aa r X i v : . [ m a t h . C O ] M a r UNAVOIDABLE VERTEX-MINORSIN LARGE PRIME GRAPHS
O-JOUNG KWON AND SANG-IL OUM
Abstract.
A graph is prime (with respect to the split decompo-sition) if its vertex set does not admit a partition p A, B q (called a split ) with | A | , | B | ě A and B induces a complete bipartite graph.We prove that for each n , there exists N such that every primegraph on at least N vertices contains a vertex-minor isomorphicto either a cycle of length n or a graph consisting of two disjointcliques of size n joined by a matching. Introduction
In this paper, all graphs are simple and undirected. We write P n and C n to denote a graph that is a path and a cycle on n vertices,respectively. We aim to find analogues of the following theorems. ‚ (Ramsey’s theorem)For every n , there exists N such that every graph on at least N vertices contains an induced subgraph isomorphic to K n or K n . ‚ (folklore; see Diestel’s book [8, Proposition 9.4.1])For every n , there exists N such that every connected graphon at least N vertices contains an induced subgraph isomorphicto K n , K ,n , or P n . ‚ (folklore; see Diestel’s book [8, Proposition 9.4.2])For every n , there exists N such that every 2 -connected graphon at least N vertices contains a topological minor isomorphicto C n or K ,n . ‚ (Oporowski, Oxley, and Thomas [15]) Date : December 11, 2018.
Key words and phrases. vertex-minor, split decomposition, blocking sequence,prime, generalized ladder.Supported by Basic Science Research Program through the National ResearchFoundation of Korea (NRF) funded by the Ministry of Science, ICT & FuturePlanning (2011-0011653).
For every n , there exists N such that every 3 -connected graphon at least N vertices contains a minor isomorphic to the wheelgraph W n on n vertices or K ,n . ‚ (Ding, Chen [9])For every integer n , there exists N such that every connectedand co-connected graph on at least N vertices contains an in-duced subgraph isomorphic to P n , K s ,n (the graph obtainedfrom K ,n by subdividing one edge once), K ,n z e , or K ,n { e z f z g where t f, g u is a matching in K ,n { e . A graph is co-connected if its complement graph is connected. ‚ (Chun, Ding, Oporowski, and Vertigan [6])For every integer n ě
5, there exists N such that every in-ternally 4-connected graph on at least N vertices contains aparallel minor isomorphic to K n , K ,n ( K ,n with a completegraph on the vertices of degree n ), T F n (the n -partition triplefan with a complete graph on the vertices of degree n ), D n (the n -spoke double wheel), D n (the n -spoke double wheel withaxle), M n (the p n ` q -rung Mobius zigzag ladder), or Z n (the p n q -rung zigzag ladder).These theorems commonly state that every sufficiently large graph hav-ing certain connectivity contains at least one graph in the list of un-avoidable graphs by certain graph containment relation. Moreover ineach theorem, the list of unavoidable graphs is optimal in the sensethat each unavoidable graph in the list has the required connectivity,can be made arbitrary large, and does not contain other unavoidablegraphs in the list.In this paper, we discuss prime graphs as a connectivity requirement.A split of a graph G is a partition p A, B q of the vertex set V p G q havingsubsets A Ď A , B Ď B such that | A | , | B | ě a P A is adjacent to a vertex b P B if and only if a P A and b P B .This concept was first studied by Cunningham [7] in his research onsplit decompositions. We say that a graph is prime if it has no splits.Sometimes we say a graph is prime with respect to split decomposition to distinguish with another notion of primeness with respect to modulardecomposition.Prime graphs play important role in the study of circle graphs (inter-section graphs of chords in a circle) and their recognition algorithms.Bouchet [2], Naji [14], and Gabor, Hsu, and Supowit [11] independentlyshowed that prime circle graphs have a unique chord diagram. This iscomparable to the fact that 3-connected planar graphs have a uniqueplanar embedding. NAVOIDABLE VERTEX-MINORS 3
Figure 1. K a K .The graph containment relation we will mainly discuss is called a vertex-minor . A graph H is a vertex-minor of a graph G if thereexist a sequence v , v , . . . , v n of (not necessarily distinct) vertices anda subset X Ď V p G q such that H “ G ˚ v ˚ v ¨ ¨ ¨ ˚ v n z X , where G ˚ v is an operation called local complementation , to take the complementgraph only in the neighborhood of v . The detailed description will begiven in Section 2.1. Vertex-minors are important in circle graphs; forinstance, Bouchet [5] proved that a graph is a circle graph if and onlyif it has no vertex-minor isomorphic to one of three particular graphs.Prime graphs have been studied with respect to vertex-minors, per-haps because local complementation preserves prime graphs, shown byBouchet [2]. In addition, he showed the following. Theorem 1.1 (Bouchet [2]) . Every prime graph on at least verticesmust contain a vertex-minor isomorphic to C . Here is the main theorem of this paper.
Theorem 7.1.
For every n , there is N such that every prime graph onat least N vertices has a vertex-minor isomorphic to C n or K n a K n . The graph K n a K n is a graph obtained by joining two copies of K n by a matching of size n , see Figure 1. This notation will be explained inSection 2.4. In addition, we show that this list of unavoidable vertex-minors in Theorem 7.1 is optimal, which will be discussed in Section 8.We will heavily use Ramsey’s theorem iteratively and so our bound N is astronomical in terms of n .The proof is splitted into two parts.(1) We first prove that for each n , there exists N such that everyprime graph having an induced path of length N contains avertex-minor isomorphic to C n . (In fact, we prove that N “ r . n s .)(2) Secondly, we prove that for each n , there exists N such thatevery prime graph on at least N vertices contains a vertex-minor isomorphic to P n or K n a K n . O-JOUNG KWON AND SANG-IL OUM x yG xG ˚ x y xG ^ xy Figure 2.
Local complementation and pivot.To prove (1), we actually prove first that every sufficiently large gen-eralized ladder, a certain type of outerplanar graphs, contains C n asa vertex-minor. This will be shown in Section 4. Then, we use thetechnique of blocking sequences developed by Geelen [13] to constructa large generalized ladder in a prime graph having a sufficiently longinduced path, shown in Section 6. Blocking sequences will be discussedand developed in Section 5. The second part (2) is discussed in Sec-tion 7, where we iteratively use Ramsey’s theorem to find a bigger con-figuration called a broom inside a graph. In Section 3, we give similartheorems of this type on vertex-minors with respect to less restrictiveconnectivity requirements.2. Preliminaries
For X Ď V p G q , let δ G p X q be the set of edges having one end in X and another end in V p G qz X . Let N G p x q be the set of the neighbors ofa vertex x in G . For X Ď V p G q , let G r X s be the induced subgraphof G on the vertex set X . For two disjoint subsets S, T of V p G q ,let G r S, T s “ G r S Y T szp E p G r S sq Y E p G r T sqq . Clearly, G r S, T s is abipartite graph with the bipartition p S, T q .2.1. Vertex-minors.
The local complementation of a graph G at avertex v is an operation to replace the subgraph of G induced by theneighborhood of v by its complement graph. In other words, to applylocal complementation at v for every pair x , y of neighbors of v , we flipthe pair x , y , where flipping means that we delete the edge if it existsand add it otherwise. We write G ˚ v to denote the graph obtained from G by applying local complementation of G at v . Two graphs are locallyequivalent if one is obtained from another by applying a sequence oflocal complementations. A graph H is a vertex-minor of G if H is aninduced subgraph of a graph locally equivalent to G .For an edge xy of a graph G , a graph obtained by pivoting an edge xy of G is defined as G ^ xy “ G ˚ x ˚ y ˚ x . Here is a direct way to see NAVOIDABLE VERTEX-MINORS 5 G ^ xy ; there are 3 kinds of neighbors of x or y ; some are adjacent toboth, some are adjacent to only x , others are adjacent to only y . Weflip the adjacency between all pairs of neighbors of x or y of distinctkinds and then swap the two vertices x and y . Two graphs are pivot-equivalent if one is obtained from another by a sequence of pivots.Thus, pivot-equivalent graphs are locally equivalent. See Figure 2 foran example of these operations.The following lemma by Bouchet provides a key tool to investigatevertex-minors. His proof is based on isotropic systems, which are somelinear algebraic objects corresponding to the equivalence classes ofgraphs with respect to local equivalence, introduced by Bouchet [1].A direct proof is given by Geelen and Oum [12]. Lemma 2.1 (Bouchet [3]; see Geelen and Oum [12]) . Let H be avertex-minor of G and let v P V p G qz V p H q . Then H is a vertex-minorof G z v , G ˚ v z v , or G ^ vw z v for a neighbor w of v . The choice of a neighbor w in Lemma 2.1 does not matter, becauseif x is adjacent to y and z , then G ^ xy “ p G ^ xz q ^ yz (see [16]).2.2. Cut-rank function.
Let A p G q be the adjacency matrix of G overthe binary field. For an X ˆ Y matrix A , if X Ď X and Y Ď Y , thenwe write A r X , Y s to denote the submatrix of A obtained by takingrows in X and columns in Y .We define ρ ˚ G p X, Y q “ rank A p G qr X, Y s . This function satisfies thefollowing submodular inequality (see Oum and Seymour [18]): Lemma 2.2 (See Oum and Seymour [18]) . For all
A, B, A , B Ď V p G q , ρ ˚ G p A, B q ` ρ ˚ G p A , B q ě ρ ˚ G p A X A , B Y B q ` ρ ˚ G p A Y A , B X B q . The cut-rank function ρ G of a graph G is defined as ρ G p X q “ ρ ˚ G p X, V p G qz X q “ rank A p G qr X, V p G qz X s . By Lemma 2.2, we have the submodular inequality: ρ G p A q ` ρ G p B q ě ρ G p A X B q ` ρ G p A Y B q for all A, B Ď V p G q .The cut-rank function is invariant under taking local complementa-tion, which makes it useful for us. Lemma 2.3 (Bouchet [4]; See Oum [16]) . If G and H are locally equiv-alent, then ρ G p X q “ ρ H p X q for all X Ď V p G q . O-JOUNG KWON AND SANG-IL OUM
Lemma 2.4 (Oum [16, Lemma 4.4]) . Let G be a graph and v P V p G q .Suppose that p X , X q , p Y , Y q are partitions of V p G qzt v u . Then wehave ρ G z v p X q ` ρ G ˚ v z v p Y q ě ρ G p X X Y q ` ρ G p X X Y q ´ . Similarly if w is a neighbor of v , then ρ G z v p X q ` ρ G ^ vw z v p Y q ě ρ G p X X Y q ` ρ G p X X Y q ´ . Lemma 2.4 is equivalent to the following lemma, which we will usein the proof of Proposition 5.3.
Lemma 2.5.
Let G be a graph and v P V p G q . Suppose that X , X , Y , Y are subsets of V p G qzt v u such that X Y X “ Y Y Y and X X X “ Y X Y “ H . Then ρ ˚ G p X , X q ` ρ ˚ G ˚ v p Y , Y qě ρ ˚ G p X X Y , X Y Y Y t v uq ` ρ ˚ G p X Y Y Y t v u , X X Y q ´ . Similarly if w P X Y X is a neighbor of v , then ρ ˚ G p X , X q ` ρ ˚ G ^ vw p Y , Y qě ρ ˚ G p X X Y , X Y Y Y t v uq ` ρ ˚ G p X Y Y Y t v u , X X Y q ´ . Proof.
Apply Lemma 2.4 with G “ G r X Y X Y t v us . (cid:3) Prime graphs.
For a graph G , a partition p A, B q of V p G q iscalled a split if | A | , | B | ě A Ď A and B Ď B suchthat x P A is adjacent to y P B if and only if x P A and y P B . Agraph is prime (with respect to the split decomposition) if it has nosplits. These concepts were introduced by Cunningham [7].Alternatively, a split can be understood with the cut-rank function ρ G . A partition p A, B q of V p G q is a split if and only if | A | , | B | ě ρ G p A q ď Lemma 2.6.
If a prime graph H on at least vertices is a vertex-minor of a graph G , then G has a prime induced subgraph G such that G has a vertex-minor isomorphic to H .Proof. We may assume that G is connected. It is enough to prove thefollowing claim: if G has a split p A, B q , then there exists a vertex v such that H is isomorphic to a vertex-minor of G z v . Let G be a graphlocally equivalent to G such that H is an induced subgraph of G . Wehave ρ H p V p H q X A q “ ρ ˚ G p V p H q X A, V p H q X B q ď ρ ˚ G p A, B q ď | V p H q X A | ď | V p H q X B | ď H is prime. By NAVOIDABLE VERTEX-MINORS 7 symmetry, let us assume | V p H q X B | ď
1. Let us choose x P B suchthat x has a neighbor in A and x P V p H q if V p H q X B is nonempty.Let H be a vertex-minor of G on A Y t x u such that H is isomorphicto a vertex-minor of H . Then H “ G ˚ v ˚ v ¨ ¨ ¨ ˚ v n zp B zt x uq for somesequence v , v , . . . , v n of vertices. We may choose H and n so that n is minimized.Suppose n ą
0. Then v n P B zt x u . Let H “ G ˚ v ˚ v ¨ ¨ ¨ ˚ v n ´ zp B zt x, v n uq . Since p A, t x, v n uq is a split of H , one of the followingholds.(i) The two vertices v n and x have the same set of neighbors in A .(ii) The vertex v n has no neighbors in A .(iii) The vertex x has no neighbors in A .If we have the case (i), then p H z v n q ˚ x “ H and therefore H isisomorphic to a vertex-minor of H z v n , contradicting our assumptionthat H is chosen to minimize n . If we have the case (ii), then H z v n “ H , contradicting the assumption too. Finally if we have the case (iii),then x is adjacent to v n in G because G is connected. Then H ˚ v n z v n is isomorphic to H ˚ v n z x . Then H z x has a vertex-minor isomorphicto H , contradicting our assumption that n is minimized. (cid:3) Constructions of graphs.
For two graphs G and H on the sameset of n vertices, we would like to introduce operations to constructgraphs on 2 n vertices by making the disjoint union of them and addingsome edges between two graphs. Roughly speaking, G a H will add aperfect matching, G b H will add the complement of a perfect matching,and G m H will add a bipartite chain graph. Formally, for two graphs G and H on t v , v , . . . , v n u , let G a H , G b H , G m H be graphs on t v , v , . . . , v n , v , v , . . . , v n u such that for all i, j P t , , . . . , n u ,(i) v i v j P E p G a H q if and only if v i v j P E p G q ,(ii) v i v j P E p G a H q if and only if v i v j P E p H q ,(iii) v i v j P E p G a H q if and only if i “ j ,(iv) v i v j P E p G b H q if and only if v i v j P E p G q ,(v) v i v j P E p G b H q if and only if v i v j P E p H q ,(vi) v i v j P E p G b H q if and only if i ‰ j ,(vii) v i v j P E p G m H q if and only if v i v j P E p G q ,(viii) v i v j P E p G m H q if and only if v i v j P E p H q ,(ix) v i v j P E p G m H q if and only if i ě j .See Figure 3 for K a K , K b K , and K m K .We will use the following lemmas. Lemma 2.7.
Let n ě be an integer. O-JOUNG KWON AND SANG-IL OUM v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v Figure 3. K a K , K b K , and K m K . (1) K n b K n has a vertex-minor isomorphic to K n ´ a K n ´ .(2) K n b K n has a vertex-minor isomorphic to K n ´ a K n ´ .Proof. (1) Let V p K n q “ V p K n q “ t v i : 1 ď i ď n u . The graph p K n b K n q ˚ v ˚ v z v z v is isomorphic to K n ´ a K n ´ .(2) Let V p K n q “ t v , v , . . . , v n u . The graph p K n b K n q ˚ v z v z v is isomorphic to K n ´ a K n ´ . By (1), K n b K n has a vertex-minorisomorphic to K n ´ a K n ´ . (cid:3) Lemma 2.8.
Let n be a positive integer.(1) The graph K n m K n is pivot-equivalent to P n .(2) The graph K n m K n is locally equivalent to P n .Proof. (1) Let P “ p p . . . p n . We can check that K n m K n can beobtained from P by pivoting p i p i ` for all i “ , , . . . , n ´ V p K n q “ V p K n q “ t v , v , . . . , v n u . Since p K n m K n q ˚ v isisomorphic to K n m K n , the result follows from (1). (cid:3) Ramsey numbers. A clique is a set of pairwise adjacent vertices.A stable set or an independent set is a set of pairwise non-adjacentvertices.We write R p n , n , . . . , n k q to denote the minimum number N suchthat in every k coloring of the edges of K N , there exist i and a clique ofsize n i whose edges are all colored with the i -th color. Such a numberexists by Ramsey’s theorem [19].3. Unavoidable vertex-minors in large graphs
We present three simple statements on unavoidable vertex-minors.These are optimal as discussed in Section 1.
Theorem 3.1. (1) For every n , there exists N such that everygraph on at least N vertices has a vertex-minor isomorphic to K n . NAVOIDABLE VERTEX-MINORS 9 p p p p p p p p p q q q q q q q q Figure 4.
An example of a generalized ladder. (2) For every n , there exists N such that every connected graphhaving at least N vertices has a vertex-minor isomorphic to K n .(3) For every n , there exists N such that every graph having at least N edges has a vertex-minor isomorphic to K n or K n a K n .Proof. (1) If a graph has no K n as a vertex-minor, then it has novertex-minor isomorphic to K n ` . So we can take N “ R p n, n ` q .(2) Let us assume that G has no vertex-minor isomorphic to K n .Then the maximum degree of G is less than ∆ “ R p n ´ , n ´ q byRamsey theorem. If | V p G q | is big enough, then it contains an inducedpath P of length 2 n ´ P n ´ has a vertex-minor isomorphic to K ,n ´ , that islocally equivalent to K n .(3) Let G be a graph having no vertex-minor isomorphic to K n or K n a K n . Each component of G has bounded number of vertices, say M , by (2). Since K n a K n is not a vertex-minor of G , G has less than n non-trivial components. (A component is trivial if it has no edges.)So G has at most ` M ˘ p n ´ q edges. (cid:3) Obtaining a long cycle in a huge generalized ladder A generalized ladder is a graph G with two vertex-disjoint paths P “ p p . . . p a , Q “ q q . . . q b ( a, b ě
1) with additional edges, called chords , each joining a vertex of P with a vertex of Q such that V p P q Y V p Q q “ V p G q , p is adjacent to q , p a is adjacent to q b , and no twochords cross. Two chords p i q j and p i q j ( i ă i ) cross if and only if j ą j . We remark that a generalized ladder is a outerplanar graphwhose weak dual is a path. We call p q the first chord and p a q b the last chord of G . Since no two chords cross, p or q has degree at most2. Similarly, p a or q b has degree at most 2. See Figure 4 for an example.We will prove the following proposition. Proposition 4.1.
Let n ě . Every generalized ladder having at least n vertices has a cycle of length n ` as a vertex-minor. Lemmas on a fan.
Let F n be a graph on n vertices with a speci-fied vertex c , called the center, such that F n z c is a path on n ´ c is adjacent to all other vertices. We call F n a fan on n vertices. Lemma 4.2.
A fan F n has a vertex-minor isomorphic to a cycle oflength n ` .Proof. Let c be the center of F n . Let v , v , . . . , v n ´ be the non-centervertices in F n forming a path. Let G “ F n ˚ v ˚ v ˚ v ¨ ¨ ¨ ˚ v n ´ .Clearly c is adjacent to v i in G if and only if i P t , n ´ u or i ” p mod 3 q and furthermore v i ´ is adjacent to v i ` in G for all i . Let H “ G zt v , v , . . . , v n ´ u . Then H is a cycle of length 3 n ´ p n ´ q . (cid:3) Lemma 4.3.
Let n ě . Let G be a graph with a vertex c such that G z c is isomorphic to an induced path P whose both ends are adjacentto c . If | V p G q | ě p n ´ q ´ , then G has a vertex-minor isomorphicto a cycle of length n ` .Proof. We may assume that n ě
3. Let P “ v v . . . v k with k ě v is adjacent to c because otherwise we replace G with G ˚ v . Similarly we may assume that v k ´ is adjacent to c .We may also assume v is adjacent to c because otherwise we replace G with G ^ v v . Similarly we may assume that v k ´ is adjacent to c .If c is adjacent to at least 3 n ´ P , then G has a vertex-minor isomorphic to F n . So by Lemma 4.2, G has a vertex-minorisomorphic to a cycle of length 2 n `
1. Thus we may assume that thenumber of neighbors of c is at most 3 n ´
2. The neighbors of c givesa partition of P into at most 3 n ´ | E p P q | ě p n ´ q ´ ´ ą p n ´ qpp n ´ q ´ q ` , there exists a subpath P of P having length at least 2 n ´ P is adjacent to c and the ends of P are adjacentto c . This together with c gives an induced cycle of length at least2 n ` (cid:3) Generalized ladders of maximum degree at most .Lemma 4.4. Let G be a generalized ladder of maximum degree . If G has at least n vertices of degree , then G has a cycle of length n ` as a vertex-minor.Proof. We proceed by induction on | V p G q | . Let P , Q be two definingpaths of G . We may assume that all internal vertices of P or Q hasdegree 3, because if P or Q has an internal vertex v of degree 2, thenwe apply the induction hypothesis to G ˚ v z v . Since p or q has degree NAVOIDABLE VERTEX-MINORS 11
2, we may assume that p has degree 2 by symmetry. We may assumethat q has degree 3 because otherwise we can apply the inductionhypothesis to G ˚ q z q . Consequently q is adjacent to p and thus foreach internal vertex q i of Q , q i is adjacent to p i ` and each internalvertex p i ` of P is adjacent to q i . Thus either a “ b and p a has degree3 or a “ b ` p a has degree 2. But if a “ b ` p a hasdegree 2, then we can apply the induction hypothesis to G ˚ p a z p a .Thus we may assume that a “ b and p a has degree 3. Since G has atleast 6 n vertices of degree 3, a ą n and b ą n . If a “ b ą n ` G z q b . Thus we mayassume that a “ b “ n ` p a has degree 3 and q b has degree2. Note that p i is adjacent to q i ´ for all i “ , . . . , n `
1. Then G ˚ p ^ p q ^ p q ¨ ¨ ¨ ^ p n ` q n zt p , p , . . . , p n ´ , q , q , . . . , q n ´ , q n ` u is isomorphic to a cycle of length 4 n ` (cid:3) Lemma 4.5.
Let G be a generalized ladder of maximum degree . If | V p G q | ě n , then G has a cycle of length n ` as a vertex-minor.Proof. Let P , Q be two defining paths of G . We may assume a ą b ą G has an induced cycle of length at least6 n ` ě n ` p x q y be the unique chord other than p q with minimum x ` y .We claim that we may assume p x ´ q ` p y ´ q ď
2. Suppose not. Then p x q y , p q and subpaths of P and Q form a cycle of length x ` y ě p , p , . . . , p x ´ , q , q , . . . , q y ´ have degree 2. By moving the firstfew vertices of P to Q or Q to P , we may assume that x ě y ě G with G ˚ p . This proves the claim.Thus the induced cycle containing p q has at most 2 edges from E p P q Y E p Q q . Similarly we may assume that the induced cycle con-taining p a q b has at most 2 edges from E p P q Y E p Q q .If G has at least 6 n vertices of degree 3, then by Lemma 4.4, weobtain a desired vertex-minor. So we may assume that G has at most6 n ´ G has at most 3 n ´ p q and p a q b . These chords give at most 3 n induced cycles of G where each edge in E p P q Y E p Q q appears in exactly one of them. Ifevery such induced cycle has length at most 4 n `
2, then | E p P q Y E p Q q | ď p n ´ qp n q ` “ n ´ n ` ă n ´ . Since | V p G q | ě n , we have | E p P q Y E p Q q | ě n ´
2. This leadsto a contradiction. (cid:3)
Generalized ladders of maximum degree . Lemma 4.6.
Let G be a generalized ladder of maximum degree at most . Let α be the number of vertices of G having degree or . Then G has a vertex-minor H that is a generalized ladder of maximum degreeat most such that | V p H q | ě α { .Proof. Let P “ p p . . . p a , Q “ q q . . . q b be the paths defining ageneralized ladder G . Let X i,j “ t p , p , . . . , p i , q , q , . . . , q j u . We mayassume α ą a “
1, then p has at least α ´ α ď
5, contradicting our assumption. Thus a ą
1. Similarly b ą P or Q has degree 2,because otherwise we can apply local complementation and remove it.Let α i,j p G q be the number of vertices in V p G qz X i,j having degree 3or 4. We will prove the following. Claim 1.
Suppose that there exist ď i ă a and ď j ă b such that δ G p X i,j q has exactly two edges and every vertex in X i,j has degree or in G . Then G has a vertex-minor H that is a generalized ladder ofmaximum degree at most such that | V p H q | ě | X i,j | ` α i,j p G q{ . Before proving Claim 1, let us see why this claim implies our lemma.First we would like to see why there exist i and j such that δ G p X i,j q hasexactly two edges. If p has degree bigger than 2, then p is adjacentto q and so G ˚ q “ G z p q . Thus we may assume that both p and q have degree 2. Keep in mind that the number of vertices of degree3 or 4 in X , may be decreased by 1 by replacing G with G ˚ q andso α , p G q ě α ´ i “ j “
1, we obtain a generalized ladder H of maximum degree at most 3 as a vertex-minor such that | V p H q | ě ` p α ´ q{ ě α {
4. This completes the proof of the lemma, assumingClaim 1.We now prove Claim 1 by induction on | V p G q | ´ | X i,j p G q | . We mayassume that every vertex in V p G qzp X i,j Y t p a , q b uq has degree 3 or 4because otherwise we can apply local complementation and delete itwhile keeping α i,j . Then p i ` is obviously adjacent to q j ` .We may assume that i ă a ´ G is a generalizedladder of maximum degree 3 if p a has degree 3 and G z q b is a generalizedladder of maximum degree 3 otherwise. Similarly we may assume j ă b ´
1. Either p i ` or q j ` has degree 4, because otherwise δ G p X i ` ,j ` q has exactly two edges. By symmetry, we may assume that p i ` hasdegree 3 and q j ` has degree 4 and therefore q j ` is adjacent to p i ` . NAVOIDABLE VERTEX-MINORS 13 If α i,j p G q ď
12, then H “ G r X i ` ,j ` s is a generalized ladder ofmaximum degree at most 3. Thus we may assume that α i,j p G q ą b ´ j ď
4, then a ´ i ď q j ` , q j ` , . . . , q b hasdegree at most 4 and each vertex in p i ` , p i ` , . . . , p a ´ has degree atleast 3. This contradicts our assumption that α i,j p G q ą
12. So we mayassume that b ´ j ě a ´ i ě R be the component of G zp E p P q Y E p Q qq containing p i ` . Be-cause of the degree condition, R is a path. We now consider six cases,see Figure 5.(a) If R has length 2 and p i ` has degree 3 in G , then G “ G ˚ p i ` z p i ` “ p G z p i ` ` p i ` p i ` ` q j ` p i ` qz p i ` q j ` is a generalizedladder of maximum degree at most 4. Every vertex in G not in X i,j has degree at most 4. Furthermore p i ` has degree 2 in G . Thus, δ G p X i ` ,j q has exactly 2 edges. Then | X i ` ,j | ` α i ` ,j p G q{ ěp | X i,j | ` q ` p α i,j p G q ´ q{ ě | X i,j | ` α i,j p G q{
4. By the inductionhypothesis, we find a desired vertex-minor H in G .(b) If R has length 2 and p i ` has degree 4 in G , then the vertex q j ` has degree 3. Then G “ G ˚ p i ` ˚ q j ` z p i ` z q j ` is a generalizedladder of maximum degree at most 4. Then δ G p X i ` ,j ` q has ex-actly two edges and α i ` ,j ` p G q ě α i,j p G q ´
6. Again, | X i ` ,j ` | ` α i ` ,j ` p G q{ ě | X i,j | ` ` p α i,j p G q ´ q{ ě | X i,j | ` α i,j p G q{ R has length 3 and q j ` has degree 3 in G , then G “ G ˚ q j ` z q j ` is a generalized ladder of maximum degree at most 4.Then δ G p X i ` ,j ` q has exactly two edges and α i ` ,j ` p G q ě α i,j p G q´
3. We deduce that | X i ` ,j ` | ` α i ` ,j ` p G q{ ě | X i,j | ` `p α i,j p G q´ q{ ě | X i,j | ` α i,j p G q{ R has length 3 and q j ` has degree 4 in G , then p i ` has de-gree 3 and G “ G ˚ q j ` ˚ p i ` z q j ` z p i ` is a generalized ladder ofmaximum degree at most 4. Then δ G p X i ` ,j ` q has exactly twoedges and α i ` ,j ` p G q ě α i,j p G q ´
7. We deduce that | X i ` ,j ` | ` α i ` ,j ` p G q{ ě | X i,j | ` ` p α i,j p G q ´ q{ ě | X i,j | ` α i,j p G q{ G has a desired vertex-minor and sodoes G .(e) If R has length 4, then G “ G ^ p i ` q j ` ˚ p i ` z p i ` z p i ` z q j ` is ageneralized ladder of maximum degree at most 4. Then δ G p X i ` ,j ` q has exactly two edges and α i ` ,j ` p G q ě α i,j p G q ´ | X i ` ,j ` | ` α i ` ,j ` p G q{ ě | X i,j | ` ` p α i,j p G q ´ q{ ě | X i,j | ` α i,j p G q{
4. Our induction hypothesis implies that G has adesired vertex-minor. p i q j p i ` p i ` p i ` q j ` q j ` X i,j p i q j ñ X i ` ,j q j ` p i ` p i ` q j ` (a) Apply G ˚ p i ` z p i ` p i q j p i ` p i ` p i ` q j ` q j ` q j ` X i,j p i q j ñ X i ` ,j ` p i ` p i ` q j ` q j ` (b) Apply G ˚ p i ` ˚ q j ` z p i ` z q j ` p i q j p i ` p i ` p i ` q j ` q j ` q j ` X i,j p i q j ñ X i ` ,j ` p i ` p i ` p i ` q j ` q j ` (c) Apply G ˚ q j ` z q j ` p i q j p i ` p i ` p i ` p i ` q j ` q j ` q j ` X i,j p i q j ñ X i ` ,j ` p i ` p i ` p i ` q j ` q j ` (d) Apply G ˚ q j ` ˚ p i ` z q j ` z p i ` p i q j p i ` p i ` p i ` p i ` q j ` q j ` q j ` X i,j p i q j ñ X i ` ,j ` p i ` p i ` q j ` q j ` (e) Apply G ^ p i ` q j ` ˚ p i ` z p i ` z q j ` z p i ` p i q j p i ` p i ` p i ` q j ` q j ` q j ` X i,j p i q j ñ X i,j ` p i ` p i ` q j ` q j ` (f) Apply G ^ p i ` q j ` z p i ` z q j ` Figure 5.
Cases in the proof of Lemma 4.6.
NAVOIDABLE VERTEX-MINORS 15 (f) If R has length at least 5, then G “ G ^ p i ` q j ` z p i ` z q j ` is ageneralized ladder of maximum degree at most 4. Then δ G p X i,j ` q has exactly two edges and α i,j ` p G q ě α i,j p G q ´ | X i,j ` | ` α i,j ` p G q{ ě | X i,j | ` ` p α i,j p G q ´ q{ “ | X i,j | ` α i,j p G q{
4. Our induction hypothesis implies that G has a desiredvertex-minor.In all cases, we find the desired vertex-minor H . This completes theproof of Claim 1. (cid:3) Lemma 4.7.
Let G be a generalized ladder of maximum degree at most . If | V p G q | ě n , then G has a cycle of length n ` as a vertex-minor.Proof. Let P , Q be two defining paths of G . We may assume a ą b ą p n ´ q{ ` ě n ` p x q y be the unique chord other than p q with minimum x ` y .We claim that we may assume p x ´ q ` p y ´ q ď
2. Suppose not. Then p x q y , p q and subpaths of P and Q form a cycle of length x ` y ě p , p , . . . , p x ´ , q , q , . . . , q y ´ have degree 2. By moving the firstfew vertices of P to Q or Q to P , we may assume that x ě y ě G with G ˚ p . This proves the claim.Thus the induced cycle containing p q has at most 2 edges from E p P q Y E p Q q . Similarly we may assume that the induced cycle con-taining p a q b has at most 2 edges from E p P q Y E p Q q .If G has at least 48 n vertices of degree 3 or 4, then by Lemma 4.6, G has a generalized ladder H as a vertex-minor such that | V p H q | ě n and H has maximum degree at most 3. By Lemma 4.5, H has a cycleof length 4 n ` G has less than 48 n vertices of degree3 or 4. We may assume that G has at least one vertex of degree atleast 3. The cycle formed by edges in E p P q Y E p Q q Y t p q , p a q b u is partitioned into less than 48 n paths whose internal vertices havedegree 2 in G . One of the paths has length greater than 192 n {p n q “ n . Then there is an induced cycle C of G containing this path. Since C does not contain p q or p a q b , C must contain two edges not in E p P q Y E p Q q Y t p q , p a q b u . Thus the length of C is at least 4 n ` (cid:3) Treating all generalized ladders.Lemma 4.8.
Let G be a generalized ladder. If G has n vertices ofdegree at least , then G has a vertex-minor H that is a generalizedladder such that the maximum degree of H is at most and H has atleast n vertices. Proof.
Let S be the set of vertices having degree at least 4. For eachvertex v in S , let P v be the minimal subpath of Q containing all neigh-bors of v in Q if v P V p P q and let P v be the minimal subpath of P containing all neighbors of v in P if v P V p Q q .Then each internal vertex of P v has degree 2 or 3 and has degree 3if and only if it is adjacent to v . We apply local complementation toeach internal vertex and delete all internal vertices of P v . It is easy tosee that the resulting graph H is a generalized ladder and moreover S Ď V p H q and every vertex in S has degree at most 4 in H . (cid:3) We are now ready to prove the main proposition of this section.
Proof of Proposition 4.1.
Let G be such a graph. If G has at least192 n vertices of degree at least 4, then by Lemma 4.8, G has a vertex-minor H having at least 192 n vertices such that H is a generalizedladder of maximum degree at most 4. By Lemma 4.7, H has a cycle oflength 4 n ` G has less than 192 n vertices of degreeat least 4. For a vertex v in P having degree at least 5, let q i , q j be two neighbors of v in Q such that if q k is a neighbor of v in Q ,then i ď k ď j . By Lemma 4.3, if j ´ i ` ě n ´
3, then G contains a cycle of length 4 n ` j ´ i ď n ´
6. The subpath of Q from q i to q j contains j ´ i ´ ď n ´ v in Q having degree at least 5. As in the proof of Lemma 4.8,we apply local complementation and delete all internal vertices of theminimal path spanning the neighbors of each vertex of degree at least5 to obtain H . Then each vertex of degree at least 5 in G will havedegree at most 4 in H . Since we remove at most p n ´ qp n ´ q vertices, | V p H q | ě | V p G q | ´ p n ´ qp n ´ q ą n . By Lemma 4.7, H has a cycle of length 4 n ` (cid:3) Blocking sequences
Let
A, B be two disjoint subsets of the vertex set of a graph G . Bythe definition of ρ ˚ G and ρ G , it is clear thatif A Ď X Ď V p G qz B, then ρ ˚ G p A, B q ď ρ G p X q . What prevents us to achieve the equality for some X ? We now presenta tool called a blocking sequence, that is a certificate to guarantee thatno such X exists. Blocking sequences were introduced by Geelen [13]. NAVOIDABLE VERTEX-MINORS 17
A sequence v , v , . . . , v m ( m ě
1) is called a blocking sequence of apair p A, B q of disjoint subsets A , B of V p G q if(a) ρ ˚ G p A, B
Y t v uq ą ρ ˚ G p A, B q ,(b) ρ ˚ G p A Y t v i u , B Y t v i ` uq ą ρ ˚ G p A, B q for all i “ , , . . . , m ´ ρ ˚ G p A Y t v m u , B q ą ρ ˚ G p A, B q ,(d) no proper subsequence of v , . . . , v m satisfies (a), (b), and (c).The condition (d) is essential for the following standard lemma. Lemma 5.1.
Let v , v , . . . , v m be a blocking sequence for p A, B q in agraph G . Let X , Y be disjoint subsets of t v , v , . . . , v m u such that if v i P X and v j P Y , then i ă j . Then ρ ˚ G p A Y X, B Y Y q “ ρ ˚ G p A, B q if and only if v R Y , v m R X , and for all i P t , , . . . , m ´ u , either v i R X or v i ` R Y .Proof. The forward direction is trivial. Let us prove the backwardimplication. Let k “ ρ ˚ G p A, B q . It is enough to prove ρ ˚ G p A Y X, B Y Y q ď k . Suppose that v R Y , v m R X , and for all i P t , , . . . , m ´ u ,either v i R X or v i ` R Y and yet ρ ˚ G p A Y X, B Y Y q ą k . We mayassume that | X | ` | Y | is chosen to be minimum. If | X | ě
2, thenwe can partition X into two nonempty sets X and X . Then bythe hypothesis, ρ ˚ G p A Y X , B Y Y q “ ρ ˚ G p A Y X , B Y Y q “ k . ByLemma 2.2, we deduce that ρ ˚ G p A Y X , B Y Y q ` ρ ˚ G p A Y X , B Y Y q ě k ` ρ ˚ G p A Y X, B Y Y q and therefore we deduce that ρ ˚ G p A Y X, B Y Y q ď k .So we may assume | X | ď
1. By symmetry we may also assume | Y | ď (cid:3) The following proposition states that a blocking sequence is a certifi-cate that ρ G p X q ą ρ ˚ G p A, B q for all A Ď X Ď V p G qz B . This appearsin almost all applications of blocking sequences. The proof uses thesubmodular inequality (Lemma 2.2). Proposition 5.2 (Geelen [13, Lemma 5.1]; see Oum [17]) . Let G bea graph and A , B be two disjoint subsets of V p G q . Then G has ablocking sequence for p A, B q if and only if ρ G p X q ą ρ ˚ G p A, B q for all A Ď X Ď V p G qz B . The following proposition allows us to change the graph to reducethe length of a blocking sequence. This was pointed out by Geelen[private communication with the second author, 2005]. A special caseof the following proposition is presented in [17].
Proposition 5.3.
Let G be a graph and A , B be disjoint subsets of V p G q . Let v , v , . . . , v m be a blocking sequence for p A, B q in G . Let ď i ď m . ‚ If m ą , then ρ ˚ G ˚ v i p A, B q “ ρ ˚ G p A, B q and a sequence v , v , . . . , v i ´ , v i ` , . . . , v m obtained by removing v i from the blocking sequence is a blockingsequence for p A, B q in G ˚ v i . ‚ If m “ , then ρ ˚ G ˚ v i p A, B q “ ρ ˚ G p A, B q ` .Proof. Let k “ ρ ˚ G p A, B q and H “ G ˚ v i .If m “
1, then by Lemma 2.5, ρ ˚ H p A, B q ` ρ ˚ G p A, B q ě ρ ˚ G p A Y t v u , B q ` ρ ˚ G p A, B
Y t v uq ´ ě k ` ρ ˚ H p A, B q ě k `
1. Since ρ ˚ H p A, B q ď ρ ˚ H p A, B
Y t v uq “ ρ ˚ G p A, B
Y t v uq ď k `
1, we deduce that ρ ˚ H p A, B q “ k ` m “ m ‰
1. First it is easy to observe that ρ ˚ H p X, Y q ď ρ ˚ G p X, Y Yt v i uq and ρ ˚ H p X, Y q ď ρ ˚ G p X Yt v i u , Y q whenever X , Y are dis-joint subsets of V p G qzt v i u , because the local complementation does notchange the cut-rank function of G r X Y Y Y t v i us . This with Lemma 5.1implies that ‚ ρ ˚ H p A, B q ď k , ‚ ρ ˚ H p A Y t v j u , B q ď k for all j P t , , . . . , m uzt i ´ , m u , ‚ ρ ˚ H p A Y t v i ´ u , B q ď k if i ‰ , m . ‚ ρ ˚ H p A, B
Y t v j uq ď k for all j P t , , . . . , m uzt , i ` u . ‚ ρ ˚ H p A, B
Y t v i ` uq ď k if i ‰ , m . ‚ ρ ˚ H p A Y t v j u , B Y t v ℓ uq ď k for all j, ℓ P t , , . . . , m uzt i u with ℓ ´ j ą
1, unless j ` “ i “ ℓ ´ B “ B Y t v i ` u if i ă m and B “ B otherwise. Then ρ ˚ G p A Yt v i u , B q “ k ` ρ ˚ G p A, B q “ k .(1) We claim that if i ą
1, then ρ ˚ H p A, B
Y t v uq ą k . By Lemma 2.5, ρ ˚ H p A, B Yt v uq` ρ ˚ G p A, B q ě ρ ˚ G p A, B Yt v , v i uq` ρ ˚ G p A Yt v i u , B q´ , and therefore we deduce that ρ ˚ H p A, B Y t v uq ě ρ ˚ G p A, B Y t v , v i uq ą k . By Lemma 2.2, ρ ˚ H p A, B Y t v i uq ` ρ ˚ H p A, B
Y t v uq ě ρ ˚ H p A, B Yt v , v i uq` ρ ˚ H p A, B q ą k . We deduce that ρ ˚ H p A, B Yt v uq ą k because ρ ˚ H p A, B Y t v i uq “ ρ ˚ G p A, B Y t v i uq “ k by Lemma 5.1.(2) By (1) and symmetry between A and B , if i ă m , then ρ ˚ H p A Yt v m u , B q ą k .Then we deduce that ρ ˚ H p A, B q ě k and therefore ρ ˚ H p A, B q “ k . NAVOIDABLE VERTEX-MINORS 19 (3) We claim that if j ă i ´
1, then ρ ˚ H p A Y t v j u , B Y t v j ` uq ą k .By Lemma 2.5, ρ ˚ H p A Y t v j u , B Y t v j ` uq ` ρ ˚ G p A Y t v j u , B qě ρ ˚ G p A Y t v j u , B Y t v j ` , v i uq ` ρ ˚ G p A Y t v j , v i u , B q ´ ą k, and therefore ρ ˚ H p A Y t v j u , B Y t v j ` uq ą k . By Lemma 2.2, ρ ˚ H p A Yt v j u , B Yt v j ` uq` ρ ˚ H p A Yt v j u , B q ě ρ ˚ H p A Yt v j u , B Yt v j ` uq` ρ ˚ H p A Yt v j u , B q ą k . Note that ρ ˚ H p A Y t v j u , B q ě ρ ˚ H p A, B q “ k . Since ρ ˚ H p A Yt v j u , B q ď ρ ˚ H p A Yt v j u , B Yt v i uq “ ρ ˚ G p A Yt v j u , B Yt v i uq ď k ,we deduce that ρ ˚ H p A Y t v j u , B Y t v j ` uq ą k .(4) By symmetry, we deduce from (3) that if i ă j ă m , then ρ ˚ H p A Y t v j u , B Y t v j ` uq ą k .(5) We claim that ρ ˚ H p A Y t v i ´ u , B q ą k . By Lemma 2.5, ρ ˚ H p A Y t v i ´ u , B q ` ρ ˚ G p A Y t v i ´ u , B qě ρ ˚ G p A Y t v i ´ u , B Y t v i uq ` ρ ˚ G p A Y t v i ´ , v i u , B q ´ ą k. Since ρ ˚ G p A Y t v i ´ u , B q “ k , we have ρ ˚ H p A Y t v i ´ u , B q ą k .This completes the proof of the lemma that v , v , . . . , v i ´ , v i ` , . . . , v m is a blocking sequence of p A, B q in G ˚ v i . (cid:3) Corollary 5.4.
Let G be a graph and A , B be disjoint subsets of V p G q .Let v , v , . . . , v m be a blocking sequence for p A, B q in G . Let ď i ď m .Suppose that v i has a neighbor w in A Y B . ‚ If m ą , then ρ ˚ G ^ v i w p A, B q “ ρ ˚ G p A, B q and the sequence v , v , . . . , v i ´ , v i ` , . . . , v m obtained by removing v i from theblocking sequence is a blocking sequence for p A, B q in G ^ v i w . ‚ If m “ , then ρ ˚ G ^ v i w p A, B q “ ρ ˚ G p A, B q ` .Proof. It follows easily from the facts that G ^ v i w “ G ˚ w ˚ v i ˚ w and ρ ˚ G p X, Y q “ ρ ˚ G ˚ w p X, Y q for all graphs G with w P X Y Y . (cid:3) Corollary 5.5.
Let G be a graph and A , B be disjoint subsets of V p G q .Let v , v , . . . , v m be a blocking sequence for p A, B q in G . Let ď i ď m .Suppose that v i and v i are adjacent and i ă i . ‚ If m ą , then ρ ˚ G ^ v i v i p A, B q “ ρ ˚ G p A, B q and the sequence v , v , . . . , v i ´ , v i ` , . . . , v i ´ , v i ` , . . . , v m obtained by removing v i and v i from the blocking sequence is a blocking sequence for p A, B q in G ^ v i v i . ‚ If m “ , then ρ ˚ G ^ v i v i p A, B q “ ρ ˚ G p A, B q ` .Proof. If v i has a neighbor w in A Y B , then G ^ v i v i “ G ^ v i w ^ wv i and this corollary follows from Corollary 5.4. So we may assume that v i has no neighbors in A Y B and similarly v i has no neighbors in A Y B .Thus i, i R t , m u and m ě v i and v i are adjacent, we may assume that i “ i `
1. Let H “ G ^ v i v i ` and k “ ρ ˚ G p A, B q . Since v i and v i ` have no neighborsin A Y B , ρ ˚ H p A, B q “ k .Then v , v , . . . , v i is a blocking sequence for p A, B
Y t v i ` uq in G by Lemma 5.1. Similarly v i ` , v i ` , . . . , v m is a blocking sequence for p A Y t v i u , B q in G .By Corollary 5.4, v , v , . . . , v i ´ is a blocking sequence for p A, B Yt v i ` uq in H . Then ρ ˚ H p A, B
Y t v uq “ ρ ˚ H p A, B
Y t v , v i ` uq ą k ,because v i ` has no neighbors of H in A .For 1 ď j ă i ´ ρ ˚ H p A Yt v j u , B Yt v j ` uq` ρ ˚ H p A Yt v j u , B Yt v i ` uq ě ρ ˚ H p A Y t v j u , B Y t v j ` , v i ` uq ` ρ ˚ H p A Y t v j u , B q ą k and therefore ρ ˚ H p A Y t v j u , B Y t v j ` uq ą k because ρ ˚ H p A Y t v j u , B q ď ρ ˚ H p A Y t v j u , B Y t v i ` uq ď k .Similarly v i ` , v i ` , . . . , v m is a blocking sequence for p A Y t v i u , B q in H . By symmetry, we deduce that ρ ˚ H p A Y t v m u , B q ą k and ρ ˚ H p A Yt v j u , B Y t v j ` uq ą k for all i ` ă j ă m .We now claim that ρ ˚ H p A Y t v i ´ u , B Y t v i ` uq ą k . By Lemma 2.2, ρ ˚ H p A Y t v i ´ u , B Y t v i ` uq ` ρ ˚ H p A Y t v i ` u , B Y t v i ` uqě ρ ˚ H p A Y t v i ´ , v i ` u , B Y t v i ` uq ` ρ ˚ H p A, B
Y t v i ` uq . Since v i ` has no neighbors in A Y B , we have ρ ˚ H p A Y t v i ` u , B Yt v i ` uq “ ρ ˚ G p A Y t v i u , B Y t v i ` uq “ k and ρ ˚ H p A, B
Y t v i ` uq “ ρ ˚ G p A, B
Y t v i ` uq “ k . Therefore ρ ˚ H p A Y t v i ´ u , B Y t v i ` uq ě ρ ˚ H p A Y t v i ´ , v i ` u , B Y t v i ` uq . By Lemma 2.5, ρ ˚ H p A Y t v i ´ , v i ` u , B Y t v i ` uq ` ρ ˚ G p A Y t v i ´ u , B Y t v i ` , v i ` uqě ρ ˚ G p A Y t v i ´ u , B Y t v i , v i ` , v i ` uq` ρ ˚ G p A Y t v i ´ , v i , v i ` u , B Y t v i ` uq ´ . By Lemma 5.1, ρ ˚ G p A Y t v i ´ , v i , v i ` u , B Y t v i ` uq ą k and ρ ˚ G p A Yt v i ´ u , B Y t v i ` , v i ` uq “ k . Therefore ρ ˚ H p A Y t v i ´ u , B Y t v i ` uq ě ρ ˚ H p A Yt v i ´ , v i ` u , B Yt v i ` uq ě ρ ˚ G p A Yt v i ´ u , B Yt v i , v i ` , v i ` uq ą k .This proves the claim.So far we have shown that the sequence v , v , . . . , v i ´ , v i ` , . . . , v m satisfies (a), (b), (c) of the definition of blocking sequences. It remainsto show (d). For j P t , , . . . , m uzt i, i ` u , ρ ˚ H p A, B Yt v j uq “ ρ ˚ G p A, B Yt v j uq “ k because v i and v i ` have no neighbors in A Y B . Similarly NAVOIDABLE VERTEX-MINORS 21 ρ ˚ H p A Yt v j u , B q “ ρ ˚ G p A Yt v j u , B q “ k for j P t , , . . . , m ´ uzt i, i ` u .For j, ℓ P t , , . . . , m uzt i, i ` u with ℓ ´ j ą
1, either ρ ˚ G p A Y t v j u , B Yt v ℓ , v i , v i ` uq “ k or ρ ˚ G p A Y t v j , v i , v i ` u , B Y t v ℓ uq “ k and therefore ρ ˚ H p A Y t v j u , B Y t v ℓ uq ď k , unless j “ i ´ ℓ “ i `
2. Thiscompletes the proof. (cid:3)
We will now prove that without loss of generality, a blocking sequencefor p A, B q is short by applying local complementation while keeping thesubgraph induced on A Y B . Proposition 5.6.
Let G be a prime graph and let A , B be disjoint sub-sets of V p G q with | A | , | B | ě . Suppose that there exist two nonemptysets A Ď A and B Ď B such that the set of all edges between A and B is t xy : x P A , y P B u . Let ℓ “ $’&’% if | A | “ | B | “ , if | A | “ or | B | “ , otherwise.Then there exists a graph G locally equivalent to G satisfying the fol-lowing.(i) G r A Y B s “ G r A Y B s .(ii) G has a blocking sequence b , b , . . . , b ℓ of length at most ℓ for p A, B q .Proof. Since G is prime, G has a blocking sequence for p A, B q by Propo-sition 5.2. Let G be the set of all graphs G locally equivalent to G suchthat G r A Y B s “ G r A Y B s . We assume that G is chosen in G so thatthe length ℓ of a blocking sequence b , b , . . . , b ℓ for p A, B q is minimized.For 1 ď i ă ℓ , N G p b i q X B “ B or H because ρ G p A Y t b i u , B q “ ă i ď ℓ , N G p b i q X A “ A or H because ρ G p A, B
Y t b i uq “ N G p b i q X p A Y B q “ N G p b j q X p A Y B q for some 1 ă i ă j ă ℓ . If b i and b j are adjacent, then G “ G ^ b i b j P G . If b i and b j are non-adjacent, then G “ G ˚ b i ˚ b j P G . In both cases, we founda graph in G having a shorter blocking sequence by Proposition 5.3 orCorollary 5.5, contradicting our assumption.If | B | “
1, then for all 1 ă i ă ℓ , N G p b i q X A “ A becauseotherwise G ˚ b i P G has a shorter blocking sequence by Proposition 5.3,contradicting our assumption. Similarly if | A | “
1, then N G p b i q X B “ B for all 1 ă i ă ℓ .By the pigeonhole principle, we deduce that ℓ ď ℓ . (cid:3) Obtaining a long cycle from a huge induced path
In this section we aim to prove the following theorem. v v v v v v v v v w w w w Figure 6.
An example of a 4-patched path of length 8.
Theorem 6.1.
If a prime graph has an induced path of length r . n s ,then it has a cycle of length n as a vertex-minor. The main idea is to find a big generalized ladder, defined in Section 4as a vertex-minor by using blocking sequences in Section 5.6.1.
Patching a path.
For 1 ď k ď n ´
2, a k -patch of an inducedpath P “ v v ¨ ¨ ¨ v n of a graph G is a sequence Q “ w , w , . . . , w k ofdistinct vertices not on P such that for each i P t , , . . . , k u ,(i) v i ` is the only vertex adjacent to w i among v i ` , v i ` , . . . , v n ,(ii) H ‰ N G p w i q X t v , . . . , v i , w , . . . , w i ´ u ‰ t v i , w i ´ u if i ą N G p w q X t v , v u “ t v u .An induced path is called k -patched if it has a k -patch. An inducedpath of length n is called fully patched if it is equipped with a p n ´ q -patch. See Figure 6 for an example.Our goal is to find a fully patched long induced path in a vertex-minor of a prime graph having a very long induced path. Lemma 6.2.
Let P “ v v . . . v m be an induced path from s “ v to t “ v m in a graph G and let H be a connected induced subgraph of G z V p P q . Let v be a vertex in V p G qzp V p H q Y V p P qq . Suppose that N G p V p H qq X V p P q “ t s u , | E p P q | ě p n ´ q ´ , and v has neighborsin both V p P qzt s u and V p H q .If G has no cycle of length n ` as a vertex-minor, then there exista graph G locally equivalent to G and an induced path P from s to t of G disjoint from V p H q satisfying the following.(i) G r V p H q Y t s us “ G r V p H q Y t s us ,(ii) N G p v q X V p H q “ N G p v q X V p H q ,(iii) P “ v v i v i ` v i ` ¨ ¨ ¨ v m for some i ,(iv) v i is the only vertex on V p P q adjacent to v in G ,(v) | E p P q | ě | E p P q | ´ p n ´ q ` .Proof. Since G has a cycle using H with s and P , G is not a forest andtherefore n ě
2. Let v “ s, v , v , . . . , v m “ t be vertices in P . Let v k be the neighbor of v with maximum k . Then G has a fan having atleast k ` H is connected and v has a neighbor in H . NAVOIDABLE VERTEX-MINORS 23 If k ě p n ´ q ´
6, then G has a fan having at least 6 p n ´ q ´ G contains a cycle of length 2 n ` G has no suchvertex-minor. Thus, k ď p n ´ q ´ G “ G ˚ v ˚ v ˚ v ¨ ¨ ¨ ˚ v k ´ and let P “ v v k ´ v k v k ` ¨ ¨ ¨ v m .(If k ď
2, then let G “ G and P “ P .) Then clearly P is an inducedpath of G and v k P N G p v q X V p P q Ď t v , v k ´ , v k u .If N G p v q X V p P q “ t v k u , then we are done by taking G “ G ˚ v k ´ and P “ v v k v k ` ¨ ¨ ¨ v m .If N G p v q X V p P q “ t v k ´ , v k u , then we can take G “ G ˚ v k ˚ v k ´ and P “ v v k ` v k ` ¨ ¨ ¨ v m .If N G p v q X V p P q “ t v , v k u , then we can take G “ G ˚ v k ´ ˚ v k and P “ v v k ` v k ` ¨ ¨ ¨ v m .Finally, if N G p v q X V p P q “ t v , v k ´ , v k u , then we can take G “ G ˚ v k ˚ v k ´ ˚ v k ` and P “ v v k ` v k ` ¨ ¨ ¨ v m .In all cases, | E p P q | ě | E p P q | ´ p k ` q ě | E p P q | ´ p n ´ q ` (cid:3) Lemma 6.3.
Let n ě . Let G be a prime graph having an inducedpath of length t . If t ě p n ´ q ´ , then there exists a graph G locallyequivalent to G having a -patched induced path of length t ´ p n ´ q ` ,unless G has a cycle of length n ` as a vertex-minor.Proof. We may choose G so that the length t of an induced path P ismaximized among all graphs locally equivalent to G . Let v , v , . . . , v m be vertices of P in this order. Since G is prime, v has a neighbor v other than v . We may assume that v is non-adjacent to v becauseotherwise we can replace G with G ˚ v .Since P is a longest induced path, v must have some neighbors in V p P qzt v , v u . We now apply Lemma 6.2 with H “ G rt v , v us , de-ducing that there exists a graph G locally equivalent to G having a1-patched induced path of length t ´ p n ´ q `
6, unless G has a cycleof length 2 n ` (cid:3) Lemma 6.4.
Let n ě . Let G be a prime graph and let P be a k -patched induced path v v ¨ ¨ ¨ v t . If t ě p n ´ q ` k , then there exists agraph G locally equivalent to G having a p k ` q -patched induced path v v ¨ ¨ ¨ v k ` v i v i ` ¨ ¨ ¨ v t of length at least t ´ p n ´ q ` with some i ą k ` , unless G has a cycle of length n ` as a vertex-minor.Proof. Let P “ v v . . . v t be an induced path of length t in G and Q “ w , w , . . . , w k be its k -patch. Suppose that G has no vertex-minor isomorphic to a cycle of length 2 n ` A “ t v , v , . . . , v k ` u Y Q . By Proposition 5.6, we may assumethat G has a blocking sequence b , b , . . . , b ℓ of length at most 4 for p A, V p P qz A q because v k ` is the only vertex in V p P qz A having neigh-bors in A .Notice that P z A is an induced path of G . We say that a blockingsequence b , b , . . . , b ℓ for p A, V p P qz A q is nice if b ℓ has a unique neighborin V p P qz A , that is also a unique neighbor of v k ` in V p P qz A .We know that b ℓ has neighbors in t v k ` , . . . , v t u by the definitionof a blocking sequence. We take H “ G r A Y Q Y t b , b , . . . , b ℓ ´ us .By Lemma 6.2, there exist a graph G ℓ locally equivalent to G andan induced path P ℓ “ v v ¨ ¨ ¨ v k ` v i v i ` ¨ ¨ ¨ v t of G ℓ for some i witha k -patch Q such that G ℓ r A Y t v k ` us “ G r A Y t v k ` us , a sequence b , b , . . . , b ℓ is a nice blocking sequence for p A, V p P ℓ qz A q in G ℓ , and | E p P ℓ q | ě t ´ p n ´ q ` r ě G locallyequivalent to G and an induced path P “ v v ¨ ¨ ¨ v k ` v i v i ` ¨ ¨ ¨ v m forsome i with a k -patch Q in G such that G r A Yt v k ` us “ G r A Yt v k ` us ,a sequence b , b , . . . , b r is a nice blocking sequence for p A, V p P qz A q in G , and | E p P q | ě t ´ p n ´ q ` ` r ´ ℓ . Such r exists because G ℓ and P ℓ satisfy the condition when r “ ℓ .We claim that r “
1. Suppose r ą b r is non-adjacent to v k ` in G . Then v i is the onlyneighbor of b r in V p P q in G and b r is adjacent to b r ´ in G . If b r ´ isnon-adjacent to v k ` , then take G “ G ˚ b r and P “ P ; in G , a se-quence b , b , . . . , b r ´ is a nice blocking sequence for p A, V p P qz A q andthe length of P is at least t ´ p n ´ q ` ` r ´ ℓ . This leads a contradic-tion to the assumption that r is minimized. Therefore b r ´ is adjacentto v k ` . Then take G “ G ˚ b r ˚ v i with P “ v v ¨ ¨ ¨ v k ` v i ` ¨ ¨ ¨ v m .Then b , b , . . . , b r ´ is a nice blocking sequence for p A, V p P qz A q in G and the length of P is at least t ´ p n ´ q ` ` r ´ ℓ ´
1. Thiscontradicts to the assumption that r is chosen to be minimum.Therefore b r is adjacent to v k ` in G . Since b r is the last vertexin the blocking sequence, b r is also adjacent to w k in G . If b r ´ isnon-adjacent to v k ` , then take G “ G ˚ v k ` ˚ b r and P “ P ; in G ,a sequence b , b , . . . , b r ´ is a nice blocking sequence for p A, V p P qz A q and the length of P is at least t ´ p n ´ q ` ` r ´ ℓ , contradictingour assumption on r . So b r ´ is adjacent to v k ` . Then we take G “ G ˚ v k ` ˚ b r ˚ v i with P “ v v ¨ ¨ ¨ v k ` v i ` ¨ ¨ ¨ v m . Then b , b , . . . , b r ´ is a nice blocking sequence for p A, V p P qz A q in G and the length of P is at least t ´ p n ´ q ` ` r ´ ℓ ´
1. This again contradicts tothe assumption on r . This proves that r “ b is a nice blocking sequence for p A, V p P qz A q in G , b has aneighbor in A in G and N G p b q X A ‰ t v k ` , w k u . In addition, v i is theonly neighbor of b among V p P qz A in G . Now it is easy to see that NAVOIDABLE VERTEX-MINORS 25 w , w , w , . . . , w k , b is a p k ` q -patch of P in G . And, since ℓ ď | E p P q | ě t ´ p n ´ q ` (cid:3) Proposition 6.5.
Let N ě be an integer. If a prime graph G on atleast vertices has an induced path of length L “ p p n ´ q ´ qp N ´ q ´ , then there exists a graph G locally equivalent to G having a fullypatched induced path of length N , unless G has a cycle of length n ` as a vertex-minor.Proof. Suppose that G has no cycle of length 2 n ` n ě G has a 1-patched path of length L ´ p n ´ q `
6. By Lemma 6.4, wemay assume that G has an p N ´ q -patched path of length L ´ p n ´ q ` ´ p N ´ qp p n ´ q ´ q “ N Thus G has a fully patched induced path of length N . (cid:3) Finding a cycle from a fully patched path.
We aim to finda cycle as a vertex-minor in a sufficiently long fully patched path.Let P “ v v ¨ ¨ ¨ v n be an induced path of a graph G with a p n ´ q -patch Q “ w w w , . . . w n ´ . Let A “ t v , v u and for i “ , . . . , n ´ A i “ t v , v , . . . , v i , w , w , . . . , w i ´ u and B i “ V p P qz A i for all i P t , , . . . , n ´ u .For i ě
1, let L p w i q be the minimum j ě ρ ˚ G p A j ` , B j ` Y t w i uq ą . Since w i is a blocking sequence for p A i , B i q , L p w i q is well defined and L p w i q ă i .We classify vertices in Q as follows. ‚ A vertex w i has Type 0 if L p w i q “ w i is adjacent to v . ‚ A vertex w i has Type 1 if L p w i q ě w i has no neighbor in A L p w i q and w i is adjacent to exactly one of v L p w i q` and w L p w i q . ‚ A vertex w i has Type 2 if L p w i q “ w i is adjacent to v ,non-adjacent to v . ‚ A vertex w i has Type 3 if L p w i q ě w i has no neighbor in A L p w i q´ and w i is adjacent to both v L p w i q and w L p w i q´ .By the definition of fully patched paths, we can deduce the followinglemma easily. Lemma 6.6.
Each vertex in Q has Type 0, 1, 2, or 3.Proof. If w i is adjacent to v , then ρ ˚ G p A , B Y t w i uq ą L p w i q “
0, implying that w i has Type 0. We may now assume that w i is non-adjacent to v and so L p w i q ą If w i has no neighbors in A L p w i q , then ρ ˚ G p A L p w i q` , B L p w i q` Y t w i uq “ ρ ˚ G p A L p w i q` z A L p w i q , B L p w i q` Y t w i uq ą
1. Thus v L p w i q` and w i cannothave the same set of neighbors in A L p w i q` z A L p w i q “ t v L p w i q` , w L p w i q u .By the definition of fully patched paths, v L p w i q` is adjacent to both v L p w i q` and w L p w i q . It follows that w i is adjacent to exactly one of v L p w i q` and w L p w i q . So w i has Type 1.Now we may assume that w i has some neighbors in A L p w i q . By defi-nition, ρ ˚ G p A L p w i q , B L p w i q Y t w i uq ď w i and v L p w i q` have the same set of neighbors in A L p w i q .Therefore, if L p w i q “
1, then w i is adjacent to v , implying that w i hasType 2. If L p w i q ą
1, then w i is adjacent to both v L p w i q and w L p w i q´ ,and so w i has Type 3. (cid:3) We say that a pair of paths P i and P i from t v , v u to t v i ` , w i u is good if(i) P i and P i are vertex-disjoint induced paths on A i ` ,(ii) for each j P t , , . . . , i ´ u , w j P V p P i qY V p P i q or v j ` P V p P i qY V p P i q ,(iii) G r V p P i qY V p P i qs` v i ` w i is a generalized ladder with two definingpaths P i and P i . Lemma 6.7.
For all i P t , , . . . , n ´ u , G has a good pair of paths P i and P i from t v , v u to t v i ` , w i u .Proof. We proceed by induction on i . If w i has Type 0, then let P i “ v v ¨ ¨ ¨ v i ` and P i “ v w i . Since v has no neighbors in t v , v , . . . , v i ` u , G r V p P i q Y V p P i qs ` v i ` w i is a generalized ladder with two defin-ing paths P i and P i . Also, V p P i q Y V p P i q Ď A i ` and for all j Pt , , . . . , i ´ u , v j ` P V p P i q . Thus, the pair p P i , P i q is good.If w i has Type 2, then let P i “ v w v v ¨ ¨ ¨ v i ` and P i “ v w i .By the definition of a patched path, v is not adjacent to w . So, v has no neighbors in t w , v , v , . . . , v i ` u , and therefore G r V p P i q Y V p P i qs ` v i ` w i is a generalized ladder with two defining paths P i and P i . Clearly, V p P i q Y V p P i q Ď A i ` . Moreover, w P V p P i q and foreach j P t , . . . , i ´ u , v j ` P V p P i q . Therefore, the pair p P i , P i q isgood.Now, we may assume that w i has Type 1 or Type 3. Since L p w i q ě G has a good pair of paths P L p w i q , P L p w i q from t v , v u to t v L p w i q` , w L p w i q u .Suppose w i has Type 1 and therefore w i is adjacent to exactly oneof v L p w i q` and w L p w i q . Let t x, y u “ t v L p w i q` , w L p w i q u such that x is NAVOIDABLE VERTEX-MINORS 27 w i (Type 1) v i ` xy v L p w i q` v L p w i q` ¨ ¨ ¨ (a) P i P i x P t v L p w i q` , w L p w i q u y P t v L p w i q` , w L p w i q uzt x u w i (Type 3) v i ` xy v L p w i q` w L p w i q v L p w i q` ¨ ¨ ¨ (b) P i P i x P t v L p w i q , w L p w i q´ u Figure 7.
Constructing a generalized ladder in a fullypatched path. The vertex w i has Type 1 in (a) and hasType 3 in (b).adjacent to w i . We may assume that the paths P L p w i q and P L p w i q endat y and x , respectively. Let P i be a path P L p w i q ` yv L p w i q` v L p w i q` ¨ ¨ ¨ v i ` and let P i be a path P L p w i q ` xw i . See Figure 7. By the inductionhypothesis, V p P L p w i q q Y V p P L p w i q q Ď A L p w i q` Ď A i ` , and for each j Pt , , . . . , L p w i q ´ u , V p P L p w i q q Y V p P L p w i q q contains w j or v j ` . Thusit follows that V p P i q Y V p P i q Ď A i ` and for each j P t , , . . . , i ´ u , V p P i q Y V p P i q contains w j or v j ` .We claim that G r V p P i q Y V p P i qs ` v i ` w i is a generalized ladderwith the defining paths P i and P i . By the induction hypothesis, it isenough to show that there are no two crossing chords xa and w i b forsome a, b P V p P i q . Since w i has no neighbor in A L p w i q and w i and y are non-adjacent, b P X “ t v k : k P t L p w i q ` , L p w i q ` , . . . , i ` uu .Since x has no neighbor in X zt v L p w i q` u , we deduce that xa and w i b cannot cross and therefore G r V p P i q Y V p P i qs ` v i ` w i is a generalizedladder. This proves that if w i has Type 1, then p P i , P i q is a good pair. Finally, suppose that w i has Type 3 and so w i is adjacent to both v L p w i q and w L p w i q´ . By symmetry, we may assume that P L p w i q ends at v L p w i q` . Let x be the predecessor of v L p w i q` in P L p w i q . Since P L p w i q is on A L p w i q` and v L p w i q` has only two neighbors v L p w i q , w L p w i q´ in A L p w i q` , either x “ v L p w i q or x “ w L p w i q´ . Let y be the predecessor of w L p w i q in P L p w i q . Let P i be a path P L p w i q ` w L p w i q v L p w i q` v L p w i q` ¨ ¨ ¨ v i ` and let P i be a path obtained from P L p w i q by removing v L p w i q` andadding xw i . See Figure 7(b). It follows from our construction and theinduction hypothesis that V p P i q Y V p P i q Ď A i ` and V p P i q Y V p P i q contains w j or v j ` for each j P t , , . . . , i ´ u .We claim that G r V p P i q Y V p P i qs ` v i ` w i is a generalized ladderwith the defining paths P i and P i . By the induction hypothesis, it isenough to prove that there are no two chords xa and w i b such that a, b P V p P i q and b precedes a in P i . Suppose not. Since w i has noneighbor in A L p w i q´ , neighbors of w i in P i are in t y, w L p w i q u Y t v k : k P t L p w i q ` , L p w i q ` , . . . , i ` uu . Since x has no neighbor in t v k : k P t L p w i q ` , L p w i q ` , . . . , i ` uu , we deduce that a “ w L p w i q and b “ y . Since w i has no neighbor in A L p w i q´ , b is one of v L p w i q and w L p w i q´ other than x . Thus w L p w i q is adjacent to both v L p w i q and w L p w i q´ . This contradicts (iii) because v L p w i q` is also adjacent to both v L p w i q and w L p w i q´ and so G r V p P L p w i q q Y V p P L p w i q qs ` v L p w i q` w L p w i q isnot a generalized ladder. (cid:3) Lemma 6.8.
If a graph has a fully patched induced path of length n ,then it has a generalized ladder having at least n ` vertices as aninduced subgraph.Proof. Let P “ v v ¨ ¨ ¨ v n be the induced path of length n with an p n ´ q -patch Q “ w w ¨ ¨ ¨ w n ´ . Lemma 6.7 provides a good pair of paths P n ´ and P n ´ from t v , v u to t v n ´ , w n ´ u such that G r V p P n ´ q Y V p P n ´ qs ` v n ´ w n ´ is a generalized ladder and V p P n ´ q Y V p P n ´ q contains w j or v j ` for each j P t , , . . . , n ´ u . Since v n is onlyadjacent to v n ´ and w n ´ in G , G “ G r V p P n ´ q Y V p P n ´ q Y t v n us is a generalized ladder. Since v , v , v n , v n ´ , w n ´ P V p G q , G has atleast p n ´ q ` “ n ` (cid:3) Now we are ready to prove the main theorem of this section.
Lemma 6.9.
Let n ě . If a prime graph has an induced path of length n , then it has a cycle of length n ` as a vertex-minor. NAVOIDABLE VERTEX-MINORS 29
Proof.
Let G be a prime graph having an induced path of length110592 n . Suppose that G has no cycle of length 4 n ` N “ n . Then p p n q ´ qp N ´ q ´ ă n . Thus by Proposition 6.5, there exists a graph G locally equivalent to G having a fully patched induced path of length N . By Lemma 6.8, G must have a generalized ladder having at least N ` G has a cycleof length 4 n ` (cid:3) Proof of Theorem 6.1.
Let k “ t n { u . Let G be a prime graph having apath of length at least 6 . n . Then G has a path of length 6 . p k q “ k , and by Lemma 6.9, G has a cycle of length 4 k ` ě n as avertex-minor. (cid:3) Main Theorem
In this section, we prove the following.
Theorem 7.1.
For every n , there is N such that every prime graph onat least N vertices has a vertex-minor isomorphic to C n or K n a K n . By Theorem 6.1, it is enough to prove the following proposition.
Proposition 7.2.
For every c , there exists N such that every primegraph on at least N vertices has a vertex-minor isomorphic to either P c or K c a K c . Here is the proof of Theorem 7.1 assuming Proposition 7.2.
Proof of Theorem 7.1.
We take c “ r . n s and apply Proposition 7.2and Theorem 6.1. (cid:3) For integers h, w, ℓ ě
1, a p h, w, ℓ q -broom of a graph G is a connectedinduced subgraph H of G such that(i) H has an induced path P of length h from some vertex v calledthe center ,(ii) P z v is a component of H z v ,(iii) H z V p P q has w components, each having exactly ℓ vertices.The path P is called a handle of H and each component of H z V p P q iscalled a fiber of H . If H “ G , then we say that G is a p h, w, ℓ q -broom.We call h , w , ℓ the height , width , length , respectively, of a p h, w, ℓ q -broom. See Figure 8. Observe that v has one or more neighbors ineach fiber. centerheight h : “ number of edges in the handlewidth w : “ number of fiberslength ℓ : “ number of vertices in each fiber Figure 8. A p h, w, ℓ q -broom.Here is the rough sketch of the proof. If a prime graph G has novertex-minor isomorphic to P c or K c a K c and G has a broom havinghuge width as a vertex-minor, then it has a vertex-minor isomorphic toa broom with larger length and sufficiently big width. So, we increasethe length of a broom while keeping its width big. If we obtain a broomof big length by repeatedly applying this process, then we will obtaina broom of larger height. By growing the height, we will eventuallyobtain a long induced path.To start the process, we need an initial broom with sufficiently bigwidth. For that purpose, we use the following Ramsey-type theorem. Theorem 7.3 (folklore; see Diestel [8, Proposition 9.4.1]) . For positiveintegers c and t , there exists N “ g p c, t q such that every connectedgraph on at least N vertices must contain K t ` , K ,t , or P c as aninduced subgraph. By Theorem 7.3, if G is prime and | V p G q | ě g p c, t ` q , then either G has an induced subgraph isomorphic to P c or G has a vertex-minorisomorphic to K ,t ` . Since a p , t, q -broom is isomorphic to K ,t ` ,we conclude that every sufficiently large prime graph has a vertex-minor isomorphic to a p , t, q -broom, unless it has an induced subgraphisomorphic to P c .7.1. Increasing the length of a broom.
We now show that if aprime graph has a broom having sufficiently large width, we can find abroom having larger length after applying local complementation andshrinking the width.In the following proposition, we want to find a wide broom of length2 when we are given a sufficiently wide broom of length 1, when thegraph has no P c or K c a K c as a vertex-minor. NAVOIDABLE VERTEX-MINORS 31
Proposition 7.4.
For all integers c ě and t ě , there exists N “ g p c, t q such that for each h ě , every prime graph having a p h, N, q -broom has a vertex-minor isomorphic to a p h, t, q -broom, K c a K c , or P c . We will use the following theorem.
Theorem 7.5 (Ding, Oporowski, Oxley, Vertigan [10]) . For every pos-itive integer n , there exists N “ f p n q such that for every bipartite graph G with a bipartition p S, T q , if no two vertices in S have the same setof neighbors and | S | ě N , then S and T have n -element subsets S and T , respectively, such that G r S , T s is isomorphic to K n a K n , K n m K n ,or K n b K n .Proof of Proposition 7.4. Let N “ f p R p w, w qq where f is the functionin Theorem 7.5, and w “ max p t ` p c ´ qp c ´ q , c ´ q . Suppose that G has a p h, g p c, t q , q -broom H . Note that every fiber of H is a singlevertex.Let S be the union of the vertex sets of all fibers of H , and x be thecenter of H . Let N G p S qzt x u “ T . Since G is prime, no two verticesin G have the same set of neighbors, and so two distinct vertices in S have different sets of neighbors in T . Since | S | “ N “ f p R p w, w qq ,by Theorem 7.5, there exist S Ď S , T Ď T such that G r S , T s isisomorphic to K R p w,w q a K R p w,w q , K R p w,w q m K R p w,w q or K R p w,w q b K R p w,w q .Since | T | ě R p w, w q , by Ramsey’s theorem, there exist S Ď S and T Ď T such that G r S , T s is isomorphic to K w a K w , K w m K w ,or K w b K w , and T is a clique or a stable set in G . If G r S , T s isisomorphic to K w m K w or K w b K w , then by Lemmas 2.7 and 2.8, G has a vertex-minor isomorphic to either P w or K w ´ a K w ´ . Since w ě c ´ c ě
3, we have P c or K c a K c . Thus we may assumethat G r S , T s is isomorphic to K w a K w .If T is a clique in G , then we can remove the edges connecting T with x by applying local complementation at some vertices in S . Thus, wecan obtain a vertex-minor isomorphic to K w a K w from G r S Y T Yt x us by applying local complementation at x and deleting x . Therefore wemay assume that T is a stable set in G .We claim that each vertex y ‰ x in the handle of H is adjacent to atmost c vertices in T , or G has K c a K c as a vertex-minor. Suppose not.If y is a neighbor of x , then by pivoting an edge of G r S , T s , we candelete the edge xy . From there, we obtain a vertex-minor isomorphicto K c a K c by applying local complementation at x and y . If y is notadjacent to x , then we obtain a vertex-minor isomorphic to K c a K c x , y x , y x , y x , y Figure 9.
Dealing with 4-vertex graphs in Lemma 7.6.by deleting all vertices in the handle other than x and y , and applyinglocal complementation at x and y . This proves the claim.By deleting at most p c ´ q h vertices in T and their pairs in S , wecan assume that no vertex other than x in the handle has a neighborin T and this broom has width at least w ´ p c ´ q h . If h ` ě c ,then we have P c as an induced subgraph. Thus we may assume that h ď c ´
3. Since w ´ p c ´ q h ě w ´ p c ´ qp c ´ q ě t , we obtain avertex-minor isomorphic to a p h, t, q -broom. (cid:3) We now aim to increase the length of a broom when the broom haslength at least 2. For a fiber F of a broom H , we say that a vertex v P V p G qz V p H q blocks F if ρ ˚ G p V p F q , p V p H qz V p F qq Y t v uq ą . If G is prime and F has at least two vertices, then G has a blocking se-quence for p V p F q , V p H qz V p F qq by Proposition 5.2 and therefore thereexists a vertex v that blocks F because we can take the first vertex inthe blocking sequence. Lemma 7.6.
Let G be a graph and let x, y be two vertices such that ρ G pt x, y uq “ and G z x z y is connected. Then there exists some se-quence v , v , . . . , v n P V p G qzt x, y u of (not necessarily distinct) verticessuch that G ˚ v ˚ v ¨ ¨ ¨ ˚ v n has an induced path of length from x to y .Proof. We proceed by induction on | V p G q | . If | V p G q | “
4, then it iseasy to check all cases to obtain a path of length 3. To do so, first ob-serve that up to symmetry, there are 2 cases in G rt x, y u , V p G qzt x, y us ;either it is a matching of size 2 or a path of length 3. In both cases, onecan find a desired sequence of vertices to apply local complementation,see Figure 9 for all possible graphs on 4-vertices up to isomorphism.Now we may assume that G has at least 5 vertices. Let A “ N G p x qzp N G p y q Y t y uq , A “ N G p x q X N G p y q , and A “ N G p y qzp N G p x q Yt x uq . Clearly ρ G pt x, y uq “ A , A , A are nonempty.We say a vertex t in G z x z y deletable if G z x z y z t is connected. If thereis a deletable vertex not in A Y A Y A , then ρ G z t pt x, y uq “ A Y A Y A . NAVOIDABLE VERTEX-MINORS 33 If | A i | ą A i has a deletable vertex t for some i “ , ,
3, then ρ G z t pt x, y uq “ A i has a deletable vertex, then | A i | “ t , t , t in G z x z y , then we mayassume A i “ t t i u . However, ρ G z t pt x, y uq “ A , A arenonempty and therefore we obtain an induced path from x to y by theinduction hypothesis.Thus we may assume that G z x z y has at most 2 deletable vertices.So G z x z y has maximum degree at most 2 because otherwise we canchoose leaves of a spanning tree of G z x z y using all edges incident witha vertex of the maximum degree. If G z x z y is a cycle, then every vertexis deletable and so G z x z y is a path. Let w be a degree-2 vertex in G z x z y . Then G ˚ w has at least 3 deletable vertices and therefore wefind a desired sequence v , v , . . . , v n such that G ˚ w ˚ v ˚ v ¨ ¨ ¨ ˚ v n hasan induced path of length 3 from x to y . (cid:3) Lemma 7.7.
Let G be a graph and let x , y be two vertices in G , andlet F , F , . . . , F c be the components of G z x z y . If ρ ˚ G pt x, y u , F i q “ forall ď i ď c , then G has a vertex-minor isomorphic to K c a K c .Proof. We proceed by induction on | V p G q | ` | E p G q | .Suppose that G r V p F i q Y t x, y us is not an induced path of length 3from x to y . By Lemma 7.6, there exists a sequence v , v , . . . , v n P V p F i q such that G r V p F i q Y t x, y us ˚ v ˚ v ¨ ¨ ¨ ˚ v n has an induced pathof length 3 from x to y . If | V p F i q | ě
3, then we delete all vertices in F i not on this path and apply the induction hypothesis. If | V p F i q | “ | E p G r V p F i q Y t x, y usq | ą | E p G r V p F i q Y t x, y us ˚ v ˚ v ˚ ¨ ¨ ¨ ˚ v n q | because two vertices in F i are connected, G rt x, y u , V p F i qs has at leasttwo edges, and G r V p F i q Y t x, y us is not an induced path of length 3from x to y . So we apply the induction hypothesis to G ˚ v ˚ v ˚ ¨ ¨ ¨ ˚ v n to obtain a vertex-minor isomorphic to K c a K c .Therefore we may assume that G r V p F i q Y t x, y us is an induced pathof length 3 from x to y for all i . Thus G ˚ x ˚ y z x z y is indeed isomorphicto K c a K c . (cid:3) Lemma 7.8.
Let t be a positive integer, and G be a bipartite graphwith a bipartition p S, T q such that every vertex in T has degree at least . Then either S has a vertex of degree at least t ` or G has aninduced matching of size at least | T | { t .Proof. We claim that if every vertex in S has degree at most t , then G has an induced matching of size at least | T | { t . We proceed by inductionon | T | . This is trivial if | T | “
0. If 0 ă | T | ď t , then we can simply pick an edge to form an induced matching of size 1. So we may assumethat | T | ą t .We may assume that T has a vertex w of degree 1, because otherwisewe can delete a vertex in S and apply the induction hypothesis. Let v be the unique neighbor of w . By the induction hypothesis, G z v z N G p v q has an induced matching M of size at least p | T | ´ t q{ t . Now M Y t vw u is a desired induced matching. (cid:3) Lemma 7.9.
Let H be a broom in a graph G having n fibers F , F , . . . , F n given with n vertices v , v , . . . , v n in V p G qz V p H q such that(1) v i blocks F j if and only if i “ j ,(2) v i has a neighbor in F j if and only if i ď j .If n ě R p c ` , c ` q , then G has a vertex-minor isomorphic to P c .Proof. If j ą i , then v i has a neighbor in F j , but v i does not block F j .Therefore, v i is adjacent to every vertex in V p F j q X N H p x q for j ą i .Since n ě R p c ` , c ` q , there exist 1 ď t ă t ¨ ¨ ¨ ă t c ` ď n suchthat t v t , v t , . . . , v t c ` u is a clique or a stable set of G . For 1 ď i ď c ` w i be a vertex in V p F t i q X N H p x q . Clearly, G rt v t , v t , . . . , v t r c { s ´ u , t w , w , . . . , w r c { s us is isomorphic to K r c { s m K r c { s .By Lemma 2.8, K r c { s m K r c { s or K r c { s m K r c { s has a vertex-minorisomorphic to P c . (cid:3) Lemma 7.10.
Let H be a broom in a graph G having n fibers F , F , . . . , F n .Let v , v , . . . , v n be vertices in V p G qz V p H q such that(1) v i blocks F j if and only if i “ j ,(2) v i has a neighbor in F j for all i and j .If n ě R p c ` , c ` q , then G has a vertex-minor isomorphic to K c a K c .Proof. If i ‰ j , then v j does not block F i and therefore N G p v j q X V p F i q “ N G p x q X V p F i q . Since n ě R p c ` , c ` q , there exist 1 ď t ă t ¨ ¨ ¨ ă t c ` ď n such that t v t , v t , . . . , v t c ` u is a clique or a stable setof G .We claim that for each 1 ď i ď c `
2, there exist a sequence w p i q , w p i q , . . . , w p i q k i of k i ě V p F t i qzp N G p x q Y N G p v t i qq and z i P V p F t i q such that z i is not adjacent to v t i in G ˚ w p i q ˚ w p i q ˚ ¨ ¨ ¨ ˚ w p i q k i but z i is adjacent to v t j in G ˚ w p i q ˚ w p i q ˚ ¨ ¨ ¨ ˚ w p i q k i for all j ‰ i .Let A p i q “ p N G p v t i qz N G p x qq X V p F t i q , A p i q “ p N G p v t i q X N G p x qq X V p F t i q and A p i q “ p N G p x qz N G p v t i qq X V p F t i q . NAVOIDABLE VERTEX-MINORS 35 If A p i q ‰ H , then a vertex z i in A p i q satisfies the claim. So wemay assume A p i q is empty. Then A p i q ‰ H and A p i q ‰ H , otherwise ρ ˚ G pt v t i , v t j u , V p F t i qq ď j ‰ i because N G p v t j q X V p F t i q “ N G p x q X V p F t i q . We choose a p i q P A p i q and a p i q P A p i q so that thedistance from a p i q to a p i q in F i is minimum.Let P i be a shortest path from a p i q to a p i q in F t i . Note that eachinternal vertex of P i is not contained in A p i q Y A p i q . After applyinglocal complementation at all internal vertices of P i , a p i q is adjacent to a p i q and v t i , and non-adjacent to v t j for all j ‰ i . So by applying onemore local complementation at a p i q if necessary, we can delete the edgesbetween a p i q and v t j for all j ‰ i . And then, z i “ a p i q satisfies the claim.Now, take G “ G ˚ w p q ˚¨ ¨ ¨˚ w p q k ˚ w p q ˚¨ ¨ ¨˚ w p q k ¨ ¨ ¨˚ w p c ` q ˚¨ ¨ ¨˚ w p c ` q k c ` .Since each w p i q k has no neighbors in t v t , v t , . . . , v t c ` u in G , applyinglocal complementation at w p i q k does not change the adjacency betweenany two vertices in t v t , v t , . . . , v t c ` u . Thus the induced subgraph of G on t z , z , . . . , z c ` uYt v t , v t , . . . , v t c ` u is isomorphic to K c ` b K c ` or K c ` b K c ` , and by Lemma 2.7, G has a vertex-minor isomorphicto K c a K c . (cid:3) Lemma 7.11.
Let H be a p h, n, ℓ q -broom in a graph G having n fibers F , F , . . . , F n given with n vertices v , v , . . . , v n in V p G qz V p H q suchthat(1) v i blocks F j if and only if i “ j ,(2) if i ‰ j , then v i has no neighbor in F j .If n ě R p t ` p c ´ qp c ´ q , c q , then G has a vertex-minor isomorphicto P c , K c a K c , or a p h, t, ℓ ` q -broom.Proof. Since n ě R p t ` p c ´ qp c ´ q , c q , there exist 1 ď t ă t ¨ ¨ ¨ ă t k ď n such that either(1) k “ c and t v t , v t , . . . , v t k u is a clique in G , or(2) k “ t ` p c ´ qp c ´ q and t v t , v t , . . . , v t k u is a stable set in G .First, we assume that k “ c and t v t , v t , . . . , v t k u is a clique. Foreach t i , since ρ ˚ G pt x, v t i u , V p F t i qq ě
2, by Lemma 7.6, there exists somesequence w , w , . . . , w n P V p F t i q of (not necessarily distinct) verticessuch that G r V p F t i q Y t x, v t i us ˚ w ˚ w ¨ ¨ ¨ ˚ w n has an induced path oflength 2 from v t i to x . By applying local complementation at x , wehave a vertex-minor isomorphic to K c a K c .Now, suppose that k “ t ` p c ´ qp c ´ q and t v t , v t , . . . , v t k u is a stable set in G . Let P be the handle of H . If h ` ě c , then we have P c as an induced subgraph. Thus we may assume that h ď c ´
3. We assume that a vertex y P V p P qzt x u adjacent to c ver-tices in t v , v , . . . , v k u . Then since ρ ˚ G pt x, y uq , V p F i q Y t v t i uq “ i , by Lemma 7.7, we have a vertex-minor isomorphic to K c a K c .Thus, every vertex in the handle other than x cannot have more than c ´ t v t , v t , . . . , v t k u . By deleting at most p c ´ q h ver-tices in t v t , v t , . . . , v t k u , we can remove all edges from V p P qzt x u to t v t , v t , . . . , v t k u . Since k ´ p c ´ q h ě k ´ p c ´ qp c ´ q ě t, we have a vertex-minor isomorphic to a p h, t, ℓ ` q -broom. (cid:3) Proposition 7.12.
For positive integers c and t , there exists N “ g p c, t q such that for all integers ℓ ě and h ě , every prime graphhaving a p h, N, ℓ q -broom has a vertex-minor isomorphic to a p h, t, ℓ ` q -broom, P c , or K c a K c .Proof. Let N “ g p c, t q “ p c ´ q m , where m “ R p m , m , m , m q , m “ R p t ` p c ´ qp c ´ q , c q , and m “ R p c ` , c ` q . Let H bea p h, N, ℓ q -broom of G . If a vertex w in V p G qz V p H q blocks c fibersof H , then for each fiber F of them, ρ ˚ G pt w, x u , V p F qq “
2. So byLemma 7.7, G has a vertex-minor isomorphic to K c a K c . Thus, avertex in V p G qz V p H q can block at most c ´ H .For each fiber F of H , there is a vertex v P V p G qz V p H q that blocks F because G is prime. Thus, by Lemma 7.8, there are g p c, t q{p c ´ q “ m vertices v , v , . . . , v m in V p G qz V p H q and fibers F , F , . . . , F m of H such that for 1 ď i, j ď m , v i blocks F j if and only if i “ j . For i ‰ j , either v i has no neighbor in F j or v i has a neighbor in F j but ρ ˚ G pt v i , x u , V p F j qq “ V p K m q “ t , , . . . , m u . We color the edges of K m such that an edge t i, j u is ‚ green if N G p v i q X V p F j q ‰ H and N G p v j q X V p F i q ‰ H , ‚ red if N G p v i q X V p F j q ‰ H and N G p v j q X V p F i q “ H , ‚ yellow if N G p v i q X V p F j q “ H and N G p v j q X V p F i q ‰ H , ‚ blue if N G p v i q X V p F j q “ N G p v j q X V p F i q “ H .Since | V p K m q | “ m “ R p m , m , m , m q , by Ramsey’s theorem, either K m has a green clique of size m , or K m has a monochromatic cliqueof size m which is red, yellow, or blue.If K m has a red clique C of size m , then for i, j P C , v i has aneighbor in F j if and only if i ď j . Since m ě R p c ` , c ` q , byLemma 7.9, G has a vertex-minor isomorphic to P c .Similarly, if K m has a yellow clique C of size m , by Lemma 7.9, G has a vertex-minor isomorphic to P c . NAVOIDABLE VERTEX-MINORS 37 If K m has a blue clique C of size m , then for distinct i, j P C , v i has a neighbor in F j . Since m “ R p c ` , c ` q , by Lemma 7.10, G has a vertex-minor isomorphic to K c a K c .If K m has a green clique C of size m , then for distinct i, j P C , v i has no neighbor in F j . Since m “ R p t ` p c ´ qp c ´ q , c q , byLemma 7.11, G has a vertex-minor isomorphic to P c , K c a K c , or a p h, t, ℓ ` q -broom. (cid:3) Increasing the height of a broom.Proposition 7.13.
For positive integers c , t , there exists N “ g p c, t q such that for h ě , every prime graph having a p h, , N q -broom has avertex-minor isomorphic to a p h ` , t, q -broom or P c .Proof. Let N “ g p c, t q “ g p c, t q where g is given in Theorem 7.3.Suppose that G has a p h, , N q -broom H and let x be the center of H .Let F be the fiber of H .Since F is connected, by Theorem 7.3, F has an induced subgraphisomorphic to P c , or F has a vertex-minor isomorphic to K t ` . Wemay assume that F has an induced subgraph F isomorphic to K t ` .Let P “ p p . . . p m be a shortest path from p “ x to F in H . Notethat m ě p m ´ is adjacent to at least one vertices of F . Let S “ N H p p m ´ q X V p F q .We claim that there exists a vertex v P V p F q such that p G ˚ v qr V p F qYt x us has an induced path of length at least m ´ x , and the lastvertex of the path has t neighbors in F which form a stable set in G .If | S | ď t , then choose p m ` P V p F qz S and we delete S z p m from F . And by applying local complementation at p m ` , we obtain a pathfrom x to p m ` such that p m ` has t neighbors in F which form astable set.If | S | ě t `
1, then by applying local complementation at p m , weobtain a path from x to p m such that p m has t neighbors in F whichform a stable set. Thus, we prove the claim.Since m ě
2, the union of the handle of H and the path in theclaim form a path of length at least h `
1, and the last vertex of thepath has t neighbors which form a stable set in F . Therefore, G has avertex-minor isomorphic to a p h ` , t, q -broom. (cid:3) Proposition 7.14.
For positive integers c , t , there exists N “ g p c, t q such that for all h ě , every prime graph having a p h, N, q -broom hasa vertex-minor isomorphic to a p h ` , t, q -broom, P c , or K c a K c . Proof.
By Proposition 7.13, there exists N depending only on c and t such that every prime graph having a p h, , N q -broom has a vertex-minor isomorphic to a p h ` , t, q -broom or P c . By applying Proposi-tion 7.12 p N ´ q times, we deduce that there exists N such that everyprime graph having a p h, N , q -broom has a vertex-minor isomorphicto a p h, , N q -broom, P c , or K c a K c . By Proposition 7.4, there ex-ists N such that every prime graph having a p h, N, q -broom has avertex-minor isomorphic to a p h, N , q -broom, P c , or K c a K c . (cid:3) We are now ready with all necessary lemmas to prove Proposition 7.2.
Proof of Proposition 7.2.
By Theorem 1.1, every prime graph on atleast 5 vertices has a vertex-minor isomorphic to C and P is a vertex-minor of C . Therefore we may assume that c ě p c ´ q times, we deduce that thereexists a big integer t depending only on c such that every prime graph G with a p , t, q -broom has a vertex-minor isomorphic to a p c ´ , , q -broom, P c , or K c a K c . Since a p c ´ , , q -broom is isomorphic to P c and a p , t, q -broom is isomorphic to K ,t ` , we conclude that everyprime graph having a vertex-minor isomorphic to K ,t ` has a vertex-minor isomorphic to P c or K c a K c . By Theorem 3.1, there exists N such that every connected graph on at least N vertices has a vertex-minor isomorphic to K ,t ` . This completes the proof. (cid:3) Why optimal?
Our main theorem (Theorem 7.1) states that sufficiently large primegraphs must have a vertex-minor isomorphic to C n or K n a K n . Butdo we really need these two graphs? To justify why we need both, weshould show that for some n , C n is not a vertex-minor of K N a K N for all N and similarly K n a K n is not a vertex-minor of C N for all N ,because C n and K n a K n are also prime. Proposition 8.1.
Let n be a positive integer.(1) K a K is not a vertex-minor of C n .(2) C is not a vertex-minor of K n a K n . Since C is a vertex-minor of C n for all n ě
7, the above propositionimplies that C n is not a vertex-minor of K N a K N when n ě
7. Similarly K n a K n is not a vertex-minor of C N for all n ě Lemma 8.2.
If a prime graph H on at least vertices is a vertex-minorof C n , then H is locally equivalent to a cycle graph. NAVOIDABLE VERTEX-MINORS 39 a v w bH J a v w bz Figure 10.
The graphs H and J . Proof.
We proceed by induction on n . If n “
5, then it is trivial. Letus assume n ą
5. Suppose | V p H q | ă | V p C n q | . By Lemma 2.1, H is avertex-minor of C n z v , C n ˚ v z v , or C n ^ vw z v for a neighbor w of v .If H is vertex-minor of C n ˚ v z v , then we can apply the inductionhypothesis because C n ˚ v z v is isomorphic to C n ´ .By Lemma 2.6, H cannot be a vertex-minor of C n z v because C n z v has no prime induced subgraph on at least 5 vertices.Thus we may assume that H is a vertex-minor of C n ^ vw z v for aneighbor w of v . Again, by Lemma 2.6, H is isomorphic to a vertex-minor of C n ´ . (cid:3) Classifying prime vertex-minors of K n a K n turns out to be moretedious. Instead of identifying prime vertex-minors of K n a K n , wefocus on characterizing prime vertex-minors on 7 vertices to prove (2)of Proposition 8.1.Instead of K n a K n , we will first consider H n . Let H n be the graphhaving two specified vertices called roots and n internally disjoint pathsof length 3 joining the roots. Let J n be the graph obtained from H n byadding a common neighbor of two roots. Then H n has 2 n ` J n has 2 n ` Lemma 8.3.
Let H be a prime vertex-minor of H n on at least ver-tices. If | V p H n q | ´ | V p H q | ě , then J n ´ has a vertex-minor isomor-phic to H .Proof. We may assume n ě
3. Since at most 2 vertices of H n havedegree other than 2, there exists v P V p H n qz V p H q of degree 2 in H n .Let w be the neighbor of v having degree 2 in H n . Let av w b be a pathof length 3 from a to b in H n such that t v, w u ‰ t v , w u . By Lemma 2.1, H is a vertex-minor of either H n z v , H n ˚ v z v or H n ^ vw z v . If H is a vertex-minor of H n ˚ v z v , then H is isomorphicto a vertex-minor of J n ´ , because H n ˚ v z v is isomorphic to J n ´ .Since w has degree 1 in H n z v , by Lemma 2.6, if H is a vertex-minorof H n z v , then H is isomorphic to a vertex-minor of H n z v z w . Since H n z v z w is isomorphic to H n ´ and H n ´ is an induced subgraph of J n ´ , H is isomorphic to a vertex-minor of J n ´ .Similarly, if H is a vertex-minor of H n ^ vw z v , then H is isomorphicto a vertex-minor of H n ^ vw z v z w . Clearly, p H n ^ vw z v z w q ^ v w isisomorphic to H n ´ . Since H n ´ is an induced subgraph of J n ´ , H isisomorphic to a vertex-minor of J n ´ , as required. (cid:3) Lemma 8.4.
Let H be a prime vertex-minor of J n on at least vertices.If | V p J n q | ´ | V p H q | ě , then H n has a vertex-minor isomorphic to H .Proof. We may assume n ě
2. Let a, b be the roots of J n , azb be thepath of length 2, and avwb be a path of length 3 from a to b .Case 1: Suppose that V p J n qz V p H q has a degree-2 vertex on a path oflength 3 from a to b . We may assume that it is v by symmetry. ByLemma 2.1, H is a vertex-minor of J n z v , J n ˚ v z v , or J n ^ vw z v .If H is a vertex-minor of J n z v , then H is isomorphic to a vertex-minorof J n z v z w by Lemma 2.6, because w has degree 1 in J n z v . Similarly,if H is a vertex-minor of J n ^ vw z v , then H is isomorphic to a vertex-minor of J n ^ vw z v z w . Clearly, J n z v z w and p J n ^ vw z v z w q ˚ z areisomorphic to J n ´ , and J n ´ is a vertex-minor of H n .If H is a vertex-minor of J n ˚ v z v , then by Lemma 2.6, H is isomorphicto a vertex-minor of J n ˚ v z v z w , which is isomorphic to J n ´ , because w and z have the same set of neighbors in J n ˚ v z v . Since J n ´ is avertex-minor of H n , H is isomorphic to a vertex-minor of H n . Thisproves the lemma in Case 1.Case 2: Suppose that z P V p J n qz V p H q . Then by Lemma 2.1, H is avertex-minor of J n z z , J n ˚ z z z , or J n ^ az z z . Since J n z z and p J n ˚ z z z q ^ vw are isomorphic to H n , we may assume that H is a vertex-minor of J n ^ az z z . However, J n ^ az z z has no prime induced subgraph on atleast 5 vertices and therefore by Lemma 2.6, H cannot be a vertex-minor of J n ^ az z z , contradicting our assumption.Case 3: Suppose that a or b is contained in V p J n qz V p H q . By symmetry,let us assume a P V p J n qz V p H q . By Lemma 2.1, H is a vertex-minor of J n z a , J n ˚ a z a , or J n ^ az z a .Since J n z a has no prime induced subgraph on at least 5 vertices, H cannot be a vertex-minor of J n z a by Lemma 2.6. NAVOIDABLE VERTEX-MINORS 41 F F F Figure 11.
Graphs F , F and F .Suppose H is a vertex-minor of J n ^ az z a . By the definition ofpivoting, b is adjacent to all vertices of N J n p a qzt z u in J n ^ az z a . Wecan remove all these edges between b and N J n p a qzt z u by applying localcomplementation on all vertices of N J n p b qzt z u in J n ^ az z a . Thus, H n islocally equivalent to J n ^ az z a , and H is isomorphic to a vertex-minorof H n .Now suppose that H is a vertex-minor of J n ˚ a z a . By the definitionof local complementation, N J n p a q forms a clique in J n ˚ a z a . So, b isadjacent to all vertices of N J n p a qzt z u in p J n ˚ a z a q ˚ z . Similarly inthe above case, by applying local complementation on all vertices of N J n p b qzt z u in p J n ˚ a z a q ˚ z , we can remove all edges between b and N J n p a qzt z u in p J n ˚ a z a q ˚ z . Finally, by pivoting vw , we can removethe edge bz , and therefore, J n ˚ a z a is locally equivalent to H n . Thus, H is isomorphic to a vertex-minor of H n . (cid:3) Let F , F , F be the graphs in Figure 11. Lemma 8.5.
Let n ě be an integer. If a prime graph H is a vertex-minor of H n and | V p H q | “ , then H is locally equivalent to F , F ,or F .Proof. We proceed by induction on n . If n “
3, then let H be a prime 7-vertex vertex-minor of H . Let axyb be a path from a root a to the otherroot b in H . By symmetry, we may assume that V p H qz V p H q “ t x u or t a u . By Lemma 2.1, H is locally equivalent to H z x , H ˚ x z x , H ^ xa z x , H z a , H ˚ a z a , or H ^ ab z a . The conclusion follows because H z x , H ^ xy z x , H z a are not prime and H ˚ x z x , H ^ ax z a , and H ˚ a z a are isomorphic to F , F , and F , respectively.Suppose n ą
3. By Lemma 8.3, every 7-vertex prime vertex-minoris also isomorphic to a vertex-minor of J n ´ . By Lemma 8.4, it isisomorphic to a vertex-minor of H n ´ . The conclusion follows from theinduction hypothesis. (cid:3) Lemma 8.6.
The graphs F , F , F are not locally equivalent to C .Proof. Suppose that F i is locally equivalent to C . Then ρ F i p X q “ ρ C p X q for all X Ď V p C q by Lemma 2.3. Let x be the vertex in the C F F F Figure 12.
List of all 3-vertex sets having cut-rank 2containing a fixed vertex x denoted by a square.center of F i , see Figure 12. By symmetry of C , we may assume that x is mapped to a particular vertex in C . Figure 12 presents all vertexsubsets of size 3 having cut-rank 2 and containing x in graphs C , F , F , F . It is now easy to deduce that no bijection on the vertex set willmap these subsets correctly. (cid:3) We are now ready to prove Proposition 8.1.
Proof of Proposition 8.1. (1) By Lemma 8.2, it is enough to check that K a K is not locally equivalent to C . This can be checked easily.(2) By applying local complementation at roots, we can easily see that H n has a vertex-minor isomorphic to K n a K n . Lemma 8.5 states thatall 7-vertex prime vertex-minors of H n are F , F , and F . Lemma 8.6proves that none of them are locally equivalent to C . Thus H n has novertex-minor isomorphic to C and therefore K n a K n has no vertex-minor isomorphic to C . (cid:3) Discussions
Vertex-minor ideals.
A set I of graphs is called a vertex-minorideal if for all G P I , all graphs isomorphic to a vertex-minor of G are also contained in I . We can interpret theorems in this paper interms of vertex-minor ideals as follows. This formulation allows us toappreciate why these theorems are optimal. Corollary 9.1.
Let I be a vertex-minor ideal. Theorem 3.1:
Graphs in I have bounded number of vertices ifand only if t K n : n ě u Ę I . Theorem 3.1:
Connected graphs in I have bounded number ofvertices if and only if t K n : n ě u Ę I . NAVOIDABLE VERTEX-MINORS 43
Theorem 3.1:
Graphs in I have bounded number of edges if andonly if t K n : n ě u Ę I and t K n a K n : n ě u Ę I . Theorem 7.1:
Prime graphs in I have bounded number of ver-tices if and only if t C n : n ě u Ę I and t K n a K n : n ě u Ę I . Rough structure.
We can also regard Theorem 7.1 as a roughstructure theorem on graphs having no vertex-minor isomorphic to C n or K n a K n as follows. The 1 -join of two graphs G , G with twospecified vertices v P V p G q , v P V p G q is the graph obtained bymaking the disjoint union of G z v and G z v and adding edges to joinneighbors of v in G with neighbors of v in G . Corollary 9.2.
For each n , there exists N such that every graph havingno vertex-minor isomorphic to C n or K n a K n can be built from graphson at most N vertices by repeatedly taking -join operation. Acknowledgment
This research was done while the authors were visiting Universityof Hamburg. The authors would like to thank Reinhard Diestel forhosting them.
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