Induced subgraphs and tree decompositions V. one neighbor in a hole

What are the unavoidable induced subgraphs of graphs with large treewidth? It is well‐known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the “basic treewidth obstructions”). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered this question in the negative. Their counterexamples, the so‐called “layered wheels,” contain wheels, where a wheel consists of a hole (i.e., an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out to be false due to Davies' recent example of graphs with large treewidth, no wheels and no basic treewidth obstructions as induced subgraphs. However, in Davies' example there exist holes and vertices (outside of the hole) with two neighbors in them. Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in graphs with large treewidth and no basic obstruction. Our main result is that graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.


Introduction
All graphs in this paper are finite and simple.Let H and G be graphs.We say G contains H if G has an induced subgraph isomorphic to H (unless stated otherwise).We say that G is H-free if G does not contain H.For a family of graphs H, we say that G is H-free if G is H-free for every H ∈ H.A tree decomposition (T, χ) of G consists of a tree T and a map χ : V (T ) → 2 V (G) such that the following hold: (i) For every vertex v ∈ V (G), there exists t ∈ V (T ) such that v ∈ χ(t).
Treewidth is an extensively-studied graph parameter, mostly due to the fact that graphs of bounded treewidth exhibit interesting structural [15] and algorithmic [9] properties.It is thus of interest to understand the unavoidable substructures emerging in graphs of large treewidth (these are often referred to as "obstructions to bounded treewidth").For instance, for each k, the (k × k)-wall, denoted by W k×k , is a planar graph with maximum degree three and with treewidth k (see Figure 1; a precise definition can be found in [3]).Every subdivision of W k×k is also a graph of treewidth k.The unavoidable subgraphs of graphs with large treewidth are fully characterized by the Grid Theorem of Robertson and Seymour, the following.

Theorem 1.1 ([14]
).There is a function f : N → N such that every graph of treewidth at least f (k) contains a subdivision of W k×k as a subgraph.Following the same line of thought, our motivation in this series is to study induced subgraph obstructions to bounded treewidth.In addition to subdivided walls mentioned above, complete graphs and complete bipartite graphs are easily observed to have arbitrarily large treewidth: the complete graph K t+1 and the complete bipartite graph K t,t both have treewidth t.Line graphs of subdivided walls form another family of graphs with unbounded treewidth, where the line graph L(F ) of a graph F is the graph with vertex set E(F ), such that two vertices of L(F ) are adjacent if the corresponding edges of G share an end.
We call a family H of graphs useful if there exists an integer c(H) such that every H-free graph has treewidth at most c(H).The discussion above can be summarized as follows: Theorem 1.2.If H is a useful family of graphs, then there exists an integer t such that H contains K t , K t,t , an induced subgraph of each subdivision of W t×t and an induced subgraph of the line graph of each subdivision of W t×t .
The following was conjectured in [1] and proved in [13]: Theorem 1.3.[13] For all k, ∆ > 0, there exists c = c(k, ∆) such that every graph with maximum degree at most ∆ and treewidth at least c contains a subdivision of W k×k or the line graph of a subdivision of W k×k as an induced subgraph.
The bounded-degree condition of Theorem 1.3 implies that K ∆+2 and K ∆+1,∆+1 are excluded.However, Theorem 1.3 does not hold if "bounded degree" is replaced by excluding K ∆+2 and K ∆+1,∆+1 , as is evidenced by the constructions of [11] and [16].Thus a natural question arises: what can replace this condition?Let us call a family F of graphs helpful if the following holds: for all t > 0, there exists c = c(t) such that every F-free graph with treewidth more than c contains K t , K t,t , a subdivision of W t×t or the line graph of a subdivision of W t×t .
A hole in a graph is an induced cycle of length at least four.The length of a hole is the number of vertices in it.A wheel is a graph consisting of a hole C and a vertex v with at least three neighbors in C (in the literature, sometimes further restrictions are placed on the location of the neighbors of v in C).In view of the prevalence of wheels in the construction of [16], one might ask if the family of all wheels is helpful.The answer to this question is negative, because of the construction of [11] (see Figure 2 for an example; we omit the precise definition), but the Figure 2. A wheel-free graph with large treewidth [11] following weaker statement is true, and it is the main result of this paper.Let T 1 be the family of all graphs consisting of a hole C and a vertex outside of C with at least two neighbors in C. The class of T 1 -free graphs was studied in [2]; in Section 6 we strengthen their results.A crucial difference between Theorem 6.3 and [2] is that in [2] only the existence of certain cutsets is shown, while we are able to guarantee that every heavy seagull is broken by a cutset of the required type (see Section 6 for details).
Our main result in this paper is the following: Theorem 1.4.The family T 1 is helpful.
In fact, we prove something stronger.In the following, the length of a path is its number of edges.A pyramid is a graph consisting of a vertex a, a triangle {b 1 , b 2 , b 3 }, and three paths P i from a to b i for 1 ≤ i ≤ 3 of length at least one, such that for i = j the only edge between P i \ {a} and P j \ {a} is b i b j , and at most one of P 1 , P 2 , P 3 has length exactly one.
A prism is a graph consisting of two triangles {a 1 , a 2 , a 3 } and {b 1 , b 2 , b 3 }, and three paths P i from a i to b i for 1 ≤ i ≤ 3, all of length at least one, and such that for i = j the only edges between P i and P j are a i a j and b i b j .
Let T 2 be the family of all graphs consisting of a hole C and a vertex outside of C with at least two non-adjacent neighbors in C, together with all prisms and all pyramids.Note that each graph in T 2 contains a graph in T 1 (so the class of T 1 -free graphs is properly contained in the class of T 2 -free graphs).We prove: Theorem 1.5.The family T 2 is helpful.
Let us next restate Theorem 1.5 more explicitly.A graph G is sparse if for every hole H of G and vertex v ∈ H, there is an edge ab of H such that N (v) ∩ H ⊆ {a, b}.A graph is very sparse if it is sparse and also (pyramid, prism)-free (thus a graph is very sparse if and only if it is T 2 -free).It follows that if G is very sparse, then G does not contain K 3,3 or the line graph of a subdivision of W 3×3 .Let F be the family of all very sparse graphs, and let F t be the family of all very sparse graphs with no clique of size at least t + 1.
We prove: Theorem 1.6.For all t > 0, there exists c = c(t) such that every graph in F t with treewidth more than c contains a subdivision of W t×t (as an induced subgraph).
The rough outline of the proof of Theorem 1.6 is as follows.Our first step is to show that if a graph in F t contains a triangle, then it admits a clique cutset.Thus it is enough to prove the result for graphs in F 2 .Now let G ∈ F 2 .A heavy seagull in G is an induced three-vertex path both of whose ends have degree at least three in G. First we prove that every heavy seagull of G is "broken" by a two-clique-separation (this means that for every heavy seagull H of G, there exist two cliques K 1 , K 2 ∈ G such that no component of G \ (K 1 ∪ K 2 ) contains H).Now the idea is to use the central bag method, developed in earlier papers in this series [3,5,6,7], to identify an induced subgraph β of G that contains no heavy seagull, and such that the treewidth of G is not much larger than the treewidth of β.The key difference between our situation here and those in the earlier papers is that the cutsets we use to break the heavy seagulls are not connected, a property that was crucial in the earlier proofs.To deal with this difficulty, we change the definition of a central bag, including in it a path between the two cliques of the cutset whose interior is in G \ β (this is in the spirit of, but different from, "marker paths" for 2-joins).We then modify the previously known central bag tools to work in this new setting.By "breaking" heavy seagulls, we arrange that in β, vertices of degree at least three appear in components of bounded size.This in turn allows us to bound the treewidth of β, and theorem follows.
1.1.Definitions and notation.Let G be a graph.For X ⊆ V (G), we denote by G[X] the induced subgraph of G with vertex set X, and In this paper we use the set X and the subgraph G every vertex of X is adjacent to every vertex of Y , and X is anticomplete to Y if there are no edges between X and Y .We use X ∪ v to mean X ∪ {v}, and X \ v to mean X \ {v}.
Given a graph G, a path in G is an induced subgraph of G that is a path.If P is a path in G, we write P = p 1 -. . .-p k to mean that p i is adjacent to p j if and only if |i − j| = 1.We call the vertices p 1 and p k the ends of P , and say that P is from p 1 to p k .The interior of P , denoted by P * , is the set P \ {p 1 , p k }.The length of a path P is the number of edges in P .
A theta is a graph T containing two vertices a, b and three paths P 1 , P 2 , P 3 from a to b of length at least two, such that P 1 \ {a, b}, P 2 \ {a, b}, P 3 \ {a, b} are pairwise disjoint and anticomplete to each other.We call a, b the ends of T .1.2.Organization of the paper.This paper is organized as follows.In Section 2, we give general background and definitions related to separations in graphs; we also discuss connections between different kinds of separations in the special case of sparse graphs.In Section 3, we reduce Theorem 1.6 to the case of triangle-free sparse graphs.In Section 4, we discuss balanced separators in graphs, and develop our main tool, Theorem 4.5, which allows us to use the central bag method.In Section 5, we prove results about two-clique-separations, which are the cutsets that will be used to form the central bag.In Section 6, we prove structural results that allow us to break every heavy seagull in a triangle-free sparse graph and produce a central bag that contains no heavy seagulls.In Section 7, we use the tools of Section 4 to prove our main result for graphs in F 2 .Finally, in Section 8, we prove Theorem 1.6.

Separations
A clique in a graph is a (possibly empty) set of pairwise adjacent vertices.We say that G admits a clique cutset if there is a cutset of G that is a clique (in particular every disconnected graph admits a clique cutset).A separation We say that G admits a star cutset if there is a proper star separation in G.
First we observe: Lemma 2.1.Let G be a sparse graph and (A, C, B) be a separation of G with A = ∅ and B = ∅.Suppose that there exist 2 ).Let P 1 be a path from x to y with P * 1 ⊆ D 1 and let P 2 be a path from x to y with Lemma 7 from [8] shows that clique cutsets do not affect treewidth.Now, by Lemma 2.1, it follows that in order to prove Theorem 1.6 it is enough to prove the following: Theorem 2.2.For all t > 0, there exists c = c(t) such that every graph in F t with treewidth more than c and with no star cutset contains a subdivision of W t×t as an induced subgraph.

Reducing to the triangle-free case
In this section we show how to deduce Theorem 1.6 from the special case of triangle-free graphs.A diamond is the graph obtained from K 4 by removing an edge.Lemma 3.1.Let G be a sparse graph and assume that G does not admit a star cutset.Then G is diamond-free.
Proof.Suppose first {a, b, c, d} is a diamond in G.We may assume that the pair ac is nonadjacent.Since b is not the center of a star cutset in G, it follows that there exists is a path from a to c with no neighbor of b in its interior.Let P be such a path.Then d is not a vertex of P , since d is adjacent to b.Moreover, a-P -c-b-a is a hole, and d has three neighbors in it, namely a, b and c, a contradiction.This proves that G is diamond-free.
We also need the following folklore result that appeared in [4]:

and that V (H) is minimal subject to inclusion. Then, one of the following holds:
(i) For some distinct i, j, k ∈ {1, 2, 3}, there exists P that is either a path from x i to x j or a hole containing the edge x i x j such that • V (H) = V (P ) \ {x i , x j }, and • either x k has two non-adjacent neighbors in H or x k has exactly two neighbors in H and its neighbors in H are adjacent.(ii) There exists a vertex a ∈ V (H) and three paths P 1 , P 2 , P 3 , where • the sets V (P 1 ) \ {a}, V (P 2 ) \ {a} and V (P 3 ) \ {a} are pairwise disjoint, and • for distinct i, j ∈ {1, 2, 3}, there are no edges between V (P i ) \ {a} and V (P j ) \ {a}, except possibly x i x j .
(iii) There exists a triangle a 1 a 2 a 3 in H and three paths P 1 , P 2 , P 3 , where P i is from • the sets V (P 1 ), V (P 2 ) and V (P 3 ) are pairwise disjoint, and • for distinct i, j ∈ {1, 2, 3}, there are no edges between V (P i ) and V (P j ), except a i a j and possibly x i x j .
Proof.We may assume that G does not admit a star cutset and G is not a complete graph.Let K be an inclusion-wise maximal clique of G with |K| > 2, and let D = G \ K. Since G does not admit a clique cutset and is not a complete graph, it follows that D is connected, non-empty, and every vertex of K has a neighbor in D. By Lemma 3.1, it follows that G does not contain a diamond.
(1) Let v ∈ D. Then v has at most one neighbor in K.
Assume that v has at least two neighbors in K, say k 1 and k 2 .Since K is a maximal clique, there exists and a minimal connected subgraph H of D containing at least one neighbor of each of x 1 , x 2 , x 3 .By (1), we have that |V (H)| ≥ 3. Now the first outcome of Lemma 3.2 gives a hole and a vertex with two non-adjacent neighbors in it, the second outcome gives a pyramid, and the third gives a prism.In all cases we get a contradiction to the fact that G ∈ F. This proves Lemma 3.3.Now, by Lemma 3.3, in order to prove Theorem 2.2 it is enough to prove: Theorem 3.4.For all k, there exists c = c(k) such that every graph in F 2 with no star cutset and with treewidth more than c contains a subdivision of W k×k as an induced subgraph.

Balanced separators and central bags
Let G be a graph, and let w : The next two lemmas show how (w, c)-balanced separators relate to treewidth.The first result was originally proved by Harvey and Wood in [12] using different language, and was restated and proved in the language of (w, c)-balanced separators in [3].A pair (G, w) is d-unbalanced if w is a weight function on G, and G has no (w, 1  2 )-balanced separator of size at most d (if there is a (w, 1  2 )-balanced separator of size at most d, we say that (G, w) is d-balanced).
Let K be an integer, let G be a graph and let K 1 , K 2 be two cliques of G, each of size at most K. Let (G, w) be a 2K-unbalanced pair.Following [7], we define the canonical twoclique-separation for {K 1 , K 2 }, as follows.Let B(K 2 )-balanced separator; consequently w(B(K 1 , K 2 )) > 1  2 , and so the choice of For the remainder of this section, let K be an integer, and let (G, w) be a 2K-unbalanced pair.Let K Clearly, if S 1 and S 2 are non-crossing, then they are loosely non-crossing.(Note that here we break the symmetry between A i and B i , and so our definition is slightly different from the classical definition of [14].) The following observation follows immediately from the definition of a canonical two-cliqueseparation.
Lemma 4.3.Assume that G does not admit a star cutset.Let K 1 , K 2 be cliques of size at most K in G such that A(K 1 , K 2 ) = ∅.Then the following hold. (1) , and so there is a path from a vertex of K 1 to a vertex of K 2 with non-empty interior in D.
Throughout this section, let S be a set of sets {K 1 , K 2 } where each of K 1 , K 2 is a clique of size at most K of G, and let T be the set of canonical two-clique-separations corresponding to members of S.Moreover, we will assume each pair of separations in T is loosely non-crossing.
We would now like to define a central bag for S. Roughly speaking, this central bag is the intersection of the heavy blocks B(S) ∪ C(S) of the separations, together with some paths that capture the important w-related information about the light blocks.In order to define it, we start by considering the connected components of the union S∈T A(S) of the light sides of the separations.We first note that, given such a component D and an S 0 ∈ T , we either have D ⊆ A(S 0 ) or D ∩ A(S 0 ) = ∅.Indeed, N (A(S 0 )) ⊆ C(S 0 ), and so if D simultaneously contains vertices in A(S 0 ) and vertices not in A(S 0 ), then D \ A(S 0 ) must contain vertices in C(S 0 ); but D \ A(S 0 ) ⊆ S∈T :S =S 0 A(S), which has empty intersection with C(S 0 ) by the loosely non-crossing property -a contradiction.
We now want to "reorganize" the A(S) by assigning each component of S∈T A(S) to a unique A(K 1 , K 2 ) in a consistent way.To that end, we fix a total order π on S, and group the components according to the π-minimal {K 1 , K 2 } to whose A(S) they belong.Specifically, for We call β a central bag for S. Note that the choice of β is not unique since the choice of the paths ).Let w β (v) = 0 for every v ∈ β where w β has not been defined yet.We call w β the weight function inherited from w. Proof.We note that, for any S 0 ⊆ S, the pair of sets
From now on, let D be a component of G \ Y .We will show that w(D) ≤ 1  2 .Since (G, w) is 2K-unbalanced, it follows that w(A(K 1 , K 2 )) < 1  2 for all {K 1 , K 2 } ∈ S, and so if D is a component of G \ β, then by (2), it follows that w(D) ≤ 1  2 .Thus we may assume that D ∩ β = ∅.

Suppose first that D
, and so w(D) < 1  2 .Therefore, we may assume that 4) holds; so we may assume that and in particular (K 1 ∪K 2 )∩D = ∅, so K 1 ∪K 2 ⊆ Y .Moreover, from (4), it follows that P * K 1 K 2 ⊆ Q k for some k ∈ {1, . . ., m}, and then T has at least three leaves, and therefore some vertex of D has degree at least three in N [D] as required.Thus we may assume that is not a path from the vertex of K 1 to the vertex of K 2 , then some vertex of D has at least three neighbors in N [D], and again theorem holds.Thus we may assume that N [D] is a path from the vertex of K 1 to the vertex of )) (such components exist because S and S ′ are active, and hence proper).Since , and ( 7) holds.
. Then in view of the last sentence before (7), this means ) is a clique, then C is the union of two cliques, say X and Y , and so (A, C, B) is a two-clique-separation of G.We claim that (A, C, B) is proper.By (7) there is a component , and therefore C ⊆ N (D).If |C| > 2, the claim follows.Since G does not admit a clique cutset, we may assume that X = {x} and Y = {y} and x is non-adjacent to y.We need to show that A is not a path from x to y. Suppose it is.Then every vertex of A has exactly two neighbors in A ∪ X ∪ Y , and each of x, y has exactly one neighbor in A. Since A(K 1 , K 2 ) ∪ A(K ′ 1 , K ′ 2 ) ⊆ A, this contradicts Lemma 4.6.This proves the claim that (A, C, B) is proper.
Observe that = ∅, the inclusion is proper and we get a contradiction to the fact that S is active.This proves (8).
, so the only possible neighbor of r lying in A(K ′ 1 , K ′ 2 ) is t.But now {s, t, r} is a triangle, contrary to the fact that G ∈ F 2 .This proves (9).
, and since . Now x has two non-adjacent neighbors in H, namely k 1 and k ′ 1 , contrary to the fact that G ∈ F 2 .

Heavy seagulls
A seagull is a graph that is a three-vertex path.Given a seagull F = a-v-u in G, an induced subgraph T of G is a theta through F if T is a theta, one of a, u is an end of T , and A heavy seagull is extendable if there is a theta through it in G.The goal of this section is to show that every heavy seagull is "broken" by some two-clique-separation.We start with a lemma.Lemma 6.1.Let G ∈ F 2 , let F = a-v 1 -u 1 be a seagull in G and let T be a theta through F in G. Let the ends of T be a, b and let the paths of T be P 1 , P 2 , P 3 where F ⊆ P 1 .Assume that T is chosen with |P 1 | minimum among all thetas through F with end a in G. Let P be a path from Proof.Suppose for a contradiction that w 3 } where w i ∈ P i .Then P contains a path Q = q 1 -. . .-q k such that q 1 has a neighbor in P 1 \ {a, v 1 , b}, q k has a neighbor in (P 2 ∪ P 3 ) \ {b, w 2 , w 3 } and Q ∩ T = ∅.We may assume that Q is chosen in such a way that k is minimum.We may also assume that q k has a neighbor s in P 2 \ {b, w 2 }.Since G ∈ F 2 , it follows that N T (q k ) = {s}.Let t be a neighbor of q 1 in P * 1 \ {v 1 }; similarly N T (q 1 ) = {t}.In particular k > 1.It follows from the minimality of k that Q * is anticomplete to T \ {w 2 , w 3 }.Moreover, since s-Q-t-P 1 -a-P 2 -s is a hole, it follows that each of w 2 , w 3 has at most one neighbor in Q.
(10) Not both w 2 and w 3 have a neighbor in Q.
Suppose not.Let i, j ∈ {1, . . ., k} be such that q i is adjacent to w 3 and q j is adjacent to w 2 .Since N T (q k ) = {s}, it follows that i, j = k.Now, w 3 -P 3 -a-P 2 -w 2 -q j -Q-q i -w 3 is a hole, and b has two neighbors in it, contrary to the fact that G ∈ F 2 .This proves (10).
Suppose not.Let i ∈ {1, . . ., k} be such that q i is adjacent to w 3 .Then, by (10), it follows that w 2 has no neighbor in Q, and so s-P 2 -b-P 1 -t-Q-s is a hole and w 3 has two neighbors b and q i in it, contrary to the fact that G ∈ F 2 .This proves (11).
Suppose w 2 has a neighbor in Q; let i ∈ {1, . . .k} be such that w 2 is adjacent to q i .Let S be the path w 1 -P 1 -t-q 1 -Q-q k .Since t = v 1 , we have that v 1 ∈ S. Now H = b-w 1 -S-q k -s-P 2 -a-P 3 -b is a hole and b, q i ∈ N H (w 2 ), contrary to the fact that G ∈ F 2 .This proves (12).
Since s = w 2 and t = v 1 the paths t-P 1 -a, t-q 1 -Q-q k -s-P 2 -a and t-P 1 -b-P 3 -a form a theta through {a, v 1 , u 1 } that contradicts the choice of T with |P 1 | minimum.
The next result allows us to use Lemma 6.1 to handle heavy seagulls.Lemma 6.2.Let G ∈ F 2 and let F be a heavy seagull in G. Assume that G does not admit a star cutset.Then F is extendable.
is stable.Since G does not admit a star cutset, it follows that for i ∈ {1, 2}, there exists a path P i from x i to u with P * i ∩ N [a] = ∅.By choosing P 1 , P 2 with P 1 ∪ P 2 minimal, and permuting the indices if necessary, we may assume that one of the following two cases holds.
(1) P * 1 ⊆ P * 2 and x 1 has a neighbor in P * 2 .(2) There exists a vertex q ∈ V (G) \ {v, a, x 1 , x 2 } and a path Q from u to q such that P i = u-Q-q-P ′ i -x i and P ′ 1 \ q is disjoint from and anticomplete to P ′ 2 \ q.
{a, u 1 }, B) of G with D a ⊆ A and D u ⊆ B. Now ( 14) follows from Lemma 2.1 applied to S. This proves (14).
Let D be the component of A(X, Y ) containing p, and let N = N (D).Then N is the union of two cliques K 1 , K 2 .
Observe that B(X, Y ) ⊆ B(K 1 , K 2 ) and D ⊆ A(K 1 , K 2 ).Since G does not admit a clique cutset, both K 1 and K 2 are non-empty.If |K 1 ∪ K 2 | ≥ 3, then D is a component of A(K 1 , K 2 ) with K 1 ∪ K 2 ⊆ N (D), and the claim holds.Thus we may assume that |K 1 | = |K 2 | = 1.Since F is heavy, it follows that deg G (p) > 2, and therefore D ∪ K 1 ∪ K 2 is not a path from K 1 to K 2 , and again the claim holds.This proves (15).

Now among all proper pairs (K
inclusion-wise minimal, and subject to that with B(K ′ 1 , K ′ 2 ) inclusion-wise maximal.Then (K ′ 1 , K ′ 2 ) is active and A(K ′ 1 , K ′ 2 ) ∩ {a, u 1 } = ∅.This proves Theorem 6.3.Proof.Let w = w(k, γ) = f (c(k, γ + 3)), where f is as in Theorem 1.1 and c is an in Theorem 1.3.Let G be a graph with treewidth at least w.By Theorem 1.1, G has a subgraph X which is isomorphic to W c(k,γ+3)×c (k,γ+3) .Let H = G[V (X)].Then H has treewidth at least c(k, γ + 3).Also, we claim that G has maximum degree at most γ+3.To see this, suppose for a contradiction that H has a vertex v of degree at least γ + 4 > 3.Then, since X has maximum degree at most 3, there are at least γ + 1 edges in E(H) \ E(X) incident with v.Moreover, for each such edge, its end distinct from v has degree at least two in X, and so degree at least 3 in H.But then v is a vertex of degree at least 3 in G with at least γ + 1 neighbors, each of degree at least 3 in G.This violates γ 3 (G) ≤ γ, and so proves the claim.Now, by Theorem 1.3, H, and so G, contains a subdivision of W k×k or the line graph of a subdivision of W k×k .
We remark that Theorem 7.1 is sharp, in the sense that the conclusion fails if the number 3 in γ 3 (G) is replaced by any larger integer.This is due to the construction of [11], in which the set of vertices of degree 4 or more is stable.Next, we deduce: Theorem 7.2.For all t, there exists M = M (t) such that every graph in F t with no heavy seagull and with treewidth more than M contains a subdivision of W t×t .
Proof.Since G contains no heavy seagull, it follows that no two vertices of degree at least three in G are at distance two in G.This implies that every connected component of the subgraph of G induced by the set of vertices of degree at least three in G is a clique, and therefore has size at most t.It follows that γ 3 (G) ≤ t − 1.Also, since G ∈ F, no induced subgraph of G is the line graph of a subdivision of W 3×3 .Now Theorem 7.2 follows from Theorem 7.1.

Lemma 4 . 1 ( 2 , 1 )
[3,12]).Let G be a graph, let c ∈ [ 1 2 , 1), and let k be a positive integer.If G has a (w, c)-balanced separator of size at most k for every weight function w on G, then tw(G) ≤1  1−c k.Lemma 4.2([10]).Let G be a graph and let k be a positive integer.If tw(G) ≤ k, then G has a (w, c)-balanced separator of size at most k + 1 for every c ∈ [ 1 and for every weight function w on G.

7 .Theorem 7 . 1 .
Proof of Theorem 3.4We begin with proving an extension of Theorem 1.3.For a graph G and positive integer d, we denote by γ d (G) the maximum degree of the subgraph of G induced by the set of vertices with degree at least d in G.For all k, γ > 0, there exists w = w(k, γ) such that every graph G with γ 3 (G) ≤ γ and treewidth more than w contains a subdivision of W k×k or the line graph of a subdivision of W k×k .