , we obtain a family of graphs H ′ with H ⪯ H ′ ⪯ G . This principle underpins splitter theorems and their v arian ts, whic h fix H by gro wing it to w ard G . Suc h to ols hav e pro v ed highly useful in the literature. The classical result, commonly known as Seymour’s splitter theorem [ 35 , 26 ], fixes the size of H when both G and H are 3 -connected. Hegde and Thomas [ 10 ] introduced a v ariant that fixes the non-em b eddabilit y of H . W e mention that Ding and Iverson [ 5 ] developed a connectivity-fixing to ol that gro ws H to certain twists , whic h do not necessarily con tain H as a minor; they used it to characterize the minor-minimal internally 4 -connected non-projective-planar graphs. W e refer to [ 36 , 27 ] for a comprehensiv e exp osition. 3.1. Seymour’s splitter theorem. Let H b e a graph and let v ∈ V ( H ) . Construct a graph H ′ from H as follows: delete v , introduce tw o adjacent vertices v 1 and v 2 , and distribute the neigh b ors of v b etw een v 1 and v 2 so th at each former neighbor of v is adjacent to exactly on e of v 1 and v 2 , and each of v 1 and v 2 receiv es at least tw o suc h neigh b ors. In particular, this requires that v has degree at least four. W e say that H ′ is obtained from H by splitting the vertex v . The reverse op eration consists of contracting an edge whose end-vertices hav e no common neigh b or. The theorem b elow is a graph-theoretic consequence of Seymour’s splitter theorem, and w as also obtained indep enden tly by Negami. Recall that a whe el is the graph formed from a cycle by adding a single vertex adjacent to ev ery v ertex of the cycle. Theorem 3.1 ([ 35 , 26 ]) . L et G and H b e 3 -c onne cte d gr aphs. Supp ose that G c ontains H as a minor and, mor e over, that if H is a whe el, then it is a lar gest whe el minor of G . Then G c an b e obtaine d fr om H by a se quenc e of op er ations, e ach of which either adds an e dge b etwe en two non-adjac ent vertic es or splits a vertex. 3.2. In ternally 4 -connected graphs. While Seymour’s splitter theorem applies to 3 -connected graphs, Johnson and Thomas [ 11 ] ask ed for an analogue tailored to in ternally 4 -connected graphs. They announced suc h a theorem in [ 11 , Section 3] (see also [ 36 ]); how ever, a pro of has not y et app eared in the literature. Here we collect sev eral of their structural results that will b e used later. Recall that if G con tains H as a minor but con tains no graph obtained from H by splitting a v ertex as a minor, then G con tains a sub division of H . No w let G and H b e graphs such that G con tains a sub division of H . Then there exists a mapping η that maps the vertices of H to distinct vertices of G and maps each edge of H to a path in G , suc h that these paths are pairwise internally disjoin t and, for every uv ∈ E ( H ) , the path η ( uv ) has end-v ertices η ( u ) and η ( v ) and no in ternal vertex of η ( uv ) lies in { η ( v ) : v ∈ V ( H ) } . Let η ( H ) denote the subgraph of G formed by the image of η ; we also say that η ( H ) is a sub division of H in G . Define η ( V ( H )) := { η ( v ) : v ∈ V ( H ) } . A vertex is called internal if it lies in V ( G ) \ η ( V ( H )) . The paths η ( e ) for e ∈ E ( H ) are called the se gments of η . F or v ∈ V ( H ) , the domain of v with resp ect to η is the set consisting of η ( v ) together with all in ternal v ertices of the segments ha ving η ( v ) as an end-vertex. The close d domain of v with resp ect to η is the set of all v ertices on the segments having η ( v ) as an end-v ertex. GRAPHS WITH NO K 3 , 4 MINOR 5 Johnson and Thomas [ 11 ] in tro duced a special t yp e of sub division, called a lexic o gr aphic al ly maximal sub division , whic h w e refer to as a JT-sub division . F or the tec hnical definition of a JT- sub division, w e refer the reader to [ 11 ]. W e mention t wo facts ab out JT-sub divisions, although the first is not needed in the proofs of this pap er. Prop osition 3.2 ([ 11 ]) . L et G and H b e gr aphs. Then the fol lowing statements hold: • If G c ontains a sub division of H , then G also c ontains a JT-sub division of H . • If G c ontains H as a sp anning sub gr aph, then this sub gr aph, when viewe d as a sub division of H , is a JT-sub division of H . The following lemma collects t wo fundamen tal prop erties of JT-sub divisions, as established in [ 11 , (5.2)] and [ 11 , (7.2)], and will be used rep eatedly in Section 4.2 . Lemma 3.3 ([ 11 ]) . L et G and H b e internal ly 4 -c onne cte d gr aphs such that G c ontains a JT- sub division η ( H ) of H . Then the fol lowing statements hold: • If G do es not c ontain, as a minor, any gr aph obtaine d fr om H by splitting a vertex, then every se gment of η ( H ) is an induc e d p ath in G . • L et u b e a vertex of H of de gr e e thr e e, and let e 1 and e 2 b e two distinct e dges of H incident with u . If ther e exists an e dge of G joining a vertex v in η ( e 1 ) − η ( u ) to an internal vertex of η ( e 2 ) , then v is adjac ent to η ( u ) in η ( e 1 ) . 3.3. Non-em b eddable extensions. Supp ose that G ⪰ H , where H is embeddable on a surface but G is not em b eddable on that surface. In this situation, G  = H , and we may exploit this to fix the non-embeddability of G relative to H . Hegde and Thomas [ 10 ] developed a p ow erful framework for addressing suc h cases. F or our purp oses, w e only need a restricted version: we assume that G is non-planar, H is planar, and b oth graphs are 4 -connected. Let H b e a 4 -connected planar graph, and let H ′ b e obtained from H by splitting a vertex v ∈ V ( H ) into tw o v ertices v 1 , v 2 ∈ V ( H ′ ) . W e say that H ′ is obtained by a planar split of v if H ′ is planar, and b y a non-planar split otherwise. Equiv alently , a split is planar if and only if the neigh b ors assigned to eac h of v 1 and v 2 form con tiguous in terv als in the cyclic order of the neigh b ors of v in the planar embedding of H . After a planar split, the facial cycles of H ′ corresp ond naturally to those of H , with all lengths preserv ed except for exactly t w o facial cycles C 1 and C 2 of H , which corresp ond to facial cycles in H ′ whose lengths increase by one; these tw o cycles in H ′ share precisely the edge v 1 v 2 . In this case, the planar split is said to b e along C 1 (and along C 2 , resp ectiv ely). The follo wing result is a w eak ened form of [ 10 , Theorem 1.2] and plays a k ey role in the pro of of Theorem 1.3 in Section 6.1 . Theorem 3.4 ([ 10 ]) . L et G and H b e 4 -c onne cte d gr aphs. Supp ose that G c ontains H as a minor, that G is non-planar, and that H is planar. Then G c ontains a gr aph H ′ as a minor such that one of the fol lowing holds: • H ′ is obtaine d fr om H by joining two non-c ofacial vertic es. • Ther e exist distinct vertic es v 1 , v 2 , v 3 , v 4 app e aring in this or der on a facial cycle of H such that H ′ is obtaine d fr om H by joining v 1 with v 3 and joining v 2 with v 4 . • H ′ is obtaine d fr om H by p erforming a non-planar split. • Ther e exist non-adjac ent c ofacial vertic es u, v of H such that H ′ is obtaine d by first p er- forming a planar split of v into v 1 , v 2 so that, in the interme diate gr aph, u and v 2 ar e non-c ofacial, and then joining u with v 2 . • Ther e exist facial cycles C 1 and C 2 of H sharing a c ommon e dge uv such that H ′ is obtaine d by p erforming a planar split of u along C 1 into u 1 , u 2 and a planar split of v along C 2 into v 1 , v 2 , with u 1 adjac ent to v 1 in the interme diate gr aph, and then joining u 2 with v 2 . 6 O.-H.S. L O • Ther e exist distinct vertic es u, v , w on a facial cycle C of H , wher e u is adjac ent to neither v nor w , such that H ′ is obtaine d by first p erforming a planar split of u along C into u 1 , u 2 so that u 1 , u 2 , v , w app e ar on the new facial cycle of the interme diate gr aph, and then joining u 1 with v and joining u 2 with w . • Ther e exist non-adjac ent vertic es u, v on a facial cycle C of H such that H ′ is obtaine d by p erforming a planar split of u into u 1 , u 2 and a planar split of v into v 1 , v 2 , b oth along C , so that u 1 , u 2 , v 1 , v 2 app e ar on the new facial cycle of the interme diate gr aph, and then joining u 1 with v 1 and joining u 2 with v 2 . 4. 4 -connected graphs with no K 3 , 4 minor In this section, we establish our principal result, Theorem 1.2 . W e begin b y presen ting the c haracterization of pro jective-planar graphs due to Maharry and Slilat y in Section 4.1 , and then pro ceed to the non-pro jective-planar case in Section 4.2 , whic h constitutes the most technical part of our analysis. The pro of of Theorem 1.2 is then completed in Section 4.3 . 4.1. The pro jective-planar case. Maharry and Slilat y established t w o c haracterizations of graphs embedded in the pro jective plane with no K 3 , 4 minor: one in terms of p atch gr aphs [ 24 ] and the other in terms of Möbius hyp erladders [ 25 ]. In this section w e recall the former. W e review the construction of patc h graphs and state the main result of [ 24 ], Theorem 4.1 . Moreov er, we prov e that ev ery patch graph is 4 -degenerate. A c onstruct is a pair ( G, P ) consisting of a multigraph G embedded in the pro jective plane and a family P of faces of length 4 in that embedding, which are called p atches of G . A construct ( G, P ) can b e transformed in to another construct ( G ′ , P ′ ) b y applying one of three op erations, called H - , Y - , and I -p atchings . The graphs depicted in Figure 2 are called the H - , Y - , and I -pie c es , resp ectiv ely; eac h is b ounded by a cycle of length 4 . Figure 2. The H -piece (left), Y -piece (middle), and I -piece (righ t). Applying the corresp onding patching adds tw o, one, or no new patc hes, resp ectiv ely , indicated b y the shaded regions. T o apply an H -patc hing to ( G, P ) , choose a patc h P ∈ P and form G ′ b y identifying the facial w alk of P with the b oundary cycle of the H -piece, allowing the latter to b e flipp ed or rotated prior to the identification. W e say that the H -piece is p atche d to P . The new family P ′ is obtained from P b y removing P and adding the t w o new patc hes contained in the H -piece. The definitions of Y - and I -patchings are analogous. The initial p atch c onstruct ( G 0 , P 0 ) is the construct in which G 0 , called the initial p atch gr aph , consists of t w o v ertices joined b y four parallel edges and is em b edded in the pro jective plane as depicted in Figure 3 , and P 0 consists of the unique face of length 4 . A p atch c onstruct is a construct obtained from the initial patch construct by a sequence of H -, Y -, and I -patc hings. A p atch gr aph is the multigraph G arising from a patch construct ( G, P ) . Maharry and Slilaty [ 24 ] prov ed the follo wing result. 1 1 In the original setting of [ 24 ], an additional patching op eration is considered. This op eration can b e simulated b y first applying a Y -patc hing and then an I -patching, follow ed by deleting an appropriate degree three vertex from the I -piece. Consequently , the main theorem of [ 24 ] remains v alid under our simplified framework. GRAPHS WITH NO K 3 , 4 MINOR 7 Figure 3. The initial patch construct ( G 0 , P 0 ) embedded in the pro jectiv e plane. The outer circle represents the cross-cap, and the shaded region is the unique face of length 4 . Theorem 4.1 ([ 24 ]) . A ny almost 4 -c onne cte d, non-planar, pr oje ctive-planar gr aph with no K 3 , 4 minor is a sub gr aph of a p atch gr aph or is isomorphic to K 6 . Mor e over, every p atch gr aph c ontains no K 3 , 4 minor. Let the pieces obtained from the H -, Y -, and I -pieces by remo ving the white v ertices indicated in Figure 2 b e called the r e duc e d H - , Y - , and I -pie c es . Th us, the reduced H -piece coincides with the H -piece, while the reduced Y - and I -pieces are obtained by deleting one and tw o vertices, resp ectiv ely . A r e duc e d p at ch gr aph is defined analogously to a patc h graph, except that only r e duc e d pie c es are used in the patc hing op erations. Equiv alently , a reduced patch graph can b e obtained from a patch graph by removing all white vertices from the pieces used in the patching op erations. It is easy to see that an y 4 -connected subgraph of a patch graph is also a subgraph of a reduced patch graph. While ev ery pro jective-planar graph is 5 -degenerate, w e establish that patch graphs hav e strictly smaller degeneracy . This prop erty will b e used in the pro of of Theorem 1.3 in Section 6.1 . Lemma 4.2. Every p atch gr aph is 4 -de gener ate. Pr o of. Observe that in a patch graph, the white vertic es originating from a Y - or I -piece alwa ys ha v e degree at most 3 . Therefore, it suffices to sho w that every reduced patch graph is 4 -degenerate. Let G b e a reduced patch graph. There exists a sequence of reduced patch graphs G 0 , G 1 , . . . , G t , where G 0 is the initial patc h graph, G t = G , and for eac h i ∈ [ t ] , the graph G i is obtained from G i − 1 b y patching a reduced R i -piece with R i ∈ { H , Y , I } to a patc h P i − 1 of G i − 1 . Moreo v er, w e ma y assume the following: • If R i = H and, in the reduced H -piece used for the patc hing, one patc h P is patc hed with a reduced I -piece while the other patch is patched with a reduced H - or Y -piece, then P = P i . • If R i = Y and the patch in the reduced Y -piece is patc hed with a reduced I -piece, then P = P i . W e prov e the assertion by induction on t . Clearly , G 0 is 4 -degenerate. Assume t ≥ 1 and that the assertion holds for all smaller v alues of t . First assume that R t = H . By deleting the t w o square vertices of the reduced H -piece (as indicated in Figure 2 ), we obtain G t − 1 . By the induction hypothesis, G t − 1 is 4 -degenerate. Since the deleted vertices hav e degree at most 3 in G t , it follows that G t is also 4 -degenerate. Next assume that R t = Y . Deleting the square v ertex of the reduced Y -piece yields G t − 1 . Again, b y the induction hypothesis, G t is 4 -degenerate. No w assume that R t = I . If t = 1 , then G t is immediately seen to b e 4 -degenerate. W e therefore assume t > 1 , and hence P t − 1 do es not b elong to the patch family of the initial patch construction. Supp ose that P t − 1 comes from a reduced H -piece, and let P denote the other patch of this piece. By our assumptions on the sequence, no reduced H - or Y -piece is patc hed to P . W e delete the 8 O.-H.S. L O square vertices of this reduced H -piece so that the first deleted v ertex has degree at most 4 and the second has degree at most 5 in G t , thereby obtaining a smaller reduced patch graph. The induction h yp othesis then implies that G t is 4 -degenerate. Finally , supp ose that P t − 1 comes from a reduced Y -piece. By our assumptions on the reduced patc h graph sequence, this reduced Y -piece was patched to P t − 2 during the reduced R t − 1 -patc hing. Deleting the square vertex of this reduced Y -piece, whic h has degree at most 4 in G t , yields either the reduced patch graph G t − 2 or that obtained from G t − 2 b y patching a reduced I -piece to P t − 2 . In either case, the induction h yp othesis applies and implies that G t is 4 -degenerate. □ 4.2. The non-pro jectiv e-planar case. Since pro jectiv e-planar graphs with no K 3 , 4 minor hav e b een characterized by Maharry and Slilaty , we extend the study to the non-pro jective-planar case. After in tro ducing a class of graphs called oloidal gr aphs , we sho w that they characterize the 4 - connected non-projective-planar graphs with no K 3 , 4 minor. 4.2.1. Thr e e A r chde ac on gr aphs and oloidal gr aphs. W e consider the graphs D 17 , E 20 , and F 4 , shown in Figure 4 with the v ertex labels as indicated. Eac h of these graphs is internally 4 -connected and app ears in Archdeacon’s list of minor-minimal non-pro jectiv e-planar graphs [ 1 ]; the graph names originate from the notation introduced in [ 9 ]. d 1 1 d 1 2 d 1 3 d 1 4 d 2 1 d 2 2 d 2 3 d 2 4 e 0 e 1 1 e 1 2 e 1 3 e 3 3 e 2 e 3 2 e 3 1 e 4 f 1 f 2 f 1 1 f 1 2 f 1 3 f 1 4 f 2 1 f 2 2 f 2 4 f 2 3 Figure 4. The internally 4 -connected non-pro jectiv e-planar graphs D 17 (left), E 20 (middle), and F 4 (righ t). The symmetries of these graphs facilitate the reduction of the case analysis. The graph D 17 has 48 automorphisms, eac h either fixing { d 1 1 , d 1 2 , d 1 3 , d 1 4 } or mapping it to { d 2 1 , d 2 2 , d 2 3 , d 2 4 } . Those automorphisms that fix { d 1 1 , d 1 2 , d 1 3 , d 1 4 } corresp ond bijectiv ely to the p er- m utations of these four vertices. The graph E 20 has 6 automorphisms, corresp onding bijectively to the p ermutations of { e 1 1 , e 1 2 , e 1 3 } . The graph F 4 has 4 automorphisms, each mapping f 1 1 to one of f 1 1 , f 1 4 , f 2 1 , or f 2 4 . The follo wing prop osition summarizes the results of [ 17 , Sections 4 and 6]. Prop osition 4.3 ([ 17 ]) . L et G b e a 4 -c onne cte d non-pr oje ctive-planar gr aph with no K 3 , 4 minor. Then the fol lowing hold: • G has a minor of D 17 , E 20 , or F 4 . • If G has a minor of D 17 but no minor of E 20 , then G c ontains a sp anning sub gr aph iso- morphic to D 17 . • If G has a minor of E 20 but no minor of F 4 , then G c ontains a sp anning JT-sub division of E 20 , and do es not c ontain as a minor any gr aph obtaine d fr om E 20 by splitting a vertex. • If G has a minor of F 4 , then G c ontains a sp anning JT-sub division of F 4 , and do es not c ontain as a minor any gr aph obtaine d fr om F 4 by splitting a vertex. W e define three classes of graphs that play a central role in our characterization of 4 -connected non-pro jectiv e-planar graphs with no K 3 , 4 minor. F or s 1 , s 2 , s ≥ 0 , let D s 1 ,s 2 , E s , and F b e the graphs depicted in Figure 5 . GRAPHS WITH NO K 3 , 4 MINOR 9 The graph D s 1 ,s 2 − { δ 1 1 , δ 1 3 , δ 2 2 , δ 2 4 } consists of the disjoint union of a path on s 1 + 2 vertices with end-v ertices δ 1 2 and δ 1 4 , and a path on s 2 + 2 v ertices with end-v ertices δ 2 1 and δ 2 3 . Moreov er, ev ery v ertex on the former (resp ectively , latter) path is adjacent to b oth δ 1 1 and δ 1 3 (resp ectiv ely , δ 2 2 and δ 2 4 ). Similarly , E s − { ε 1 1 , ε 1 3 , ε 2 , ε 3 1 , ε 3 2 , ε 3 3 , ε 4 } is a path on s + 2 vertices with end-v ertices ε 0 and ε 1 2 , where eac h vertex on the path is adjacen t to b oth ε 1 1 and ε 1 3 . These paths are referred to as the spines of D s 1 ,s 2 and the spine of E s , resp ectiv ely . W e denote the spines of D s 1 ,s 2 b y σ 1 0 σ 1 1 · · · σ 1 s 1 σ 1 s 1 +1 and σ 2 0 σ 2 1 · · · σ 2 s 2 σ 2 s 2 +1 , where δ 1 2 = σ 1 0 , δ 1 4 = σ 1 s 1 +1 , δ 2 1 = σ 2 0 , and δ 2 3 = σ 2 s 2 +1 . The spine of E s is denoted by σ 0 σ 1 · · · σ s σ s +1 , where ε 0 = σ 0 and ε 1 2 = σ s +1 . A spine is trivial if it has length one. δ 1 1 δ 1 4 δ 2 1 δ 2 4 δ 1 2 δ 1 3 δ 2 2 δ 2 3 ε 1 1 ε 1 3 ε 3 3 ε 2 ε 3 1 ε 4 ε 3 2 ε 0 ε 1 2 ϕ 1 ϕ 2 ϕ 1 1 ϕ 1 2 ϕ 1 3 ϕ 1 4 ϕ 2 1 ϕ 2 2 ϕ 2 4 ϕ 2 3 Figure 5. The 4 -connected non-pro jective-planar graphs D s 1 ,s 2 (left), E s (middle), and F (right). W e sho w that the graphs defined ab o ve are 4 -connected, non-pro jective-planar, and contain no K 3 , 4 minor. Lemma 4.4. L et G b e a gr aph in which v 1 and v 2 ar e adjac ent vertic es of de gr e e at le ast four. L et H b e obtaine d fr om G by c ontr acting the e dge v 1 v 2 . If H is 4 -c onne cte d, then G is also 4 -c onne cte d. Pr o of. Supp ose for a con tradiction that G has a k -cut S with k < 4 . Denote by v ∈ V ( H ) the v ertex obtained by contracting v 1 v 2 . Define S ′ := S if { v 1 , v 2 } ∩ S = ∅ , and S ′ := ( S \ { v 1 , v 2 } ) ∪ { v } otherwise. Since each of v 1 and v 2 has degree at least four, no comp onent of G − S consists of precisely one of v 1 , v 2 . It follo ws that S ′ is a ( k − 1) - or k -cut of H , a contradicti on. □ Lemma 4.5. F or any s 1 , s 2 ≥ 0 , the gr aph D s 1 ,s 2 is 4 -c onne cte d and non-pr oje ctive-planar. F or any s ≥ 0 , the gr aph E s is 4 -c onne cte d and non-pr oje ctive-planar. The gr aph F is 4 -c onne cte d and non-pr oje ctive-planar. Pr o of. By successively con tracting edges of the spines (resp ectiv ely , the spine), Lemma 4.4 im- plies that D s 1 ,s 2 (resp ectiv ely , E s ) is 4 -connected whenev er D 0 , 0 (resp ectiv ely , E 0 ) is 4 -connected. It therefore remains to verify th at eac h of D 0 , 0 , E 0 , and F is 4 -connected, whic h can be done straigh tforw ardly . The non-pro jective-planarit y of the graphs follows from the facts that D s 1 ,s 2 and E s con tain D 0 , 0 and E 0 as minors, resp ectiv ely , and that D 0 , 0 , E 0 , and F con tain the non-pro jectiv e-planar graphs D 17 , E 20 , and F 4 as spanning subgraphs, resp ectively . □ Lemma 4.6. L et G b e a gr aph with distinct vertic es v 1 , v 2 , w 0 , w 1 , . . . , w s , w s +1 , wher e s > 2 , such that P := w 0 w 1 . . . w s w s +1 is an induc e d p ath, every vertex of P has de gr e e four and is adjac ent to b oth v 1 and v 2 , and v 1 and v 2 ar e adjac ent. L et H b e the gr aph obtaine d fr om G by c ontr acting w 0 w 1 . If G c ontains K 3 , 4 as a minor, then H also c ontains K 3 , 4 as a minor. Pr o of. Supp ose, for a contradiction, that H has no K 3 , 4 minor. Denote the v ertex set of K 3 , 4 b y X ∪ Y , where | X | = 3 and | Y | = 4 , suc h that tw o v ertices of K 3 , 4 are adjacen t if and only if they b elong to different partite sets X and Y . 10 O.-H.S. LO Clearly , we ma y assume that G is connected. Let µ b e a spanning mo del of K 3 , 4 in G . Note that contracting any edge of P in G yields a graph isomorphic to H . F or any i ∈ { 0 , 1 , . . . , s } and an y v ∈ V ( K 3 , 4 ) , w e ha ve { w i , w i +1 } ⊆ µ ( v ) ; otherwise, the graph H , obtained from G b y con tracting the edge w i w i +1 , w ould contain a K 3 , 4 minor. F or an y i ∈ [ s ] and j ∈ [ 2] , if there exists v ∈ V ( K 3 , 4 ) suc h that { w i , v j } ⊆ µ ( v ) , then, since ev ery neigh b or of w i other than v j is also a neigh b or of v j , the graph obtained from G b y deleting w i w ould con tain a K 3 , 4 minor. This is a con tradiction, because the graph obtained from G b y remo ving w i and joining w i − 1 and w i +1 is isomorphic to H . Therefore, for eac h i ∈ [ s ] , there exists v ∈ V ( K 3 , 4 ) such that µ ( v ) = { w i } . W e claim that ev ery suc h vertex v b elongs to Y . Without loss of generality , assume that i < s . If the claim fails, then v ∈ X has degree four in K 3 , 4 . Since w i has degree four in G , it follo ws that { v 1 , v 2 , w i − 1 , w i +1 } ⊆ S u ∈ Y µ ( u ) . Consequen tly , there exists w ∈ Y such that µ ( w ) = { w i +1 } . Then w is adjacen t to at most t w o v ertices of X , which is imp ossible. F rom the ab o v e discussion, w e conclude that for each i ∈ [ s ] , there exists y i ∈ Y such that µ ( y i ) = { w i } . Since s > 2 , the vertex y 2 ∈ Y is adjacen t to at most t w o v ertices of X , a con tradiction. □ Lemma 4.7. F or any s 1 , s 2 ≥ 0 , the gr aph D s 1 ,s 2 + { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } do es not c ontain K 3 , 4 as a minor. F or any s ≥ 0 , the gr aph E s + { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } do es not c ontain K 3 , 4 as a minor. The gr aph F + { φ 1 3 φ 2 3 } do es not c ontain K 3 , 4 as a minor. Pr o of. Supp ose, to the contrary , that for some s 1 , s 2 ≥ 0 , the graph D s 1 ,s 2 + { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } con tains a K 3 , 4 minor. Choose suc h a graph with s 1 + s 2 minim um. One can directly v erify (preferably with a computer) that D 2 , 2 + { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } contains no K 3 , 4 minor. At least one of s 1 and s 2 is greater than 2 ; otherwise, D s 1 ,s 2 + { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } would b e a minor of D 2 , 2 + { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } , whic h con tains no K 3 , 4 minor. Without loss of generalit y , assume s 1 > 2 . Ev ery vertex on the spine with end-vertices δ 1 2 and δ 1 4 has degree four and is adjacent to b oth δ 1 1 and δ 1 3 , and moreov er δ 1 1 and δ 1 3 are adjacen t. Therefore, contracting the edge σ 1 s 1 σ 1 s 1 +1 pro duces a graph that con tains a K 3 , 4 minor b y Lemma 4.6 . This con tradicts the minimality of s 1 + s 2 , since the resulting graph is isomorphic to D s 1 − 1 ,s 2 + { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } . Similarly , supp ose to the contrary that for some s ≥ 0 , the graph E s + { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } contains a K 3 , 4 minor. Cho ose suc h an s minimum. One can directly v erify that E 2 + { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } con tains no K 3 , 4 minor. Hence s > 2 . Again, we ma y contract a spine edge to obtain a graph isomorphic to E s − 1 + { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } . By the minimalit y of s , this graph con tains no K 3 , 4 minor, con tradicting Lemma 4.6 . Finally , one can directly chec k that the graph F + { φ 1 3 φ 2 3 } do es not contain K 3 , 4 as a minor. □ A graph G is oloidal if one of the following holds: • G is isomorphic to D 0 ,s 2 + A with s 2 ∈ { 0 , 1 } , and A ⊆ { δ 1 1 δ 2 2 , δ 1 1 δ 2 3 , δ 1 1 δ 2 4 } or A ⊆ { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } . • G is isomorphic to D s 1 ,s 2 + A with s 1 , s 2 ≥ 0 , s 1 + s 2 ≥ 2 , and A ⊆ { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } . • G is isomorphic to E s + A with s ≥ 0 and A ⊆ { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } . • G is isomorphic to F + A with A ⊆ { φ 1 3 φ 2 3 } . Note that for s ≥ 0 , E s + { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } is isomorphic to D 0 ,s +1 + { δ 1 1 δ 2 2 , δ 1 1 δ 2 3 , δ 1 1 δ 2 4 } . Prop osition 4.8. Every oloidal gr aph is 4 -c onne cte d and non-pr oje ctive-planar and c ontains no K 3 , 4 minor. Pr o of. Lemma 4.5 implies that every oloidal graph is 4 -connected and non-pro jective-planar. Observ e that D 0 , 1 + { δ 1 1 δ 2 2 , δ 1 1 δ 2 3 , δ 1 1 δ 2 4 } is isomorphic to E 0 + { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } , where the v ertices δ 1 1 , δ 1 2 , δ 1 3 , δ 1 4 , δ 2 1 , δ 2 2 , δ 2 3 , δ 2 4 , σ 2 1 corresp ond to ε 3 2 , ε 3 1 , ε 4 , ε 3 3 , ε 1 2 , ε 1 1 , ε 2 , ε 1 3 , ε 0 , resp ectively . Thus, it follows from Lemma 4.7 that every oloidal graph has no K 3 , 4 minor. □ GRAPHS WITH NO K 3 , 4 MINOR 11 W e will establish the conv erse of Prop osition 4.8 . Theorem 4.9. A gr aph is 4 -c onne cte d, non-pr oje ctive-planar, and has no K 3 , 4 minor if and only if it is oloidal. T o pro ve Theorem 4.9 , w e apply Prop osition 4.3 . T o this end, in Section 4.2.2 (resp ectively , Section 4.2.3 ) w e study the structure of 4 -connected graphs with no K 3 , 4 minor that con tain a spanning JT-sub division of E 20 (resp ectiv ely , F 4 ). Finally , we complete the pro of of Theorem 4.9 in Section 4.2.4 . 4.2.2. Containing a sp anning sub division of E 20 . In this section we study the structure of 4 - connected graphs that hav e no K 3 , 4 minor, no F 4 minor, and no minor of an y graph obtained from E 20 b y splitting a vertex, y et con tain a spanning JT-sub division of E 20 . Our purp ose is to establish Prop osition 4.16 , beginning with a sequence of lemmas, some of which ha v e already b een giv en in [ 17 ]. A collection of graphs is presented in T able 1 . The graph in ro w i and column j is denoted by ( i, j ) . These graphs are frequen tly referenced in our arguments. F or instance, given a graph G with no K 3 , 4 minor that contains a spanning sub division η ( E 20 ) of E 20 , we can deduce that no vertex in the domain of e 0 is adjacen t to a vertex in the domain of e 4 ; otherwise, G ⪰ (2 , 2) ⪰ K 3 , 4 , con tradicting the assumption. By a sligh t abuse of notation, we ma y simply write: “by (2 , 2) , no v ertex in the domain of e 0 is adjacen t to a vertex in the domain of e 4 . ” T able 1. A collection of graphs. The graphs in the first row con tain F 4 as a spanning subgraph, while all others con tain K 3 , 4 as a minor. 1 2 3 4 5 6 1 2 3 Lemma 4.10 ([ 17 ]) . L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 , F 4 , nor any gr aph obtaine d fr om E 20 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( E 20 ) of E 20 . Then the se gment η ( e ) has no internal vertex for any e ∈ { e 0 e 1 1 , e 0 e 1 2 , e 0 e 1 3 , e 1 1 e 1 2 , e 1 2 e 1 3 , e 1 3 e 1 1 } . 12 O.-H.S. LO Lemma 4.11. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 , F 4 , nor any gr aph obtaine d fr om E 20 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( E 20 ) of E 20 . Then the se gment η ( e ) has no internal vertex for any e ∈ { e 0 e 2 , e 3 1 e 2 , e 3 2 e 2 , e 3 3 e 2 } . Pr o of. Supp ose, to the con trary , that for some e ∈ { e 0 e 2 , e 3 1 e 2 , e 3 2 e 2 , e 3 3 e 2 } , the segment η ( e ) has an in ternal vertex v . By Lemma 3.3 , eac h segment of η ( E 20 ) is an induced path in G . Since G is 4 -connected, the v ertex v has at least tw o neighbors outside η ( e ) . Case 1. e = e 0 e 2 . If follo ws that some neighbor of v lies in the domain of e 1 j or e 3 j for some j ∈ [ 3] , or in the domain of e 4 . If a neighbor of v lies in the domain of e 1 j with j ∈ [3] , then G m ust contain (1 , 1) as a minor, since the graph obtained from η ( E 20 ) by adding the edge joining v and its neighbor in the domain of e 1 j , contracting the domain of e 1 j in to a new vertex w , and finally deleting the edge joining w and η ( e 0 ) , is a sub division of (1 , 1) . (Such details are often omitted if obvious.) Hence G contains F 4 as a minor, whic h is a contradiction. Similarly , if a neigh b or of v lies in the domain of e 3 j for some j ∈ [ 3] , or in the domain of e 4 , then G must contain a minor of (2 , 1) or (2 , 2) , and hence a minor of K 3 , 4 , again a contradiction. Case 2. e = e 3 i e 2 for some i ∈ [3] . It follo ws that an y neighbor of v do es not lie in the domain of e 0 or e 1 j with j ∈ [3] \ { i } ; otherwise, G w ould contain a minor of (2 , 1) or (3 , 3) , and hence a K 3 , 4 minor, which is imp ossible. Similarly , an y neigh b or of v do es not lie in the domain of e 3 j with j ∈ [3] \ { i } ; otherwise, one obtains a sub division of (2 , 1) from η ( E 20 ) by adding the edge joining v and the domain of e 3 j , con tracting the domain of e 3 j in to a new vertex w 1 , contracting η ( e 2 e 3 6 − i − j ) in to a new vertex w 2 , and finally deleting the edge joining w 1 and w 2 . Th us, an y neigh b or of v must lie in η ( e 3 i e 1 i ) − η ( e 3 i ) or η ( e 3 i e 4 ) − η ( e 3 i ) . In fact, by Lemma 3.3 , the vertex v has exactly tw o neighbors outside η ( e ) , namely the neighbors of η ( e 3 i ) in η ( e 3 i e 1 i ) and η ( e 3 i e 4 ) . Moreo ver, since G is 4 -connected, η ( e 3 i ) has a neighbor in the domain of e 0 , or in the domain of e 1 j or e 3 j for some j ∈ [3] \ { i } . This implies that G contains (2 , 1) , (2 , 3) , or (3 , 4) as a minor, a con tradiction. This completes the pro of. □ Lemma 4.12. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 nor any gr aph obtaine d fr om E 20 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( E 20 ) of E 20 . Then, for any i ∈ [3] , the union of the se gments η ( e 1 i e 3 i ) and η ( e 3 i e 4 ) is an induc e d p ath in G . Pr o of. By Lemma 3.3 , every segment of η ( E 20 ) is an induced path in G . Supp ose, to the contrary , that for some i ∈ [3] , the union of the segmen ts η ( e 1 i e 3 i ) and η ( e 3 i e 4 ) is not an induced path in G . That is, there exists an edge joining a v ertex in η ( e 1 i e 3 i ) − η ( e 3 i ) to a v ertex in η ( e 3 i e 4 ) − η ( e 3 i ) . Since G is 4 -connected, η ( e 3 i ) has a neighbor in the domain of e 0 , or in the domain of e 1 j or e 3 j for some j ∈ [ 3] \ { i } . This implies that G contains (2 , 1) , (2 , 6) , or (3 , 1) as a minor, a contradiction. □ Lemma 4.13. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 , F 4 , nor any gr aph obtaine d fr om E 20 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( E 20 ) of E 20 . Then η ( e 4 ) is adjac ent to η ( e 2 ) and has de gr e e four in G . Pr o of. By Lemma 3.3 , every segment of η ( E 20 ) is an induced path in G . Since G is 4 -connected, η ( e 4 ) has a neighbor u outside the closed domain of e 4 . By (2 , 2) , u  = η ( e 0 ) . By Lemmas 4.10 , 4.11 , and 4.12 , w e conclude that u = η ( e 2 ) . This completes the pro of. □ GRAPHS WITH NO K 3 , 4 MINOR 13 Lemma 4.14 ([ 17 ]) . L et H b e a gr aph obtaine d fr om E 20 by sub dividing two indep endent e dges, e ach with one new vertex, and then joining these two new vertic es. Then H c ontains K 3 , 4 or F 4 as a minor. Lemma 4.15. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 , F 4 , nor any gr aph obtaine d fr om E 20 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( E 20 ) of E 20 , and η ( e 1 i e 3 i ) or η ( e 3 i e 4 ) , with i ∈ [3] , c ontains an internal vertex v . Then v is adjac ent to η ( e 2 ) and η ( e 3 j ) for some j ∈ [3] \ { i } and has de gr e e four in G . Pr o of. By Lemma 3.3 , every segment of η ( E 20 ) is an induced path in G . By Lemma 4.13 , η ( e 2 ) and η ( e 4 ) are adjacent. Without loss of generality , assume i = 1 . So v is an internal v ertex of η ( e ) , where e is either e 1 1 e 3 1 or e 3 1 e 4 . Clearly , v has at least t w o neigh b ors outside η ( e ) . Let u b e such a neigh b or. Case 1. e = e 1 1 e 3 1 . By (2 , 1) and (1 , 2) , u do es not lie in the domain of e 0 or e 1 j with j ∈ { 2 , 3 } . Moreo v er, b y Lemmas 4.12 and 4.14 , u is not in the domain of e 4 . It follows from Lemma 4.11 that u must b e η ( e 2 ) , η ( e 3 2 ) , or η ( e 3 3 ) . Supp ose v is adjacen t to b oth η ( e 3 2 ) and η ( e 3 3 ) . It is clear that η ( e 3 1 ) has a neigh b or w outside the closed domain of e 3 1 , and w is not in the domain of e 0 . Th us w must lie in the domain of e 1 2 or e 1 3 , or in the domain of e 3 2 or e 3 3 , which is imp ossible b y (2 , 4) and (3 , 2) . Therefore, v is adjacent to η ( e 2 ) and η ( e 3 j ) for some j ∈ { 2 , 3 } and has degree four in G . Case 2. e = e 3 1 e 4 . By (2 , 1) and (3 , 5) , u do es not lie in the domain of e 0 , e 1 2 , or e 1 3 . Consequen tly , b y Lemma 4.12 , u is not in the domain of e 1 1 . It follo ws from Lemmas 3.3 and 4.11 that u m ust be η ( e 2 ) or the neigh b or of η ( e 4 ) in η ( e 3 2 e 4 ) or η ( e 3 3 e 4 ) . In fact, v is not adjacent to b oth the neigh b ors of η ( e 4 ) in η ( e 3 2 e 4 ) and in η ( e 3 3 e 4 ) , b y (3 , 6) . This implies that v has degree four and is adjacent to η ( e 2 ) and the neigh b or of η ( e 4 ) in η ( e 4 e 3 j ) for some j ∈ { 2 , 3 } . Without loss of generalit y , assume j = 3 . Denote by w the neigh b or of η ( e 4 ) in η ( e 4 e 3 3 ) . W e claim th at w = η ( e 3 3 ) ; in other words, the segment η ( e 4 e 3 3 ) has no in ternal v ertex. Supp ose otherwise. By the abov e, w is adjacent to η ( e 2 ) . Note that, b y Lemma 3.3 , v and w are neighbors of η ( e 4 ) in η ( e 3 1 e 4 ) and η ( e 3 3 e 4 ) , resp ectively . Moreov er, η ( e 3 2 ) is adjacen t to neither v nor w (as the neigh b ors of v and w ha v e already b een determined). It is clear that η ( e 3 2 ) has a neigh b or outside the closed domain of e 3 2 , and, by (2 , 1) , this neighbor do es not lie in the domain of e 0 . By symmetry , it suffices to consider the cases where η ( e 3 2 ) joins the domain of e 1 1 or the domain of e 3 1 . If η ( e 3 2 ) joins the domain of e 1 1 , then by Lemma 4.10 , η ( e 3 2 ) is adjacent to a v ertex in η ( e 1 1 e 3 1 ) − η ( e 3 1 ) . It is clear that η ( e 3 1 ) has a neigh b or outside the closed domain of e 3 1 . By (2 , 1) , (2 , 4) , (2 , 5) , and (3 , 2) , this neigh b or of η ( e 3 1 ) do es not lie in the domain of e 0 , e 1 2 , e 1 3 , or e 3 2 . If that neigh b or lies in η ( e 3 3 e 4 ) − { w , η ( e 4 ) } , then G contains a minor of (2 , 5) (this can b e seen by contracting the union of η ( e 1 3 e 3 3 ) and η ( e 3 3 e 4 ) − { w , η ( e 4 ) } ). As η ( e 3 1 ) and w are not adjacent, we ha ve considered all cases. In eac h case, we obtain a con tradiction to the assumption that G has no K 3 , 4 minor. If η ( e 3 2 ) joins the domain of e 3 1 , then G would contain a minor of (3 , 2) , a con tradiction. This completes the pro of. □ Prop osition 4.16. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 , F 4 , nor any gr aph obtaine d fr om E 20 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( E 20 ) of E 20 . Then one of the fol lowing holds: • G c ontains D 0 , 1 as a sp anning sub gr aph such that e ach of δ 1 2 , δ 1 4 , and σ 2 1 has de gr e e four in G . 14 O.-H.S. LO • G c ontains D 0 ,s 2 , with s 2 = | V ( G ) | − 8 ≥ 2 , as a sp anning sub gr aph such that every vertex on either spine has de gr e e four in G . • G c ontains E 0 as a sp anning sub gr aph such that e ach of ε 0 and ε 4 has de gr e e four in G . Pr o of. Recall that, by Lemma 3.3 , every segment of η ( E 20 ) is an induced path and, by Lemma 4.13 , η ( e 4 ) has degree four and is adjacen t to η ( e 2 ) . W e consider t w o cases, depending on whether η ( E 20 ) has any internal vertices. Case 1. | V ( G ) | = | V ( E 20 ) | . In this case, there is no in ternal vertex. By (2 , 1) , each η ( e 3 i ) with i ∈ [3] has a neigh b or that is η ( e 1 j ) or η ( e 3 j ) for some j ∈ [3] \ { i } . Case 1.1. There exist distinct i, j ∈ [3] such that η ( e 3 i ) is adjacent to η ( e 1 j ) . Without loss of generality , assume i = 1 and j = 2 . By (2 , 4) and (2 , 5) , η ( e 3 2 ) is adjacen t to η ( e 3 3 ) . Hence G con tains E 0 as a spanning subgraph. More precisely , E 0 is isomorphic to the graph obtained from η ( E 20 ) b y adding the edges η ( e 3 1 ) η ( e 1 2 ) , η ( e 3 2 ) η ( e 3 3 ) , and η ( e 2 ) η ( e 4 ) , where the v ertices ε 0 , ε 1 1 , ε 1 2 , ε 1 3 , ε 2 , ε 3 1 , ε 3 2 , ε 3 3 , ε 4 corresp ond to η ( e 1 1 ) , η ( e 1 2 ) , η ( e 0 ) , η ( e 1 3 ) , η ( e 3 1 ) , η ( e 3 2 ) , η ( e 2 ) , η ( e 3 3 ) , η ( e 4 ) , respectively . The v ertex ε 4 , corresponding to η ( e 4 ) , has degree four in G b y Lemma 4.13 . Observ e that E 0 has an automorphism mapping ε 0 , ε 1 1 , ε 1 2 , ε 1 3 , ε 2 , ε 3 1 , ε 3 2 , ε 3 3 , ε 4 to ε 4 , ε 3 2 , ε 3 1 , ε 3 3 , ε 2 , ε 1 2 , ε 1 1 , ε 1 3 , ε 0 , resp ectively . Consequently , by Prop osition 3.2 , w e may consider another spanning JT-sub division η ′ ( E 20 ) of E 20 with η ′ ( e 4 ) = η ( e 1 1 ) , and conclude that ε 0 , corresponding to η ( e 1 1 ) , has degree four in G . Case 1.2. There do not exist distinct i, j ∈ [3] such that η ( e 3 i ) is adjacent to η ( e 1 j ) . Without loss of generalit y , we assume η ( e 3 1 ) η ( e 3 2 ) , η ( e 3 2 ) η ( e 3 3 ) ∈ E ( G ) . In fact, D 0 , 1 is isomorphic to the graph obtained from η ( E 20 ) by adding the edges η ( e 3 1 ) η ( e 3 2 ) , η ( e 3 2 ) η ( e 3 3 ) , and η ( e 2 ) η ( e 4 ) , where the v ertices δ 1 1 , δ 1 2 , δ 1 3 , δ 1 4 , δ 2 1 , δ 2 2 , δ 2 3 , δ 2 4 , σ 2 1 corresp ond to η ( e 1 1 ) , η ( e 1 2 ) , η ( e 1 3 ) , η ( e 0 ) , η ( e 3 1 ) , η ( e 3 2 ) , η ( e 3 3 ) , η ( e 2 ) , η ( e 4 ) , respectively . So G con tains D 0 , 1 as a spanning subgraph. By Lemma 4.13 , the vertex σ 2 1 , corresponding to η ( e 4 ) , has degree four in G . By (2 , 1) and (2 , 2) , the vertex δ 1 4 , corresp onding to η ( e 0 ) , has degree four in G . Note that the subgraph of G isomorphic to D 0 , 1 admits an automorphism that swaps η ( e 0 ) and η ( e 1 2 ) , sw aps η ( e 2 ) and η ( e 3 2 ) , and fixes all other vertices. Therefore, again by (2 , 1) and (2 , 2) , w e deduce that the v ertex δ 1 2 , corresponding to η ( e 1 2 ) , also has degree four in G . Case 2. | V ( G ) | > | V ( E 20 ) | . By Lemmas 4.10 , 4.11 , and 4.15 , there exist distinct i, j ∈ [3] such that η ( e ) , where e = e 1 i e 3 i or e 3 i e 4 , has an in ternal v ertex v adjacen t to η ( e 2 ) and η ( e 3 j ) . Without loss of generality , assume i = 1 and j = 2 . Case 2.1. e = e 1 1 e 3 1 . By (2 , 4) , (2 , 5) , and Lemma 4.15 , η ( e 1 2 e 3 2 ) has no internal vertex. If η ( e 3 2 e 4 ) had an in ternal vertex, then by Lemma 4.15 , that in ternal vertex would b e adjacen t to η ( e 2 ) and G would contain as a minor the graph obtained from E 20 b y sub dividing e 1 1 e 3 1 and e 1 2 e 3 2 , eac h with one new vertex, and joining these new v ertices with an edge. This yields a con tradiction b y Lemma 4.14 . Hence η ( e 3 2 e 4 ) has no internal vertex. By (3 , 2) and Lemma 4.15 , it is straigh tforw ard to show that any internal v ertex of η ( e 1 1 e 3 1 ) , η ( e 3 1 e 4 ) , η ( e 1 3 e 3 3 ) , or η ( e 3 3 e 4 ) is adjacent to η ( e 3 2 ) . By (2 , 1) , (2 , 4) , (2 , 5) , (3 , 2) , Lemma 4.11 , and the fact that η ( e 1 2 e 3 2 ) and η ( e 3 2 e 4 ) hav e no internal v ertex, w e ha v e that η ( e 3 1 ) has degree four and is adjacen t to η ( e 3 2 ) . By (3 , 2) and by the fact that no in ternal v ertex of η ( e 1 1 e 3 1 ) is adjacent to η ( e 3 3 ) , η ( e 3 3 ) do es not join the domain of e 3 1 . By (2 , 1) , η ( e 3 3 ) do es not join the domain of e 0 . By (1 , 2) and b y the fact that no in ternal vertex of η ( e 1 1 e 3 1 ) is adjacen t to η ( e 3 3 ) , we ha v e that η ( e 3 3 ) joins neither the domain GRAPHS WITH NO K 3 , 4 MINOR 15 of e 1 1 nor that of e 1 2 . (T o see this, consider another sub division η ′ ( E 20 ) of E 20 suc h that η ′ ( e 3 1 ) = v , η ′ ( e 4 ) = η ( e 3 1 ) , and η ′ ( e 3 3 ) = η ( e 4 ) ; then η ( e 3 3 ) b ecomes an in ternal vertex of η ′ ( e 1 3 e 3 3 ) .) Therefore, b y Lemma 4.11 and the fact that η ( e 1 2 e 3 2 ) and η ( e 3 2 e 4 ) hav e no internal vertex, η ( e 3 3 ) has degree four and is adjacent to η ( e 3 2 ) . Th us, G contains D 0 ,s 2 , where s 2 = | V ( G ) | − 8 ≥ 2 , as a spanning subgraph. This subgraph is obtained from η ( E 20 ) b y adding the edges η ( e 2 ) η ( e 4 ) , η ( e 3 1 ) η ( e 3 2 ) , η ( e 3 2 ) η ( e 3 3 ) , and the edges inciden t to internal v ertices. The vertices δ 1 1 , δ 1 2 , δ 1 3 , δ 1 4 , δ 2 1 , δ 2 2 , δ 2 3 , δ 2 4 corresp ond to η ( e 1 1 ) , η ( e 1 2 ) , η ( e 1 3 ) , η ( e 0 ) , the neighbor of η ( e 1 1 ) in η ( e 1 1 e 3 1 ) , η ( e 3 2 ) , the neighbor of η ( e 1 3 ) in η ( e 1 3 e 3 3 ) , η ( e 2 ) , resp ectiv ely , and the non-trivial spine, with end-v ertices δ 2 1 and δ 2 3 , corresp onds to the union of η ( e 1 1 e 3 1 ) − η ( e 1 1 ) , η ( e 3 1 e 4 ) , η ( e 4 e 3 3 ) , and η ( e 3 3 e 1 3 ) − η ( e 1 3 ) . As sho wn ab ov e, ev ery vertex of the non-trivial spine has degree four. Recall that no internal v ertex is adjacent to η ( e 0 ) or η ( e 1 2 ) . Note that the subgraph isomorphic to D 0 ,s 2 admits an automorphism that swaps η ( e 0 ) and η ( e 1 2 ) , sw aps η ( e 2 ) and η ( e 3 2 ) , and fixes all other v ertices. Hence, b y (2 , 1) and (2 , 2) , we conclude that the v ertices δ 1 4 and δ 1 2 of the trivial spine, corresponding to η ( e 0 ) and η ( e 1 2 ) , respectively , b oth hav e degree four. Case 2.2. None of η ( e 1 1 e 3 1 ) , η ( e 1 2 e 3 2 ) , or η ( e 1 3 e 3 3 ) has an internal vertex, and e = e 1 1 e 3 1 . Most argumen ts are similar to those in Case 2.1. Recall that v is an internal v ertex of η ( e 3 1 e 4 ) and is adjacent to η ( e 3 2 ) . It follows from Lemmas 4.15 and 4.14 that η ( e 3 2 e 4 ) has no internal vertex. Moreo v er, by Lemma 4.15 and (3 , 2) , any in ternal v ertex of η ( e 3 1 e 4 ) or η ( e 3 3 e 4 ) is adjacent to η ( e 3 2 ) . Using (2 , 1) , (1 , 2) , and (3 , 2) , one can deduce that b oth η ( e 3 1 ) and η ( e 3 3 ) hav e degree four and are adjacen t to η ( e 3 2 ) . Therefore, we conclude that G contains D 0 ,s 2 with s 2 = | V ( G ) | − 8 ≥ 2 as a spanning subgraph, suc h that δ 1 1 , δ 1 2 , δ 1 3 , δ 1 4 , δ 2 1 , δ 2 2 , δ 2 3 , δ 2 4 corresp ond to η ( e 1 1 ) , η ( e 1 2 ) , η ( e 1 3 ) , η ( e 0 ) , η ( e 3 1 ) , η ( e 3 2 ) , η ( e 3 3 ) , η ( e 2 ) , resp ectiv ely , and the non-trivial spine, with end-v ertices δ 2 1 and δ 2 3 , corresp onds to the union of η ( e 3 1 e 4 ) and η ( e 4 e 3 3 ) . Ev ery vertex on a non-trivial spine has degree four. Moreo ver, by the same argumen t as in Case 2.1, the vertices on the trivial spine also hav e degree four. □ 4.2.3. Containing a sp anning sub division of F 4 . In this section we analyze 4 -connected graphs with no K 3 , 4 minor, with no minor of any graph obtained from F 4 b y splitting a vertex, and con taining a spanning JT-subdivision of F 4 . Our goal is to prov e Prop osition 4.30 ; to this end, we establish a sequence of lemmas. Analogously to the previous section, we prepare a collection of graphs in T able 2 , using the same con v entions as b efore. Lemma 4.17. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 nor any gr aph obtaine d fr om F 4 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( F 4 ) of F 4 . F or any i ∈ [2] , η ( f i 2 ) joins to at le ast one of η ( f i 1 f i ) − η ( f i ) , η ( f i 1 f i 3 ) − η ( f i 3 ) , η ( f i 4 f i ) − η ( f i ) , or η ( f i 4 f i 3 ) − η ( f i 3 ) . Pr o of. By Lemma 3.3 , every segmen t of η ( F 4 ) is an induced path in G . As G is 4 -connected, η ( f i 2 ) has a neighbor outside the closed domain of f i 2 . By (4 , 1) , (4 , 2) , and (4 , 3) , η ( f i 2 ) m ust join to one of η ( f i 1 f i ) − η ( f i ) , η ( f i 1 f i 3 ) − η ( f i 3 ) , η ( f i 4 f i ) − η ( f i ) , or η ( f i 4 f i 3 ) − η ( f i 3 ) . □ Lemma 4.18. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 nor any gr aph obtaine d fr om F 4 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( F 4 ) of F 4 . F or any i ∈ [2] , no vertex in the domain of f i joins to any vertex in the domain of f 3 − i or in the domain of f 3 − i j with j ∈ [4] . Pr o of. Let v b e a vertex in the domain of f i . 16 O.-H.S. LO T able 2. A collection of graphs, eac h con taining K 3 , 4 as a minor. 1 2 3 4 5 6 4 5 6 7 8 By (4 , 2) and (4 , 4) , v do es not join to the domain of f 3 − i or f 3 − i 2 . W e claim that v do es not join the domain of f 3 − i 3 . Supp ose otherwise. If v were an internal v ertex of η ( f 1 f 1 1 ) , then G w ould contain a minor of (5 , 4) (to see this, contract η ( f 1 f 1 4 ) ), a contradiction. Th us v is not an in ternal v ertex of η ( f 1 f 1 1 ) , and, by symmetry , not of η ( f 1 f 1 4 ) . Hence v lies in η ( f 1 f 1 2 ) − η ( f 1 2 ) . By Lemma 4.17 , G would then again con tain a minor of (5 , 4) , a con tradiction. By symmetry , it remains to show that v do es not join any vertex in η ( f i 1 f 3 − i 4 ) − η ( f i 1 ) . Supp ose otherwise. As G con tains neither (4 , 3) nor (7 , 1) as a minor, v lies in η ( f i f i 1 ) − η ( f i 1 ) . Clearly , η ( f i 1 ) has a neigh b or u outside the closed domain of f i 1 . By (5 , 5) , (5 , 6) , (4 , 5) , (4 , 3) , and (4 , 6) , u is not in the domain of f i 2 , f i 4 , f 3 − i 1 , f 3 − i 2 , or f 3 − i 3 . Thus u lies in η ( f 3 − i f 3 − i 4 ) − η ( f 3 − i 4 ) . By Lemma 4.17 , η ( f i 2 ) (respectively , η ( f 3 − i 2 ) ) has a neighbor either in η ( f i 1 f i ) − η ( f i ) or η ( f i 1 f i 3 ) − η ( f i 3 ) (resp ectively , in η ( f 3 − i 1 f 3 − i ) − η ( f 3 − i ) or η ( f 3 − i 1 f 3 − i 3 ) − η ( f 3 − i 3 ) ), or in η ( f i 4 f i ) − η ( f i ) or η ( f i 4 f i 3 ) − η ( f i 3 ) (respectively , in η ( f 3 − i 4 f 3 − i ) − η ( f 3 − i ) or η ( f 3 − i 4 f 3 − i 3 ) − η ( f 3 − i 3 ) ). GRAPHS WITH NO K 3 , 4 MINOR 17 By (5 , 5) , if η ( f i 2 ) (resp ectively , η ( f 3 − i 2 ) ) has a neigh b or in η ( f i 1 f i ) − η ( f i ) or η ( f i 1 f i 3 ) − η ( f i 3 ) (resp ectiv ely , in η ( f 3 − i 4 f 3 − i ) − η ( f 3 − i ) or η ( f 3 − i 4 f 3 − i 3 ) − η ( f 3 − i 3 ) ), then v (repsectively , u ) must b e an in ternal vertex of η ( f i f i 1 ) (respectively , of η ( f 3 − i f 3 − i 4 ) ). W e use the ab o v e observ ation to analyze the different cases according to the neighbors of η ( f i 2 ) and η ( f 3 − i 2 ) . If η ( f i 2 ) has a neigh b or in η ( f i 1 f i ) − η ( f i ) or η ( f i 1 f i 3 ) − η ( f i 3 ) , and η ( f 3 − i 2 ) has a neigh b or in η ( f 3 − i 4 f 3 − i ) − η ( f 3 − i ) or η ( f 3 − i 4 f 3 − i 3 ) − η ( f 3 − i 3 ) , then v is an internal v ertex of η ( f i f i 1 ) and u is an in ternal v ertex of η ( f 3 − i f 3 − i 4 ) ; hence G contains a minor of (6 , 6) (to see this, contract η ( f i f i 4 ) and η ( f 3 − i f 3 − i 1 ) ), a contradiction. If η ( f i 2 ) has a neigh b or in η ( f i 1 f i ) − η ( f i ) or η ( f i 1 f i 3 ) − η ( f i 3 ) , and η ( f 3 − i 2 ) has a neigh b or in η ( f 3 − i 1 f 3 − i ) − η ( f 3 − i ) or η ( f 3 − i 1 f 3 − i 3 ) − η ( f 3 − i 3 ) , then G also con tains a minor of (6 , 6) (to see this, con tract η ( f i f i 4 ) ), a con tradiction. The case where η ( f i 2 ) has a neigh b or in η ( f i 4 f i ) − η ( f i ) or η ( f i 4 f i 3 ) − η ( f i 3 ) , and η ( f 3 − i 2 ) has a neighbor in η ( f 3 − i 4 f 3 − i ) − η ( f 3 − i ) or η ( f 3 − i 4 f 3 − i 3 ) − η ( f 3 − i 3 ) , can b e handled similarly . Finally , if η ( f i 2 ) has a neighbor in η ( f i 4 f i ) − η ( f i ) or η ( f i 4 f i 3 ) − η ( f i 3 ) , and η ( f 3 − i 2 ) has a neighbor in η ( f 3 − i 1 f 3 − i ) − η ( f 3 − i ) or η ( f 3 − i 1 f 3 − i 3 ) − η ( f 3 − i 3 ) , then one directly sees that G contains a minor of (6 , 6) , a contradiction. This completes the pro of. □ Lemma 4.19. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 nor any gr aph obtaine d fr om F 4 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( F 4 ) of F 4 . Then the se gment η ( e ) has no internal vertex for any e ∈ { f 1 1 f 1 3 , f 1 2 f 1 3 , f 1 4 f 1 3 , f 2 1 f 2 3 , f 2 2 f 2 3 , f 2 4 f 2 3 } . Pr o of. Supp ose to the contrary that η ( e ) contains an internal v ertex v . Clearly , v has at least tw o neigh b ors outside η ( e ) . By symmetry , it suffices to treat the cases e = f 1 1 f 1 3 and e = f 1 2 f 1 3 . Case 1. e = f 1 1 f 1 3 . The vertex v joins neither the domain of f 1 2 nor f 1 4 since G does not contain, as a minor, an y graph obtained from F 4 b y splitting the v ertex f 1 3 . Moreo v er, v do es not join the domain of any of f 2 1 , f 2 2 , f 2 3 b y (7 , 2) , (4 , 3) , and (7 , 3) . By Lemma 4.18 , v also do es not join the domain of f 2 . Th us, by Lemma 3.3 and the fact that G is 4 -connected, v has exactly t wo neighbors outside η ( e ) , namely the neighbor of η ( f 1 1 ) in η ( f 1 1 f 1 ) and the neighbor of η ( f 1 1 ) in η ( f 1 1 f 2 4 ) . It is clear that η ( f 1 1 ) has a neigh b or u outside the closed domain of f 1 1 . By (7 , 4) , (7 , 5) , (5 , 2) , (4 , 3) , and (5 , 1) , u does not lie in the domain of f 1 2 , f 1 4 , f 2 1 , f 2 2 , or f 2 3 . Hence u must lie in the domain of f 2 , whic h contradicts Lemma 4.18 . Case 2. e = f 1 2 f 1 3 . As G do es not contain as a minor a graph obtained from F 4 b y splitting f 1 3 , v do es not join the domain of f 1 1 or f 1 4 . By (4 , 2) , (4 , 3) , and (4 , 1) , v do es not join the domain of f 2 , f 2 1 , f 2 2 , or f 2 4 . By Lemma 3.3 and the fact that G is 4 -connected, v has exactly t w o neighbors outside η ( e ) , namely the neighbor of η ( f 1 2 ) in η ( f 1 2 f 1 ) and the neighbor of η ( f 1 2 ) in η ( f 1 2 f 2 3 ) . By Lemma 4.17 , η ( f 1 2 ) joins the union of η ( f 1 j f 1 ) − η ( f 1 ) and η ( f 1 j f 1 3 ) − η ( f 1 3 ) for some j ∈ { 1 , 4 } . This implies that G con tains a minor of (5 , 4) (to see this, contract η ( f 1 f 1 5 − j ) ), a con tradiction. □ Lemma 4.20. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 nor any gr aph obtaine d fr om F 4 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( F 4 ) of F 4 . Then for any i ∈ [2] , η ( f i ) is adjac ent to η ( f i 3 ) and has de gr e e four in G . Pr o of. It is clear that η ( f i ) has a neighbor outside the closed domain of f i . By Lemma 4.18 and Lemma 4.19 , the only p ossible such neighbor is η ( f i 3 ) . Th us η ( f i ) has degree four and is adjacent to η ( f i 3 ) . □ 18 O.-H.S. LO Lemma 4.21. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 nor any gr aph obtaine d fr om F 4 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( F 4 ) of F 4 , and that η ( f i f i j ) , with i ∈ [2] and j ∈ { 1 , 4 } , c ontains an internal vertex v . Then η ( f i f i 5 − j ) has no internal vertex, and v is adjac ent to η ( f i 5 − j ) and η ( f i 3 ) and has de gr e e four in G . Pr o of. It suffices to prov e the case where v lies in η ( e ) with e = f 1 f 1 1 . Recall that η ( f 1 ) η ( f 1 3 ) , η ( f 2 ) η ( f 2 3 ) ∈ E ( G ) by Lemma 4.20 . By Lemmas 3.3 , 4.18 , and 4.19 , v has at least tw o neighbors outside η ( e ) , eac h of which is either the neigh b or of η ( f 1 ) in η ( f 1 f 1 2 ) , the neighbor of η ( f 1 ) in η ( f 1 f 1 4 ) , or η ( f 1 3 ) . By (8 , 1) , v is not adjacent to b oth neighbors of η ( f 1 ) in η ( f 1 f 1 2 ) and in η ( f 1 f 1 4 ) . Hence v has exactly t w o neigh b ors outside η ( e ) , and one of them is η ( f 1 3 ) . Case 1. v is adjacent to the neigh b or of η ( f 1 ) in η ( f 1 f 1 2 ) . W e show that this case cannot o ccur. It is clear that η ( f 1 1 ) (resp ectively , η ( f 1 4 ) ) has a neigh b or outside the closed domain of f 1 1 (re- sp ectiv ely , f 1 4 ). By (5 , 3) , (8 , 2) , (4 , 3) , and Lemma 4.18 , η ( f 1 1 ) do es not join to any domain of f 1 2 , f 1 4 , f 2 2 , or f 2 . Thus, using Lemma 4.19 , we conclude that η ( f 1 1 ) joins to η ( f 2 1 ) or η ( f 2 3 ) . By (8 , 1) and (8 , 2) , η ( f 1 4 ) do es not join to the domain of f 1 1 . By (5 , 3) , (4 , 3) , and Lemma 4.18 , η ( f 1 4 ) do es not join to any domain of f 1 2 , f 2 2 , or f 2 . It then follows from Lemma 4.19 that η ( f 1 4 ) joins to η ( f 2 3 ) or η ( f 2 4 ) . Hence w e ma y consider the cases dep ending on where η ( f 1 1 ) and η ( f 1 4 ) join. Case 1.1. Both η ( f 1 1 ) and η ( f 1 4 ) join to η ( f 2 3 ) . By Lemmas 4.17 and 4.19 , η ( f 2 2 ) has a neighbor either in η ( f 2 1 f 2 ) − η ( f 2 ) or in η ( f 2 4 f 2 ) − η ( f 2 ) . W e assume the former; the latter is analogous. By Lemma 4.18 and b y (4 , 3) , (5 , 1) , (5 , 2) , (6 , 3) , and (5 , 3) , an y neigh b or of η ( f 2 4 ) outside the closed domain of f 2 4 cannot join the domain of f 1 , f 1 2 , f 1 3 , f 1 4 , f 2 1 , or f 2 2 . Hence η ( f 2 4 ) has no neighbor outside the closed domain of f 2 4 , a contradiction. Case 1.2. At least one of η ( f 1 1 ) and η ( f 1 4 ) does not join to η ( f 2 3 ) . If either η ( f 1 1 ) or η ( f 1 4 ) joins to η ( f 2 3 ) , then G con tains (5 , 2) as a minor. Otherwise, G con tains a minor of (6 , 1) . In either case this is imp ossible. Case 2. v is adjacent to the neigh b or of η ( f 1 ) in η ( f 1 f 1 4 ) . Denote b y u the neighbor of η ( f 1 ) in η ( f 1 f 1 4 ) . W e claim that η ( f 1 f 1 4 ) has no internal vertex and hence u = η ( f 1 4 ) , which completes the pro of. Supp ose otherwise. By the observ ations ab ov e, v and u are adjacent, and each has degree four and joins to η ( f 1 3 ) . In particular, neither v nor u joins to η ( f 1 2 ) . By Lemma 3.3 , v is the neighbor of η ( f 1 ) in η ( f 1 f 1 1 ) . It then follows from Lemma 4.17 that G contains a minor of (5 , 3) , a con tradiction. □ Lemma 4.22. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 nor any gr aph obtaine d fr om F 4 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( F 4 ) of F 4 . F or any i ∈ [2] , η ( f i 2 ) is adjac ent to either η ( f i 1 ) or η ( f i 4 ) and has de gr e e four in G . Pr o of. It follows directly from Lemmas 3.3 , 4.17 , 4.19 , and 4.21 that an y neighbor of η ( f i 2 ) outside the closed domain of f i 2 m ust be η ( f i 1 ) or η ( f i 4 ) . By Lemma 4.20 and (5 , 3) , η ( f i 2 ) is adjacent to exactly one of η ( f i 1 ) or η ( f i 4 ) , and has degree four. □ Lemma 4.23. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 nor any gr aph obtaine d fr om F 4 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( F 4 ) of F 4 , and that η ( f i f i 2 ) , with i ∈ [2] , c ontains an internal vertex v . Then v is adjac ent to η ( f i 3 ) and either η ( f i 1 ) or η ( f i 4 ) , and has de gr e e four in G . Pr o of. It follo ws immediately from Lemmas 3.3 , 4.18 , 4.19 , 4.21 , 4.20 , and (5 , 3) . □ Lemma 4.24 ([ 17 ]) . L et H b e a gr aph obtaine d fr om F 4 by sub dividing two indep endent e dges, e ach with one new vertex, and then joining these two new vertic es. Then H c ontains K 3 , 4 as a minor. GRAPHS WITH NO K 3 , 4 MINOR 19 Lemma 4.25. L et H b e the gr aph obtaine d fr om F 4 by sub dividing f 1 1 f 2 4 with a new vertex v and adding four e dges v f 1 3 , f 1 f 1 3 , f 1 1 f 1 2 , and f 1 4 u , wher e u ∈ { f 1 1 , f 1 2 , f 2 2 , v } . Then H c ontains K 3 , 4 as a minor. Pr o of. Acc ording as u is f 1 1 , f 1 2 , f 2 2 , or v , the graph H con tains, resp ectiv ely , (6 , 3) , (5 , 3) , (4 , 3) , or (8 , 2) as a minor. Hence H contains a K 3 , 4 minor. □ Lemma 4.26. L et H b e the gr aph obtaine d fr om F 4 by sub dividing f 1 1 f 2 4 with a new vertex v and adding four e dges v f 1 3 , f 1 f 1 3 , f 1 2 f 1 4 , and f 1 1 u , wher e u ∈ { f 1 2 , f 2 1 , f 2 2 , f 2 3 } . Then H c ontains K 3 , 4 as a minor. Pr o of. Acc ording as u is f 1 2 , f 2 1 , f 2 2 , or f 2 3 , the graph H contains, resp ectiv ely , (5 , 3) , (5 , 2) , (4 , 3) , or (5 , 1) as a minor. Hence H contains a K 3 , 4 minor. □ Lemma 4.27. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 nor any gr aph obtaine d fr om F 4 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( F 4 ) of F 4 , and that η ( f i 1 f 3 − i 4 ) , with i ∈ [2] , c ontains an internal vertex v . Then v is adjac ent either to η ( f i 3 ) and η ( f i 4 ) , or to η ( f 3 − i 3 ) and η ( f 3 − i 1 ) , and has de gr e e four in G . Pr o of. Without loss of generality , assume i = 1 . By Lemma 3.3 , v has at least t w o neighbors outside η ( f 1 1 f 2 4 ) . By Lemmas 4.18 , 4.19 , and 4.24 , and b y (4 , 3) , any such neighbor m ust b e η ( f 1 3 ) , η ( f 1 4 ) , η ( f 2 1 ) , or η ( f 2 3 ) . Supp ose to the contrary that the lemma fails. Then v is adjacen t to η ( f 1 3 ) and η ( f 2 3 ) , to η ( f 1 3 ) and η ( f 2 1 ) , to η ( f 1 4 ) and η ( f 2 3 ) , or to η ( f 1 4 ) and η ( f 2 1 ) . W e sho w th at each p ossibilit y is imp ossible. Case 1. v is adjacent to η ( f 1 3 ) and η ( f 2 3 ) . Recall from Lemma 4.22 that η ( f 1 2 ) is adjacent to exactly one of η ( f 1 1 ) and η ( f 1 4 ) , and that η ( f 2 2 ) is adjacen t to exactly one of η ( f 2 1 ) and η ( f 2 4 ) . Accordingly , depending on the p ossible c hoices of these neigh b ors, we distinguish the following cases. Case 1.1. η ( f 1 2 ) joins to η ( f 1 1 ) and η ( f 2 2 ) joins to η ( f 2 4 ) . By Lemmas 4.20 , 4.25 , 4.18 , 4.19 , and 3.3 , η ( f 1 4 ) joins to η ( f 2 3 ) or η ( f 2 4 ) , and η ( f 2 1 ) joins to η ( f 1 3 ) or η ( f 1 1 ) , whic h forces G to con tain (5 , 1) , (5 , 2) , or (6 , 1) as a minor, a contradiction. Case 1.2. η ( f 1 2 ) joins to η ( f 1 4 ) and η ( f 2 2 ) joins to η ( f 2 1 ) . By Lemmas 4.20 , 4.26 , 4.18 , 4.19 , and 3.3 , η ( f 1 1 ) joins to η ( f 1 4 f 1 ) − η ( f 1 ) and η ( f 2 4 ) joins to η ( f 2 1 f 2 ) − η ( f 2 ) , whic h is imp ossible by (6 , 5) . Case 1.3. η ( f 1 2 ) joins to η ( f 1 1 ) and η ( f 2 2 ) joins to η ( f 2 1 ) , or η ( f 1 2 ) joins to η ( f 1 4 ) and η ( f 2 2 ) joins to η ( f 2 4 ) . Without loss of generality , w e assume the former holds. Then, b y Lem- mas 4.20 , 4.25 , 4.26 , 4.18 , 4.19 , and 3.3 , η ( f 1 4 ) joins to η ( f 2 3 ) or η ( f 2 4 ) , and η ( f 2 4 ) joins to η ( f 2 1 f 2 ) − η ( f 2 ) . Consequently , G contains a minor of (6 , 3) or (7 , 1) , a contradiction. Case 2. v is adjacent to η ( f 1 4 ) and η ( f 2 1 ) . Then G contains (7 , 6) as a minor, a contradiction. Case 3. v is adjacent to η ( f 1 3 ) and η ( f 2 1 ) , or to η ( f 1 4 ) and η ( f 2 3 ) . By symmetry , it suffices to consider the case where v is adjacent to η ( f 1 3 ) and η ( f 2 1 ) . By Lemma 4.22 , η ( f 1 2 ) joins to either η ( f 1 1 ) or η ( f 1 4 ) . If it joins to η ( f 1 1 ) , then, as in the previous argumen ts, η ( f 1 4 ) joins to η ( f 2 3 ) or η ( f 2 4 ) , forcing G to contain (5 , 2) or (6 , 1) as a minor. If it joins to η ( f 1 4 ) , then, again as b efore, η ( f 1 1 ) joins to η ( f 1 4 f 1 ) − η ( f 1 ) , forcing G to contain (6 , 2) as a minor. In either case we obtain a con tradiction. □ Lemma 4.28. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 nor any gr aph obtaine d fr om F 4 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( F 4 ) of 20 O.-H.S. LO F 4 , and that η ( f i 2 f 3 − i 3 ) , with i ∈ [2] , c ontains an internal vertex v . Then v is adjac ent to η ( f i 3 ) and either η ( f i 1 ) or η ( f i 4 ) , and has de gr e e four in G . Pr o of. It is clear that v has at least t wo neighbors outside η ( f i 2 f 3 − i 3 ) . Moreov er, by (4 , 1) , (4 , 3) , Lemma 4.18 , and Lemma 4.19 , any neigh b or of v outside η ( f i 2 f 3 − i 3 ) must b e one of η ( f i 1 ) , η ( f i 3 ) , or η ( f i 4 ) . By (5 , 3) , v cannot join to b oth η ( f i 1 ) and η ( f i 4 ) . This establishes the lemma. □ Lemma 4.29. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 nor any gr aph obtaine d fr om F 4 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( F 4 ) of F 4 , and η ( f i 2 ) is adjac ent to η ( f i j ) with i ∈ [ 2] and j ∈ { 1 , 4 } . Then η ( f i 5 − j ) is not adjac ent to any vertex outside the close d domain of f i 5 − j exc ept for η ( f i j ) , η ( f 3 − i 3 ) , and η ( f 3 − i 5 − j ) . Pr o of. By Lemmas 4.19 , 4.21 , 4.23 , 4.27 , and 4.28 , any internal vertex adjacent to η ( f i 5 − j ) must lie in some segment η ( e ) with e ∈ { f i f i j , f i f i 2 , f i j f 3 − i 5 − j , f i 2 f 3 − i 3 } and must b e adjacent to η ( f i 3 ) . How ev er, since η ( f i 2 ) η ( f i j ) ∈ E ( G ) , it follows from Lemma 4.20 , (8 , 2) , and (5 , 3) that no in ternal v ertex is adjacen t to η ( f i 5 − j ) . By Lemmas 4.18 and 4.22 , and by (4 , 3) , any neighbor of η ( f i 5 − j ) outside the closed domain of f i 5 − j m ust b e η ( f i j ) , η ( f 3 − i 3 ) , or η ( f 3 − i 5 − j ) . □ Prop osition 4.30. L et G b e a 4 -c onne cte d gr aph that c ontains neither K 3 , 4 nor any gr aph obtaine d fr om F 4 by splitting a vertex as a minor. Supp ose G c ontains a sp anning JT-sub division η ( F 4 ) of F 4 . Then one of the fol lowing holds: • G c ontains D s 1 ,s 2 , with s 1 , s 2 ≥ 1 and s 1 + s 2 = | V ( G ) | − 8 , as a sp anning sub gr aph such that every vertex on either spine has de gr e e four in G . • G c ontains E s , with s = | V ( G ) | − 9 ≥ 1 , as a sp anning sub gr aph such that every vertex on the spine, as wel l as e ach of ε 3 1 , ε 3 3 , ε 4 , has de gr e e four in G . • G c ontains F as a sp anning sub gr aph such that the vertic es φ 1 , φ 1 1 , φ 1 2 , φ 1 4 , φ 2 , φ 2 1 , φ 2 2 , φ 2 4 have de gr e e four in G . Pr o of. Recall that for i ∈ [2] , η ( f i ) has degree four and η ( f i ) η ( f i 3 ) ∈ E ( G ) by Lemma 4.20 , and η ( f i 2 ) has degree four and joins to exactly one of η ( f i 1 ) and η ( f i 4 ) b y Lemma 4.22 . So it suffices to consider the cases where η ( f 1 2 ) η ( f 1 1 ) , η ( f 2 2 ) η ( f 2 1 ) ∈ E ( G ) and where η ( f 1 2 ) η ( f 1 1 ) , η ( f 2 2 ) η ( f 2 4 ) ∈ E ( G ) . Case 1. η ( f 1 2 ) η ( f 1 1 ) , η ( f 2 2 ) η ( f 2 1 ) ∈ E ( G ) . It follows from Lemmas 4.21 , 4.23 , 4.27 , 4.28 , and 4.29 that, for each i ∈ [2] , η ( f i f i 1 ) contains no internal v ertex, and every internal vertex in η ( f i f i 2 ) , η ( f i f i 4 ) , η ( f i 2 f 3 − i 3 ) , or η ( f i 4 f 3 − i 1 ) has degree four and joins to η ( f i 1 ) and η ( f i 3 ) . This determines the neighborho o ds of all internal v ertices, since ev ery segmen t not mentioned ab ov e contains no in ternal v ertex b y Lemma 4.19 . By Lemmas 4.18 , 4.22 , and 4.29 , and by (4 , 3) and (6 , 4) , it follows that for each i ∈ [2] any neigh b or of η ( f i 4 ) outside the closed domain of f i 4 m ust b e η ( f i 1 ) or η ( f 3 − i 3 ) . Moreo v er, by (6 , 3) one readily deduces that η ( f 1 4 ) and η ( f 2 4 ) hav e degree four, and that either η ( f 1 4 ) η ( f 1 1 ) , η ( f 2 4 ) η ( f 2 1 ) ∈ E ( G ) or η ( f 1 4 ) η ( f 2 3 ) , η ( f 2 4 ) η ( f 1 3 ) ∈ E ( G ) . Case 1.1. η ( f 1 4 ) η ( f 2 3 ) , η ( f 2 4 ) η ( f 2 3 ) ∈ E ( G ) . Then there is no internal vertex; otherwise G w ould con tain (6 , 3) or (5 , 2) as a minor. Hence | V ( G ) | = | V ( F 4 ) | . In fact, F is isomorphic to the subgraph of G obtained from η ( F 4 ) by adding the edges η ( f 1 f 1 3 ) , η ( f 1 1 f 1 2 ) , η ( f 1 4 f 2 3 ) , η ( f 2 f 2 3 ) , η ( f 2 1 f 2 2 ) , and η ( f 2 4 f 1 3 ) , where φ 1 , φ 1 1 , φ 1 2 , φ 1 3 , φ 1 4 , φ 2 , φ 2 1 , φ 2 2 , φ 2 3 , φ 2 4 corresp ond to η ( f 1 ) , η ( f 1 4 ) , η ( f 1 2 ) , η ( f 1 3 ) , η ( f 1 1 ) , η ( f 2 ) , η ( f 2 4 ) , η ( f 2 2 ) , η ( f 2 3 ) , and η ( f 2 1 ) , resp ectiv ely . By this corresp ondence, the v ertices φ 1 , φ 1 1 , φ 1 2 , φ 2 , φ 2 1 , and φ 2 2 , corresp onding to η ( f 1 ) , η ( f 1 4 ) , η ( f 1 2 ) , η ( f 2 ) , η ( f 2 4 ) , and η ( f 2 2 ) , eac h hav e degree four. F urthermore, by (5 , 1) and (5 , 2) , the vertices φ 1 4 and φ 2 4 , corresponding to η ( f 1 1 ) and η ( f 2 1 ) , also hav e degree four. GRAPHS WITH NO K 3 , 4 MINOR 21 Case 1.2. η ( f 1 4 ) η ( f 1 1 ) , η ( f 2 4 ) η ( f 2 1 ) ∈ E ( G ) . It follo ws that D s 1 ,s 2 , with s 1 , s 2 ≥ 1 and s 1 + s 2 = | V ( G ) | − 8 , is isomorphic to the subgraph of G obtained from η ( F 4 ) by adding the edges η ( f 1 f 1 3 ) , η ( f 1 1 f 1 2 ) , η ( f 1 1 f 1 4 ) , η ( f 2 f 2 3 ) , η ( f 2 1 f 2 2 ) , and η ( f 2 1 f 2 4 ) , together with all edges incident with the in ternal v ertices of η ( F 4 ) . The v ertices δ 1 1 , δ 1 2 , δ 1 3 , δ 1 4 , δ 2 1 , δ 2 2 , δ 2 3 , and δ 2 4 corresp ond to η ( f 1 1 ) , the neigh b or of η ( f 2 3 ) in η ( f 1 2 f 2 3 ) , η ( f 1 3 ) , the neigh b or of η ( f 2 1 ) in η ( f 1 4 f 2 1 ) , the neigh b or of η ( f 1 1 ) in η ( f 2 4 f 1 1 ) , η ( f 2 3 ) , the neigh b or of η ( f 1 3 ) in η ( f 2 2 f 1 3 ) , and η ( f 2 1 ) , resp ectiv ely . Moreov er, for each i ∈ [2] , the union of η ( f 3 − i 3 f i 2 ) − η ( f 3 − i 3 ) , η ( f i 2 f i ) , η ( f i f i 4 ) , and η ( f i 4 f 3 − i 1 ) − η ( f 3 − i 1 ) forms the spine whose end-vertices are the neighbor of η ( f 3 − i 3 ) in η ( f i 2 f 3 − i 3 ) and the neighbor of η ( f 3 − i 1 ) in η ( f i 4 f 3 − i 1 ) . By Lemmas 4.20 , 4.21 , 4.22 , 4.23 , 4.27 , and 4.28 , together with the ab ov e discussion, every v ertex on either spine has degree four. Case 2. η ( f 1 2 ) η ( f 1 1 ) , η ( f 2 2 ) η ( f 2 4 ) ∈ E ( G ) . By Lemma 4.29 , (5 , 1) , (5 , 2) , (6 , 1) , and (6 , 5) , either η ( f 1 4 ) η ( f 1 1 ) ∈ E ( G ) or η ( f 2 1 ) η ( f 2 4 ) ∈ E ( G ) . Without loss of generalit y , assume η ( f 1 4 ) η ( f 1 1 ) ∈ E ( G ) and η ( f 2 1 ) η ( f 2 4 ) / ∈ E ( G ) . So η ( f 1 4 ) has degree four. By Lemmas 4.19 , 4.21 , 4.23 , 4.27 , 4.28 , and 4.29 , and by (6 , 5) , every in ternal vertex m ust lie in η ( f 2 3 f 1 2 ) , η ( f 1 2 f 1 ) , η ( f 1 f 1 4 ) , or η ( f 1 4 f 2 1 ) , and b e adjacent to η ( f 1 1 ) and η ( f 1 3 ) . F urthermore, by Lemma 4.29 and the assumption that η ( f 2 1 ) η ( f 2 4 ) / ∈ E ( G ) , the v ertex η ( f 2 1 ) has no neigh b ors among the internal vertices and is adjacent to at least one of η ( f 1 1 ) and η ( f 1 3 ) . Case 2.1. η ( f 2 1 ) η ( f 1 1 ) ∈ E ( G ) . Then E s , where s = | V ( G ) | − 9 , is isomorphic to the graph obtained from η ( F 4 ) by adding the edges η ( f 1 ) η ( f 1 3 ) , η ( f 2 ) η ( f 2 3 ) , η ( f 1 2 ) η ( f 1 1 ) , η ( f 2 2 ) η ( f 2 4 ) , η ( f 1 4 ) η ( f 1 1 ) , and η ( f 2 1 ) η ( f 1 1 ) , together with all edges incident with the internal vertices of η ( F 4 ) . More precisely , the vertices ε 0 , ε 1 1 , ε 1 2 , ε 1 3 , ε 2 , ε 3 1 , ε 3 2 , ε 3 3 , and ε 4 corresp ond to the neigh b or of η ( f 2 1 ) in η ( f 2 1 f 1 4 ) , η ( f 1 1 ) , the neighbor of η ( f 2 3 ) in η ( f 2 3 f 1 2 ) , η ( f 1 3 ) , η ( f 2 1 ) , η ( f 2 4 ) , η ( f 2 3 ) , η ( f 2 2 ) , and η ( f 2 ) , resp ectively , and the union of η ( f 2 3 f 1 2 ) − η ( f 2 3 ) , η ( f 1 2 f 1 ) , η ( f 1 f 1 4 ) , and η ( f 1 4 f 2 1 ) − η ( f 2 1 ) forms the spine, whose end-v ertices are the neighbor of η ( f 2 1 ) in η ( f 2 1 f 1 4 ) and the neighbor of η ( f 2 3 ) in η ( f 2 3 f 1 2 ) . By Lemmas 4.20 , 4.21 , 4.22 , 4.23 , 4.27 , and 4.28 , together with the observ ation that η ( f 1 4 ) has degree four, every vertex in the spine has degree four. Moreov er, by Lemmas 4.20 and 4.22 , the v ertices ε 4 and ε 3 3 , corresp onding to η ( f 2 ) and η ( f 2 2 ) , also ha v e degree four. Finally , since η ( f 2 4 ) is adjacen t to neither an y vertex on the spine nor to η ( f 2 1 ) , it follows from (6 , 3) that ε 3 1 , corresp onding to η ( f 2 4 ) , has degree four. Case 2.2. η ( f 2 1 ) η ( f 1 3 ) ∈ E ( G ) . Observ e that the graph obtained from η ( F 4 ) by adding the edges η ( f 1 ) η ( f 1 3 ) , η ( f 2 ) η ( f 2 3 ) , η ( f 1 2 ) η ( f 1 1 ) , η ( f 2 2 ) η ( f 2 4 ) , and η ( f 1 4 ) η ( f 1 1 ) , together with all edges incident to internal vertices, admits an automorphism that sw aps η ( f 1 1 ) and η ( f 1 3 ) , sw aps η ( f 2 2 ) and η ( f 2 4 ) , and fixes all other v ertices. Consequen tly , w e may pro ceed exactly as in Case 2.1, except that η ( f 1 3 ) and η ( f 2 2 ) now pla y the roles of η ( f 1 1 ) and η ( f 2 4 ) , respectively . □ 4.2.4. Pr o of of The or em 4.9 . In this section we complete the characterization of 4 -connected non- pro jectiv e-planar graphs with no K 3 , 4 minor b y sho wing that they are precisely the oloidal graphs. As in the previous sections, we assem ble in T able 3 a family of graphs, adopting the same conv en tions as before. Pr o of of The or em 4.9 . The sufficiency follo ws from Prop osition 4.8 . It therefore remains to prov e the necessit y . Let G b e a 4 -connected non-pro jectiv e-planar graph that contains no K 3 , 4 minor. 22 O.-H.S. LO T able 3. A collection of graphs, eac h con taining K 3 , 4 as a minor. 1 2 3 4 5 6 9 It follows from Prop ositions 4.3 , 4.16 , and 4.30 that ev ery 4 -connected non-pro jective-planar graph with no K 3 , 4 minor falls into one of the six cases listed b elow. In each case we show that G is oloidal. Case 1. G contains D 0 , 0 as a spanning subgraph. If δ 1 i δ 2 j ∈ E ( G ) \ E ( D 0 , 0 ) for some distinct i, j ∈ [4] , then, by (9 , 1) an d (9 , 2) , no edge in E ( G ) \ E ( D 0 , 0 ) is inciden t with δ 2 i or δ 1 j . Consequently , there exist i ∈ [2] and j ∈ [4] suc h that E ( G ) \ E ( D 0 , 0 ) ⊆ { δ i j δ 3 − i k : k ∈ [ 4] \ { j }} , or there exist distinct i, j ∈ [4] suc h that E ( G ) \ E ( D 0 , 0 ) ⊆ { δ 1 i δ 2 k , δ 1 j δ 2 k : k ∈ [ 4] \ { i, j }} . By the symmetry of D 0 , 0 , we can conclude that G is isomorphic to D 0 , 0 + A , where A ⊆ { δ 1 1 δ 2 2 , δ 1 1 δ 2 3 , δ 1 1 δ 2 4 } or A ⊆ { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } . Thus, G is oloidal. Case 2. G contains D 0 , 1 as a spanning subgraph suc h that eac h of δ 1 2 , δ 1 4 , and σ 2 1 has degree four in G . Using the arguments from Case 1, together with the assumption that eac h of δ 1 2 , δ 1 4 , and σ 2 1 has degree four in G , one can readily show that there exists i ∈ { 1 , 3 } such that E ( G ) \ E ( D 0 , 1 ) ⊆ { δ 1 i δ 2 j : j ∈ [4] \ { i }} , or that E ( G ) \ E ( D 0 , 1 ) ⊆ { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } . By the symmetry of D 0 , 1 , we can conclude that G is isomorphic to D 0 , 1 + A , where A ⊆ { δ 1 1 δ 2 2 , δ 1 1 δ 2 3 , δ 1 1 δ 2 4 } or A ⊆ { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } . Thus, G is oloidal. Case 3. G contains D s 1 ,s 2 , with s 1 + s 2 ≥ 2 , as a spanning subgraph such that every v ertex on either spine has degree four in G . As every vertex on either spine has degree four in G , we ha v e E ( G ) \ E ( D s 1 ,s 2 ) ⊆ { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } . W e conclude that G is isomorphic to D s 1 ,s 2 + A , where A ⊆ { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } , and hence oloidal. Case 4. G contains E 0 as a spanning subgraph such that eac h of ε 0 and ε 4 has degree four in G . It follo ws from (9 , 3) and (9 , 4) that each of ε 1 2 and ε 3 1 has degree four in G . Moreov er, again b y (9 , 3) and (9 , 4) , if ε 2 is adjacen t to one of ε 1 3 and ε 3 3 , then the other has degree four in G . On the other hand, it follows from (9 , 5) that G do es not contain b oth edges ε 1 1 ε 3 3 and ε 1 3 ε 3 2 . Therefore, we conclude that E ( G ) \ E ( E 0 ) is a subset of { ε 1 1 ε 3 2 , ε 1 1 ε 3 3 , ε 3 3 ε 2 } or { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } . It is straightforw ard to verify that E 0 + { ε 1 1 ε 3 2 , ε 1 1 ε 3 3 , ε 3 3 ε 2 } ∼ = E 0 + { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } . Hence G is isomorphic to E 0 + A for some A ⊆ { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } . Thus, G is oloidal. Case 5. G contains E s , with s ≥ 1 , as a spanning subgraph such that every vertex on the spine, as w ell as each of ε 3 1 , ε 3 3 , ε 4 , has degree four in G . By the assumption that ev ery vertex on the spine, as well as eac h of ε 3 1 , ε 3 3 , and ε 4 , has degree four in G , we conclude that E ( G ) \ E ( E s ) ⊆ { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } . Therefore, G is isomorphic to E s + A for some A ⊆ { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } , and is oloidal. GRAPHS WITH NO K 3 , 4 MINOR 23 Case 6. G contains F as a spanning subgraph suc h that the v ertices φ 1 , φ 1 1 , φ 1 2 , φ 1 4 , φ 2 , φ 2 1 , φ 2 2 , φ 2 4 ha v e degree four in G . It is immediate that E ( G ) \ E ( F ) ⊆ { φ 1 3 φ 2 3 } . Hence, G is isomorphic to F + A for some A ⊆ { φ 1 3 φ 2 3 } , and therefore G is oloidal. □ 4.3. Pro of of Theorem 1.2 . W e now pro ceed to merge the results of the pro jective-planar and non-pro jectiv e-planar cases, thereb y establishing the pro of of our principal characterization theo- rem. Pr o of of The or em 1.2 . By Theorems 4.1 and 4.9 , a graph G is 4 -connected and has no K 3 , 4 minor if and only if, dep ending on whether G is planar, non-planar but pro jective-planar, or non-pro jective- planar, it is resp ectiv ely a 4 -connected planar graph, a 4 -connected non-planar subgraph of a patch graph or isomorphic to K 6 , or an oloidal graph. It remains to note that the class of 4 -connected subgraphs of patc h graphs coincides with the class of 4 -connected subgraphs of reduced patc h graphs. This completes the pro of. □ 5. Genera ting all graphs with no K 3 , 4 minor The 4 -connected graphs with no K 3 , 4 minor ha ve b een characterized in Theorem 1.2 . In this section, we sho w that general graphs with no K 3 , 4 minor can b e constructed from the 4 -connected ones. W e first sho w ho w 3 -connected graphs with no K 3 , 4 minor can b e reduced to the 4 -connected case. Then all graphs with no K 3 , 4 minor can b e generated from the 3 -connected ones via the standard clique sum construction. W e b egin with th e following observ ation. Lemma 5.1. L et G b e a 3 -c onne cte d gr aph with a 3 -cut S , and let v 1 , v 2 ∈ S b e two non-adjac ent vertic es. If G has no K 3 , 4 minor, then G + v 1 v 2 also has no K 3 , 4 minor. Pr o of. Supp ose, to the con trary , that G + v 1 v 2 has a K 3 , 4 minor. Let µ b e a spanning mo del of K 3 , 4 in G + v 1 v 2 . Denote the v ertex set of K 3 , 4 b y X ∪ Y , where | X | = 3 and | Y | = 4 , so that t wo vertices are adjacen t if and only if they b elong to different partite sets. Since G has no K 3 , 4 minor, there exists w ∈ V ( K 3 , 4 ) such that { v 1 , v 2 } ⊆ µ ( w ) , or, without loss of generalit y , there exist vertices x ∈ X and y ∈ Y such that v 1 ∈ µ ( x ) and v 2 ∈ µ ( y ) . Case 1. There exists w ∈ V ( K 3 , 4 ) suc h that { v 1 , v 2 } ⊆ µ ( w ) . W e claim that there exists a comp onent A of G − S such that, for ev ery vertex v ∈ V ( K 3 , 4 ) with µ ( v ) ∩ S = ∅ , we hav e µ ( v ) ⊆ V ( A ) . Supp ose otherwise. Then there exist t w o distinct comp onents A and A ′ of G − S and vertices v , v ′ ∈ V ( K 3 , 4 ) such that µ ( v ) ⊆ V ( A ) and µ ( v ′ ) ⊆ V ( A ′ ) , with neither in tersecting S . Consequently , v and v ′ b elong to the same partite set, either b oth in X or b oth in Y . This is imp ossible, since A and A ′ are separated by S , while { v 1 , v 2 } ⊆ µ ( w ) for some w ∈ V ( K 3 , 4 ) . This pro v es the claim. W e now define a spanning mo del µ ′ of K 3 , 4 in G . Cho ose an arbitrary comp onen t ˜ A of G − S distinct from A . F or ev ery vertex v ∈ V ( K 3 , 4 ) with µ ( v ) ⊆ V ( A ) , set µ ′ ( v ) := µ ( v ) . Let z b e the v ertex of K 3 , 4 suc h that µ ( z ) contains the third v ertex of S . Set µ ′ ( w ) := µ ( w ) ∪ V ( ˜ A ) , and, if z  = w , set µ ′ ( z ) := µ ( z ) \ V ( ˜ A ) . This yields a spanning mo del of K 3 , 4 in G , con tradicting our assumption. Case 2. There exist vertices x ∈ X and y ∈ Y such that v 1 ∈ µ ( x ) and v 2 ∈ µ ( y ) . W e pro ceed similarly to Case 1. By the same argument, there exists a comp onent A of G − S suc h that µ ( v ) ⊆ V ( A ) for every vertex v ∈ V ( K 3 , 4 ) with µ ( v ) ∩ S = ∅ . W e again deriv e a contradiction b y defining a spanning mo del µ ′ of K 3 , 4 in G . Cho ose an arbitrary comp onent ˜ A of G − S with ˜ A  = A . F or every v ertex v ∈ V ( K 3 , 4 ) with µ ( v ) ⊆ V ( A ) , 24 O.-H.S. LO set µ ′ ( v ) := µ ( v ) . Let z ∈ V ( K 3 , 4 ) b e suc h that µ ( z ) con tains the third vertex of S . If x  = z and z ∈ X , set µ ′ ( x ) := µ ( x ) \ V ( ˜ A ) , µ ′ ( y ) := µ ( y ) ∪ V ( ˜ A ) , and µ ′ ( z ) := µ ( z ) \ V ( ˜ A ) . If x = z , set µ ′ ( x ) := µ ( x ) ∪ V ( ˜ A ) and µ ′ ( y ) := µ ( y ) \ V ( ˜ A ) . The cases y  = z with z ∈ Y and y = z are treated analogously . □ A k -clique of G is a subgraph of G on k vertices in which every pair of v ertices is adjacen t. W e sa y that a 3 -connected graph G is clamp e d if every 3 -cut of G induces a 3 -clique. Lemma 5.1 immediately yields the following corollary . Corollary 5.2. Every 3 -c onne cte d gr aph with no K 3 , 4 minor is a sp anning sub gr aph of a clamp e d 3 -c onne cte d gr aph with no K 3 , 4 minor. W e follow the approach of [ 21 ], with a sligh t mo dification, to decomp ose a clamp ed 3 -connected graph into graphs that are either 4 -connected or isomorphic to K 4 . W e only give an outline of the framew ork and refer to [ 21 ] for further details. The decomp osition is based on the following observ ation. Let G b e a clamp ed 3 -connected graph. If G is neither 4 -connected nor isomorphic to K 4 , then G has a 3 -cut S , and there exist subgraphs G 1 and G 2 of G such that V ( G 1 ) ∩ V ( G 2 ) = S , E ( G 1 ) ∪ E ( G 2 ) = E ( G ) , and E ( G 1 ) ∩ E ( G 2 ) = E ( G [ S ]) . Since an y 3 -cut of G 1 or G 2 is also a 3 -cut of G , b oth G 1 and G 2 are clamped 3 -connected. Therefore, an y clamped 3 -connected graph that is neither 4 -connected nor isomorphic to K 4 can b e decomposed into tw o smaller clamp ed 3 -connected graphs. This motiv ates the follo wing structure. Let G b e the family of subgraphs of G that are either edge-maximal 4 -connected, or isomorphic to K 4 and not contained in an y edge-maximal 4 -connected subgraph of G . Let S b e the family of 3 -cliques in G that are induced by 3 -cuts. Define T to b e the graph with v ertex set G ∪ S such that every edge has one end-v ertex in G and the other in S , and where H ∈ G is adjacent to ∆ ∈ S if and only if H con tains ∆ as a subgraph of G . Then T is a tree, and we call ( T , G , S ) the clamp e d tr e e de c omp osition of G . The graph G can b e reconstructed by first taking the disjoint union of the graphs in G and then, for eac h ∆ ∈ S , iden tifying the copies of ∆ in those graphs from G that are adjacen t to ∆ in T . F or H ∈ G , define the closur e of H , denoted by b H , to b e the graph obtained from H b y adding, for each ∆ ∈ S that is adjacen t to H in T , a new v ertex v H ∆ adjacen t to the three vertices of ∆ . Define b G := { b H : H ∈ G } . Since ev ery cycle of length three in a 4 -connected planar graph is a facial cycle, it follows that for an y H ∈ G , the graph H is planar if and only if its closure b H is planar. W e need the follo wing tw o lemmas. The first is an immediate consequence of [ 31 , (2.4)], and the second follo ws from [ 24 , Prop osition 3.2]. Lemma 5.3 ([ 31 ]) . L et G b e a 3 -c onne cte d non-planar gr aph with a vertex v of de gr e e thr e e. Then G c ontains a sub division of K 3 , 3 that c ontains v and in which v has de gr e e thr e e. Lemma 5.4 ([ 24 ]) . L et H 1 b e a gr aph c ontaining a 3 -clique ∆ 1 , and let H 2 b e a gr aph c ontaining a 3 -clique ∆ 2 . Supp ose that the gr aph obtaine d fr om H 1 by adding a new vertex adjac ent to al l vertic es of ∆ 1 is planar, and that the gr aph obtaine d fr om H 2 by adding a new vertex adjac ent to al l vertic es of ∆ 2 has no K 3 , 4 minor. Then the gr aph obtaine d fr om the disjoint union of H 1 and H 2 by identifying ∆ 1 with ∆ 2 c ontains no K 3 , 4 minor. The following prop osition c haracterizes the clamp ed 3 -connected graphs with no K 3 , 4 minor in terms of their clamp ed tree decomp ositions. Prop osition 5.5. L et G b e a clamp e d 3 -c onne cte d gr aph, and let ( T , G , S ) b e its clamp e d tr e e de c omp osition. Then G c ontains no K 3 , 4 minor if and only if every closur e b H ∈ b G c ontains no K 3 , 4 minor and one of the fol lowing holds: • Exactly one vertex of S has de gr e e thr e e in T , al l other vertic es of S have de gr e e two in T , and every gr aph in G is planar. GRAPHS WITH NO K 3 , 4 MINOR 25 • Every vertex of S has de gr e e two in T , and at most one gr aph in G is non-planar. Pr o of. W e first pro ve the necessit y . F or every H ∈ G , w e hav e b H ⪯ G , since it is a subgraph of the graph obtained from G by con tracting ev ery comp onen t of G − V ( H ) . Hence, b H contains no K 3 , 4 minor. If some ∆ ∈ S has degree at least four in T , then the union U of the neigh b ors of ∆ in T forms a subgraph of G that contains a K 3 , 4 minor. More precisely , one obtains K 3 , 4 b y con tracting eac h comp onen t of U − V (∆) and deleting the edges of ∆ . Therefore, ev ery vertex of S has degree t w o or three in T . If there are distinct vertices ∆ 1 , ∆ 2 ∈ S of degree three in T , then the union of the neigh b ors of ∆ 1 and ∆ 2 together with the elements of G lying on the path of T joining ∆ 1 and ∆ 2 con tains a K 3 , 4 minor (as there are three disjoin t paths betw een ∆ 1 and ∆ 2 in G ). Hence, there is at most one v ertex of S of degree three in T . Case 1. Exactly one vertex of S has degree three in T . Let ∆ 1 ∈ S b e this v ertex. Suppose, to the con trary , that there exists a non-planar graph H ∈ G . Let ∆ 2 b e the neighbor of H on the path in T joining ∆ 1 and H . Let e H b e the subgraph of G suc h that H is a subgraph of e H , e H − V (∆ 2 ) is connected, and, sub ject to these conditions, | E ( e H ) | is maximized. Since H is non-planar, the graph obtained from e H by adding a vertex v adjacen t to the vertices of ∆ 2 is also non-planar and hence, by Lemma 5.3 , con tains a sub division η ( K 3 , 3 ) of K 3 , 3 that con tains v and in whic h v has degree three. Since there are three d isjoin t paths in G joining ∆ 1 and ∆ 2 , it follo ws that the union of these three paths, the elements of G con taining ∆ 1 , and η ( K 3 , 3 ) − v contains a K 3 , 4 minor, a contradiction. Therefore, ev ery graph in G is planar. Case 2. No vertex of S has degree three in T . Supp ose, to the contrary , that there exist t w o non-planar graphs H 1 , H 2 ∈ G . F or i ∈ [2] , let ∆ i b e the neigh b or of H i on the path in T joining H 1 and H 2 , and let f H i b e the subgraph of G suc h that H i is a subgraph of f H i , f H i − V (∆ i ) is connected, and, sub ject to these conditions, | E ( f H i ) | is maximized. As b efore, Lemma 5.3 implies that the graph obtained from f H i b y adding a vertex v i adjacen t to the vertices of ∆ i con tains a subdivision η i ( K 3 , 3 ) of K 3 , 3 that contains v i and in which v i has degree three. As there exist three disjoint paths in G joining ∆ 1 and ∆ 2 , it follows that the union of η 1 ( K 3 , 3 ) − v 1 , η 2 ( K 3 , 3 ) − v 2 , and these three paths con tains a K 3 , 4 minor, a contradiction. Hence, at most one graph in G is non-planar. W e now pro ve the sufficiency . By assumption, for eac h H ∈ G , the closure b H con tains no K 3 , 4 minor. Moreo v er, b H is planar if and only if H is planar. Recall that G can b e constructed b y identifying 3 -cliques in S among the graphs H ∈ G . Equiv alen tly , for each 3 -clique ∆ ∈ S , one identifies the copies of ∆ in the closures b H ∈ b G that con tain ∆ , and then remov es from each such b H the vertex adjacent to ∆ that was added in forming the closure (denoted v H ∆ in the definition). After all iden tifications are p erformed and the added v ertices are remo v ed, the closures are merged in to the graph G . This reform ulation is con venien t for applying Lemma 5.4 . It remains to show that, in each of the following tw o cases, the constructed graph G has no K 3 , 4 minor. Case 1. At most one graph in G is non-planar and ev ery v ertex of S has degree tw o in T . Let H 1 ∈ G b e such that ev ery other graph in G has a planar closure. Order the graphs in G as H 1 , H 2 , . . . , H t , where t = |G | , so that for each i ∈ [ t ] the union of H 1 , . . . , H i forms a clamp ed 26 O.-H.S. LO 3 -connected subgraph of G . Starting from c H 1 , w e successively merge c H 2 , . . . , c H t in this order via 3 -clique identifications, deleting t wo degree three vertices at eac h step (since each vertex of S has degree t wo in T ). As c H 1 con tains no K 3 , 4 minor and c H 2 , . . . , c H t are planar, Lemma 5.4 implies that the resulting graph G con tains no K 3 , 4 minor. Case 2. Every graph in G is planar, exactly one v ertex ∆ ∈ S has degree three in T , and all other v ertices of S hav e degree tw o in T . Let H 1 and H 2 b e tw o neighbors of ∆ in T , and let v 1 and v 2 b e the degree three v ertices of c H 1 and c H 2 , resp ectively , that w ere added in forming the closures and are adjacen t to the v ertices of ∆ . Let e H b e obtained from the disjoint union of c H 1 and c H 2 b y identifying the copies of ∆ and deleting v 2 . W e claim that e H contains no K 3 , 4 minor. Supp ose otherwise, and let µ b e a spanning model of K 3 , 4 in e H . Denote the v ertex set of K 3 , 4 b y X ∪ Y , where | X | = 3 and | Y | = 4 , and where t w o v ertices are adjacent if and only if they b elong to different partite sets X and Y . Since e H − v 1 is planar, w e hav e v 1 ∈ µ ( x ) for some x ∈ X . As v 1 has degree three in e H while x has degree four in K 3 , 4 , at least one neighbor of v 1 lies in µ ( x ) . Let u be suc h a neighb or. As ev ery neighbor of v 1 other than u is also a neigh b or of u , it follows immediately that e H − v 1 has a K 3 , 4 minor, whic h is imp ossible. No w, as in Case 1, w e may successively merge e H with the graphs in b G other than c H 1 and c H 2 to obtain G . By Lemma 5.4 , the resulting graph con tains no K 3 , 4 minor, since e H has no K 3 , 4 minor and all graphs in b G are planar. □ Let H 1 , H 2 b e tw o graphs, each containing a k -clique. By identifying these k -cliques in the disjoin t union of H 1 and H 2 , we obtain a k -clique sum of H 1 and H 2 . The following prop osition is straigh tforw ard (cf. [ 24 ]); we omit the pro of. Prop osition 5.6. L et H 1 , H 2 b e gr aphs e ach c ontaining a k -clique, wher e k ∈ [2] . Then H 1 and H 2 c ontain no K 3 , 4 minor if and only if any k -clique sum of H 1 and H 2 c ontains no K 3 , 4 minor. Prop ositions 5.6 and 5.5 show ho w a connected graph without a K 3 , 4 minor can b e reduced to 4 - connected graphs without a K 3 , 4 minor. T ogether with Theorem 1.2 , this yields a c haracterization of graphs with no K 3 , 4 minor. 6. Applica tions In this section w e derive three consequences of Theorem 1.2 concerning edge density conditions forcing the presence of certain minors, hamiltonian-connectedness, and em b eddability on the torus. In particular, we establish Theorems 1.3 , 1.4 , and 1.5 in Sections 6.1 , 6.2 , and 6.3 , resp ectiv ely . 6.1. Edge densit y and K 3 , 4 minors. A classical problem in graph minor theory is to determine, for a graph class G and a graph H , a low er b ound on the av erage degree (resp ectively , the minimum degree) that forces every graph in G to contain H as a minor. The following theorem is due to Jørgensen [ 12 ], and concerns K 3 , 4 minors in the class of 3 - connected graphs. Theorem 6.1 ([ 12 ]) . Every 3 -c onne cte d gr aph G with at le ast 3 | V ( G ) | − 3 e dges is either isomorphic to K 6 or c ontains K 3 , 4 as a minor. Mor e over, every gr aph with minimum de gr e e at le ast six c ontains K 3 , 4 as a minor. Both statemen ts in Theorem 6.1 are tight. As mentioned (without explicit construction) in [ 12 ], there exist infinitely many 3 -connected graphs G with 3 | V ( G ) | − 4 edges that contain no K 3 , 4 minor. Note also that there are planar graphs with minimum degree exactly fiv e. W e complement Theorem 6.1 with the following result. GRAPHS WITH NO K 3 , 4 MINOR 27 Corollary 6.2. Ther e exist infinitely many 4 -c onne cte d gr aphs G with 3 | V ( G ) | − 4 e dges that c ontain no K 3 , 4 minor. Mor e over, every 4 -c onne cte d non-planar gr aph with minimum de gr e e at le ast five is either isomorphic to K 6 or c ontains K 3 , 4 as a minor. Pr o of. F or the first statemen t, consider G := D s 1 ,s 2 + { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } , with s 1 , s 2 ≥ 0 . Then G has no K 3 , 4 minor and has 3 | V ( G ) | − 4 edges. F or the second statement, we apply Theorem 1.2 . It suffices to observ e that b oth patc h graphs and oloidal graphs are 4 -degenerate. The 4 -degeneracy of patc h graphs follows from Lemma 4.2 , and the same prop erty for oloidal graphs is easily verified. □ W e now pro ceed to the pro of of Theorem 1.3 , using the following theorem of Mader [ 20 ]. Theorem 6.3 ([ 20 ]) . Every gr aph with minimum de gr e e at le ast five c ontains K − 6 or the ic osahe dr on as a minor. Pr o of of The or em 1.3 . By Theorem 6.3 and Corollary 6.2 , it suffices to show that every 4 -connected non-planar graph G containing the icosahedron as a minor also contains K − 6 as a minor. Denote b y H the icosahedron. Then G and H are 4 -connected, with G non-planar and H planar. This p ermits the application of Theorem 3.4 . Theorem 3.4 lists seven p ossible extensions. Since all facial cycles of H ha ve length three, only three of these need consideration. Hence G con tains a minor H ′ obtained from H in one of the follo wing w a ys: • H ′ arises from joining t w o non-cofacial vertices of H . By symmetry , H ′ is isomorphic to one of the tw o graphs depicted in Figure 6 . • H ′ arises from performing a non-planar split on H . In this case, H ′ is isomorphic to the graph in Figure 7 . • There exist facial cycles C 1 and C 2 of H sharing an edge uv , and H ′ is obtained by splitting u along C 1 in to u 1 , u 2 and v along C 2 in to v 1 , v 2 , with u 1 adjacen t to v 1 in the intermediate graph, follo w ed b y joining u 2 to v 2 . Then H ′ con tains the graph of Figure 8 as a spanning subgraph. In eac h of the three cases, H ′ con tains K − 6 as a minor. This concludes the pro of. □ Figure 6. The tw o graphs obtained from the icosahedron b y joining t wo non- cofacial v ertices. 6.2. Hamiltonian-connectedness. The follo wing theorem is due to Kaw arabay ashi and Ozeki [ 15 ]. T ogether with Theorem 1.2 , it yields a direct pro of of Theorem 1.4 , for which w e pro vide only a brief sketc h. 28 O.-H.S. LO Figure 7. The graph obtained from the icosahedron by a non-planar split. u 2 u 1 v 2 v 1 Figure 8. A common spanning subgraph of graphs obtained from the icosahedron b y planar splits along facial cycles C 1 and C 2 sharing an edge uv , with u split in to u 1 , u 2 and v split into v 1 , v 2 , u 1 adjacen t to v 1 , and u 2 joined to v 2 . Theorem 6.4 ([ 15 ]) . Every 4 -c onne cte d pr oje ctive-planar gr aph is hamiltonian-c onne cte d. Sketch of the pr o of of The or em 1.4 . Patc h graphs and K 6 are pro jectiv e-planar. Hence, by Theo- rems 6.4 and 1.2 , it suffices to sho w that the edge-minimal oloidal graphs D s 1 ,s 2 , E s , and F are hamiltonian-connected. The v erification for these cases is routine. □ 6.3. T oroidal em b eddings without K 3 , 4 minors. In what follows, w e sho w that Theorem 1.5 is a consequence of Theorem 1.2 . Pr o of of The or em 1.5 . W e apply Theorem 1.2 . It is well known that K 6 em b eds on the torus. Next, every non-planar patch graph admits a toroidal em b edding. Indeed, each such graph can b e obtained from a planar graph bounded by a facial cycle of length four (corresp onding to the facial w alk of length four in the initial patch graph) b y iden tifying each pair of non-consecutiv e v ertices on that cycle. It then follows immediately that the resulting graph can b e em b edded on the torus. It remains to sho w that every oloidal graph embeds on the torus. Suc h embeddings are sho wn in Figure 9 for the edge-maximal cases, namely D s 1 ,s 2 + { δ 1 1 δ 2 2 , δ 1 1 δ 2 3 , δ 1 1 δ 2 4 } , D s 1 ,s 2 + { δ 1 1 δ 2 2 , δ 2 2 δ 1 3 , δ 1 3 δ 2 4 , δ 2 4 δ 1 1 } , E s + { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } , and F + { φ 1 3 φ 2 3 } . The embedding of E s + { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } is omitted, since E s + { ε 1 1 ε 3 2 , ε 1 3 ε 3 2 , ε 1 3 ε 2 } ∼ = D 0 ,s +1 + { δ 1 1 δ 2 2 , δ 1 1 δ 2 3 , δ 1 1 δ 2 4 } . 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School of Ma thema tical Sciences, Key Labora tor y of Intelligent Computing and Applica tions (Ministr y of Educa tion), Tongji University, Shanghai 200092, China Email addr ess : ohsolomon.lo@gmail.com