Obstruction for bounded branch-depth in matroids

DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid Un,2n or the cycle matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. As a corollary, we prove their conjecture for matroids representable over a fixed finite field and quasi-graphic matroids, where the uniform matroid is not an option.


§1. Introduction
Motivated by the notion of tree-depth of graphs, DeVos, Kwon, and Oum [3] introduced the branch-depth of a matroid M as follows. Recall that the connectivity function λ M of a matroid M is defined as λ M pXq " rpXq`rpEpM q Xq´rpEpM qq, where r is the rank function of M . A decomposition is a pair pT, σq of a tree T with at least one internal node and a bijection σ from EpM q to the set of leaves of T . For an internal node v of T , the width of v is defined as where P v is the partition of EpM q into sets induced by components of T´v under σ´1.
The width of a decomposition pT, σq is defined as the maximum width of its internal nodes.
The radius of pT, σq is the radius of T . A decomposition is a pk, rq-decomposition if its width is at most k and its radius is at most r. The branch-depth of a matroid M is defined to be the minimum integer k for which M admits a pk, kq-decomposition if EpM q has more than one element, and is defined to be 0 otherwise.
It is well known that graphs of large tree-depth contains a long path as a subgraph (see the book of Nešetřil and Ossona de Mendez [14, Proposition 6.1]). DeVos, Kwon, and Oum [3] made an analogous conjecture for matroid branch-depth as follows. Since the cycle matroid of a path graph has branch-depth at most 1, paths no longer are obstructions for small branch-depth. Instead, they use the cycle matroid of fans.
The fan matroid M pF n q is the cycle matroid of a fan F n , which is the union of a star K 1,n together with a path with n vertices through the leaves of the star, see Figure 1.1. Note that the path with 2n´1 vertices is a fundamental graph of M pF n q. We write U n,2n to denote the uniform matroid of rank n on 2n elements. Now, here is the conjecture. Conjecture 1.1 (DeVos, Kwon, and Oum [3]). For every positive integer n, there is an integer d such that every matroid of branch-depth at least d contains a minor isomorphic to M pF n q or U n,2n .
Our main theorem verifies their conjecture for matroids of bounded branch-width as follows. Note that U n,2n has large branch-width if n is big and so U n,2n will not appear in the following theorem. This allows us to obtain the following corollary for matroids representable over a fixed finite field, since we can use a well-known grid theorem for matroids of high branch-width by Geelen, Gerards, and Whittle [9]. In a big picture, our proof follows the strategy of Kwon, McCarty, Oum, and Wollan [12].
As branch-width is small, we can find, in every large set, a large subset having small connectivity function value. We use that recursively to find a long path in a fundamental graph, which results a minor isomorphic to the fan matroid.
The paper is organized as follows. In Section 2, we will introduce our notations and a few results for matroids, branch-depth, and branch-width. In Section 3, we will discuss the concept of twisted matroids introduced by Geelen, Gerards, and Kapoor [8]. In Section 4, we prove our main theorem, Theorem 1.2 by finding a 'lollipop' inside a twisted matroid.
In Section 5, we prove its consequences to matroids representable over a fixed finite fields and quasi-graphic matroids. §2. Preliminaries

Set systems.
A set system S is a pair pE, Pq of a finite set of E and a subset P of the power set of E.
We call E the ground set of S and may denote it by EpSq.
For i P t1, 2u let S i " pE i , P i q be set systems. A map ϕ : E 1 Ñ E 2 is an isomorphism between S 1 and S 2 if it is bijective and P P P 1 if and only if ϕpP q P P 2 . We say S 1 and S 2 are isomorphic if there is such an isomorphism.
Given two sets X and Y , we denote by Given a set system S " pE, Pq and a subset X Ď E we define P X :" tP X : P P Pu and P|X :" tP Ď X : P P Pu.
Given an integer n, we write rns for the set ti : 1 ď i ď nu of positive integers up to n.
A matroid M is a set system pE, Bq satisfying the following properties: (B1) B is non-empty.
(B2) For every B 1 , B 2 P B and every x P B 1 B 2 , there is an element y P B 2 B 1 such that pB 1 txuq Y tyu P B.
An element of B is called a base of M . We denote the set of bases of a matroid M by BpM q.
A set X is independent if it is a subset of a base, and we denote the set of independent sets of M by IpM q. A set X is dependent if it is not independent.
A circuit is a minimal dependent set, and we denote the set of circuits of M by CpM q.
The rank of a set X in a matroid M , denoted by r M pXq, is defined as the size of a maximum independent subset of X. We write rpM q to denote r M pEpM qq, the rank of M .
The rank function satisfies the submodular inequality: for all X, Y Ď EpM q, The dual matroid of M , denoted by M˚, is the matroid on EpM q where a set B is a base of M˚if and only if EpM q B is a base of M . It is well known that For a subset X of EpM q, we write M X to denote the matroid pEpM q X, where B 1 is the set of maximal elements of IpM q|pEpM q Xq. This operation is called the deletion. The contraction is defined as M {X " pM˚ Xq˚. The restriction is defined for some disjoint subsets X and Y of EpM q.
The connectivity function λ M of a matroid M is defined as It is easy to check that λ M pXq " λ M˚p Xq.
The connectivity function satisfies the following three inequalities.
A matroid M is connected if λ M pXq ‰ 0 for all non-empty proper subsets X of EpM q.
A component of a matroid M with |EpM q| ‰ 0 is a minimal non-empty set X such that λ M pXq " 0, and the empty set is the unique component of the empty matroid p∅, t∅uq.
We use the above lemma to prove the following useful lemma.
be the one-node graph on tr i u and take σ i : C i Ñ tr i u.
We construct a decomposition pT, σq of M by letting T be a tree obtained from the disjoint union of all T i 's by adding a new node r and adding edges rr i for all i P rts, letting σ map v P C i to σ i pvq, and appending |X Y Y | leaves to r and assigning each element of X Y Y to a distinct leaf attached to r with the map σ. Then pT, σq has radius at most m´|X Y Y |.
Furthermore the width of pT, σq is at most and if any of them has branch-depth equal to m`1, then it has a pm, m`1q-decomposition.
If |X| ą 1, then let pT 1 , σ 1 q be a pm, m`1q-decomposition of M {Y 1 |X and let r 1 be a node of T 1 within distance m`1 from every node of T 1 . If |X| " 1, then let pT 1 , σ 1 q be the one-node tree on tr 1 u and take σ 1 : X Ñ tr 1 u.
Similarly, if |Y | ą 1, then let pT 2 , σ 2 q be a pm, m`1q-decomposition of M {X 1 |Y and let r 2 be a node of T 2 within distance m`1 from every node of T 2 . If |Y | " 1, then let pT 2 , σ 2 q be the one-node tree on tr 2 u and take σ 2 : Y Ñ tr 2 u.
Let T be a tree obtained from the disjoint union of T 1 and T 2 by adding a new node r and adding two edges rr 1 and rr 2 . Let σ be the bijection from X Y Y to the set of leaves of T induced by σ 1 and σ 2 . Then pT, σq is a decomposition of radius at most m`2.
Furthermore by Lemma 2.3, the width of pT, σq is at most m`k. Thus, the branch-depth of M is at most maxpm`k, m`2q.

Branch-width.
Robertson and Seymour [19] introduced the concept of branch-width. A subcubic tree is a tree such that every node has degree 1 or 3. Here is a classical lemma on branch-width. For the completeness of this paper, we include its proof. An equivalent lemma appears in [ Since |EpT q| ă |V pT q|, there is a node v of T having no outgoing edges. Since k ě 1, every edge incident with a leaf is oriented away from the leaf and therefore v is an internal node.
However v has degree 3 and so |Z| ď 3k, contrary to the assumption that |Z| ą 3k.
The following lemma is well known and is an easy consequence of the definitions. The following statements about the fundamental graph are well known and are easy consequences of the relevant definitions. Hence, binary matroids are completely determined by its fundamental graph and a colour class of any proper 2-colouring of that fundamental graph, which is the base of the matroid.
For general matroids, such a complete determination fails; two distinct matroids may have the same fundamental graph with respect to the same base. But one can ask how a fundamental graph with respect to some base will change when doing base exchange.
Note that if GpM, Bq has an edge uv, then B 1 :" B tu, vu is a base of M . The operation of constructing GpM, B 1 q from GpM, Bq is called a pivot on uv.
Note that for all pairs tx, yu, the first three rules of the above proposition determine the adjacencies between x and y in G 1 from G. This is not true of the fourth rule. However, if in addition to the edge set of the fundamental graph we were given a list of 'hyperedges' tx, y, u, vu for which B tx, y, u, vu is a base, then we could apply all four rules.
As an extension of that idea, Geelen, Gerards, and Kapoor [8] introduced twisted matroids, which can in a sense be viewed as 'fundamental hypergraphs'. We introduce their machinery in the next subsection.

Twisted matroids.
Let S " pE, Pq be a set system and let X Ď E. We define the twist of S by X as Moreover, we define the restriction of S to X as SrXs :" pE, P|Xq.

Remark 3.3.
Let S " pE, Pq be a set system and let X, Y Ď E. Then A twisted matroid W is a set system pE, Fq satisfying the following properties: We call E the ground set of W and may denote it by EpW q. We call the elements of F feasible (with respect to W ), and may denote the set F by FpW q. We call a set B which satisfies (T3) a base of W . We denote by BpW q the set of bases of W .
We observe that (T3) implies that every feasible set has even size. And in fact it is enough to restrict out attention to feasible sets of size two, as the following proposition will show. is equivalent to the following axiom.
Proof. Assume (T3 1 ) holds and let B Ď E be as required. Suppose for a contradiction that (T3) does not hold and let F P F be a set of minimum size violating (T3).
Let X, Y P tB, pE Bqu with |X X F | ă |Y X F |. With (T1), by applying (T2) to ∅, F , and some e P F , there is an f P F such that te, f u " ∅ te, f u P F and hence e ‰ f by (T3 1 ). By (T3 1 ), exactly one of e or f is in B, so X X F is non-empty. Applying (T2) again to F , ∅, and some x P X X F , there is some z P F such that contradicting that F was the smallest counterexample to (T3).
Note that this axiomatic definition of twisted matroids does not coincide with the original definition of Geelen, Gerards, and Kapoor [8], in which they defined twisted matroids to be the twist M˚B of a matroid M with a base B of M . The following proposition establishes together with Remark 3.3 the equivalence of these definitions.  Note that Then e P F 1 F 2 and hence by (T2) there is an For a twisted matroid W we define Then the following statements are true.
(i) X P FpW q if and only if B X P BpM q.
(ii) The fundamental graph GpM, Bq coincides with the fundamental graph GpW q.
(iii) pM˚, E Bq P MpW q.
Proof. For (i), suppose X P F. Then M has a base B 1 such that X " B B 1 . By the properties of the symmetric difference we obtain For (iii), note that for pM, Bq P MpW q we have BpM˚q " BpM q EpM q " pF Bq EpM q " F pEpM q Bq.
For (iv), suppose M is connected. It follows that GpM, Bq " GpW q is connected and hence every proper 2-colouring of GpW q has B and E B as its colour classes. In particular, M has a base B such that pM˚BqrEpU qs " N˚pB X EpU qq " U .
Proof. For (i), let pM, B 1 q P MpW q and X Ď E and F P F such that U " pW˚F qrXs.
Then M " W˚B 1 " W˚pB 1 F F q " pW˚F q˚pB 1 F q, and hence for B :" B 1 F we have pM, Bq P MpW˚F q. By Proposition 3.7(iv), we have that U˚pB X Xq is a minor of M , as desired. pW˚F qrEpN qs˚pB X EpN qq " N " U˚B 2 " U˚pB X EpN qq.
Lastly, let us remark that the minor relation of twisted matroids is transitive.
Proposition 3.9. Let W " pE, Fq be a twisted matroid, let X 1 Ď X Ď E, let F P F and let F 1 P FpW˚F q|X. Then F F 1 P F and pW˚pF F 1 qqrX 1 s " pppW˚F qrXsq˚F 1 qrX 1 s.

Proof.
We have that pW˚F q˚F 1 is a twisted matroid by Proposition 3.7(i). Since pW˚F q˚F 1 " W˚pF F 1 q, we have that F F 1 P F again by Proposition 3.7(i). Now by Proposition 3.7(iii) we have pppW˚F qrXsq˚F 1 qrX 1 s " pppW˚F q˚F 1 qrXsqrX 1 s " pW˚pF F 1 qqrX 1 s.

3.4.
More on the fundamental graph and twisted matroids. (i) (Brualdi [2]) If X P F, then GpW qrXs has a perfect matching.
We deduce the following two propositions easily from the above proposition. Proof. Let X Ď E. Let G denote the common fundamental graph with respect to B, which by Proposition 3.6(ii) is equal to GpM 1˚B q " GpM 2˚B q. Since G is a forest, every induced subgraph has at most one perfect matching and so for i P r2s by Proposition 3.10, It follows thatB :" B F is a base of M by Proposition 3.6(i). Note that B X EpN q " pB ppB X EpN qq B 1 qq X EpN q " B 1 , and hence pM˚BqrEpN qs " N˚B 1 . Therefore GpM,BqrEpN qs " GpN, B 1 q by Propositions 3.6(ii) and 3.7(ii), which is the required path.
Conversely, suppose there is a base B of M and a set X Ď EpM q such that GpM, BqrXs is a path on 2n´1 vertices, starting and ending in B. Let W :" M˚B and U :" pM˚BqrXs.
Since U is a minor of W , for some B 1 P BpU q the matroid N :" U˚B 1 is a minor of M by Proposition 3.8(i). Again, by Propositions 3.6(ii) and 3.7(ii), we have that GpM, BqrXs " GpU q " GpN, B 1 q.
Since GpM, BqrXs is a forest, by Proposition 3.11, there is a unique matroid with base B 1 whose fundamental graph is GpM, BqrXs, and that matroid is isomorphic to M pF n q, as desired.
We will need the following result about the change of the fundamental graph of a twisted matroid when twisting with a feasible set, which in particular will not change for the vertices not involved in the twist. Proof. If te, xu P F F , then F 1 :" te, xu F P F. Since e P F 1 , by (T1) and (T2) there is a y P F 1 such that te, yu P F. Now by the premise of this proposition, y " x, as desired.
If te, xu P F, then by applying (T1) and (T2) for W˚F , there is a y P te, xu F for which te, yu P F F . So by the previous paragraph, te, yu P F and hence y R F . But then y " x, as desired.
If a matroid property is invariant under component-wise duality, then for a twisted matroid W that property will be shared by every matroid associated with W . So for such properties we are justified to call these properties of the twisted matroid.
For example, we have the following proposition. Given these definitions, the related results for matroids in Section 2 also hold for twisted matroids, and we will apply them for twisted matroids without further explanation.

§4. Lollipop minors of twisted matroids
In this section we complete the proof of Theorem 1.2. To do this, we introduce the following class of twisted matroids. (2) G is connected; (3) GrS Y tzus is a path with terminal vertex z; (4) GrCs is a connected component of G´z; and (5) LrCs has branch-depth at least b.
We call the tuple pS, z, Cq the witness of L, and the twisted matroid LrCs the candy of L.
In order to prove Theorem 1.2, we prove the following theorem.  We also remark that Theorem 1.2 implies Theorem 4.2 because for all non-negative integers a and b, there is an integer n such that for some base B of M pF n q the twisted matroid M pF n q˚B is an pa, bq-lollipop.
The reason for considering lollipops as opposed to fan matroids is that it allows an inductive approach to find lollipop minors in twisted matroids of sufficiently high branchdepth. If we find a lollipop whose candy has sufficiently high branch-depth, then we can iteratively find another lollipop as a minor of the candy.
Since lollipops are defined as twisted matroids, the choice of a base of the original matroid is important. However, the following result allows us a large amount of flexibility in exchanging parts of the base of the matroid associated with the candy. Proof. Let G :" GpLq and G 1 :" GpL˚F q. By Proposition 3.7(i), there is a matroid M associated with both L and L˚F . Since G is connected, so is M by Proposition 3.1, and hence so is G 1 .
By Proposition 3.7(iii), pL˚F qrCs is equal to LrCs˚F , and hence has branch-depth at least b and is connected. Now by Proposition 3.13, the neighbourhood of each s P S is the same in G and G 1 .
Hence GrS Y tzus " G 1 rS Y tzus, and no s P S has a neighbour in C in G 1 . Hence G 1 rCs is indeed a component of G 1´z .

The induction.
As mentioned in the last subsection, we aim to prove Theorem 4.2 by induction on a.
For the start of the induction we consider the following lemma. Proof. By Lemma 2.4, W has a component C such that W rCs has branch-depth at least b`1. Let z P C be arbitrary. By Lemma 2.5, W rC tzus has a connected com- For the induction step, the following two lemmas are the main tools we will need. (2) the neighbourhood of z in G :" GpLq is disjoint from C 1 .
Then there exist a set S 1 Ě S and an element z Proof. There is a shortest path P from z to C 1 in GrC Y tzus. Let x be the unique vertex in V pP q X C 1 , let z 1 be the neighbour of x in P , and let S 1 :" S Y pV pP q tx, z 1 uq. Now |S 1 | ě |S|`1 ě a`1, since z has no neighbour in C 1 . Hence L 1 :" LrS 1 Y tz 1 u Y C 1 s is an pa`1, b 1 q-lollipop witnessed by pS 1 , z 1 , C 1 q, as desired. (2) every minor of W of branch-depth at least g i contains an pa, g i`1 q-lollipop as a minor for all i ă .
Then there is a feasible set F and for each i P r s there is a set E i " S i 9 Ytz i u 9 YC i such that for W 1 :" W˚F the following properties hold.
(i) L i :" W 1 rE i s is an pa, g i q-lollipop witnessed by pS i , z i , C i q for all i P r s; and Proof. Let W 0 :" W and let C 0 :" EpW q. For i P r s let L 1 i be an pa, g i q-lollipop with candy W i such that L 1 i is a minor of W i´1 . Note that L 1 i exists by the premise of the lemma. Let pS i , z i , C i q be the witness of L 1 i and let Let F 1 0 :" ∅. For i P r s, let F i´1 P FpW i´1 q be such that L 1 i " pW i´1˚Fi´1 qrE i s, and recursively define F 1 i :" F 1 i´1 F i´1 . We now prove the following.
Claim. For i P r s, we have Proof of Claim. For i " 1, (a) and (b) follow from the fact that F 1 Therefore (a) and (b) follow from Proposition 3.9. For i " 1, (c) follows from the fact that F 1 " F 1 ´1 F ´1 , and hence F 1 Hence, (c) follows from the inductive hypothesis and the fact that both C ´pi´1q and F ´i are subsets of C ´i .
Define F :" F 1 . For i P r s, we have F F 1 i P FpW˚F q by (a). Therefore, by (b) and (c), we have Hence, by Corollary 4.4, L i :" pW˚F qrE i s is an pa, g i q-lollipop witnessed by pS i , z i , C i q, as required.
Combining these two lemmas will be the heart of the induction step, as noted in the following corollary.

Corollary 4.8. In the situation of Lemma 4.7, additionally let b be a non-negative integer and assume that
p˚q there is an i P r s, a set C Ď C Ď C i , and a feasible setF P FpL i q|C i such that (1) pL i˚F qrCs is connected and has branch-depth at least b; and (2) the neighbourhood of z in GpL i˚F q is disjoint from C.
Then W contains an pa`1, bq-lollipop as a minor.
Up to this point, we have not used the fact that the twisted matroid has bounded branchwidth. In the next subsection we will prove the following lemma, which will complete the proof of Theorem 4.2. The following two lemmas are the final tools we will need for this proof. and W rCs has branch-depth at least b`w´1. Then for some X Ď Z and Y Ď C, the following hold.
(i) |X| ě k`1; (ii) W rY s is connected and has branch-depth at least b; (iii) λ W rXYY s pXq ă w.
Proof. Since W has branch-width at most w, so does W 1 :" W rZ Y Cs by Lemma 2.8.
Hence by Lemma 2.7, there is a bipartition pX 1 , Y 1 q of EpW 1 q with λ W 1 pX 1 q ă w such that |Z X X 1 | ą k and |Z X Y 1 | ą k. By Lemma 2.6, without loss of generality W rY 1 X Cs has a component Y of branch-depth at least b. Let X :" X 1 X Z. Since λ W 1 pX 1 q ă w, it follows from Lemma 2.3 that λ W rXYY s pXq ă w. Proof. Let B 1 be a base of W . We set B 2 :" EpW q B 1 , as well as M 1 :" W˚B 1 and M 2 :" W˚B 2 . Note that M1 " M 2 . Now we observe that Hence r M 1 {pB 1 XXq pX B 1 q`r M 2 {pXXB 2 q pX B 2 q ă w. Since |X| ě w, for some base Proof. By applying Lemma 4.10 to W 1 , Z :" tz i : i P r su, and C , there are sets X Ď Z and Y Ď C such that (i) |X| ě w; (ii) W 1 rY s is connected and has branch-depth at least b`w; (iii) λ W 1 rXYY s pXq ă w.
By applying Lemma 4.11 to W 2 :" W 1 rX Y Y s there exists a base B of W 2 and a set O Ď X B of size at most w such that O is a circuit in pW 2˚B q{pX X Bq. It follows that O is a circuit in the restriction of that matroid to pO Y Y q, which we call M . Note that since B EpM q " B X X, we get M " pW 2 rEpM qsq˚pB X Y q by Proposition 3.7(iv).
Since M |Y " W 1 rY s˚pB X Y q, it follows from (ii) that M has branch-depth at least b`w. NowF :" pB X EpM qq B is feasible with respect to W 2 rEpM qs by Proposition 3.6(i), and since z i RF we getF P FpL i q|C i . By Propositions 3.6(ii) and 3.7(ii), GpM,Bq " GpW 1 rEpM qs˚F q " GpL i˚F qrEpM qs.
Hence, by our choice ofB the neighbourhood of z i in GpL i˚F q is O z i , which is disjoint from C. And since pL i˚F qrCs " pM˚BqrCs, we obtain condition p˚q of Corollary 4.8, as desired.

Proof of Theorem 4.2.
We now prove Theorem 4.2, which completes the proof of Theorem 1.2.

Definition 4.12.
For each integer w ą 2, we define a function f w : N 2 Ñ N for all nonnegative integers a and b we set :" 3w´2 and define a sequence pg i : 0 ď i ď q as follows.
We set and for a ě 1 we set Since neither U 2,q`2 nor U q,q`2 is representable over GFpqq, we will instead show the following stronger corollary, implying Corollary 1.3.

Corollary 5.1. For any positive integers n and q, there is an integer d such that every
matroid having no minor isomorphic to U 2,q`2 or U q,q`2 with branch-depth at least d contains a minor isomorphic to M pF n q.
The mˆn grid is the graph with vertices tpi, jq : i P rms, j P rnsu, where pi, jq and pi 1 , j 1 q are adjacent if and only if |i´i 1 |`|j´j 1 | " 1. The above corollary is obtained by using the following theorem of Geelen, Gerards, and Whittle [9], because the cycle matroid of the nˆn grid contains M pF n q as a minor. Geelen, Gerards, and Whittle [7] introduced the class of quasi-graphic matroids, which includes the classes of graphic matroids, bicircular matroids, frame matroids, and lift matroids. We will show that quasi-graphic matroids of large branch-depth contain large fan minors, as a corollary of Theorem 1.2.
Though the original definition of quasi-graphic matroids is due to Geelen, Gerards, and Whittle [7], we present the equivalent definition of Bowler, Funk, and Slilaty [1]. Let G be a graph. A tripartition pB, L, Fq of cycles of G into possibly empty sets is called proper if it satisfies the following properties.
(i ) B satisfies the theta property: if C 1 , C 2 are two cycles in B such that EpC 1 q EpC 2 q is the edge set of a cycle C, then C is in B.
(ii ) Whenever L is in L and F is in F, there is at least one common vertex of L and F .
A cycle is balanced if it is in B and unbalanced otherwise. Let X be a subset of EpGq.
If the subgraph GrXs contains no unbalanced cycle, then we say that X and GrXs are balanced, and otherwise we say they are unbalanced. A theta graph is a subgraph consisting of three internally disjoint paths joining two distinct vertices. We define a matroid M " M pG, B, L, Fq by describing its circuits as follows: a subset X of EpGq is a circuit of M if and only if X is the edge set of one of the following.
(2) An unbalanced theta graph.  Proof. We may assume that G has at least 2 edges. Let pT, Lq be a branch-decomposition of the graph G with width at most w. This means that whenever e is an edge of T , there are at most w vertices incident with both sides of a partition pA e , B e q of EpGq induced by the components of T´e under L´1. We will demonstrate that λ M pA e q ď w`1 for every edge e, and then pT, Lq will certify the branch-width of M to be at most w`2.
Let X be a subset of EpGq. Let cpXq denote the number of connected components in the subgraph GrXs, and let bpXq denote the number of these components that are balanced.
Moreover, let pXq be 1 if GrXs contains a cycle in L, and otherwise set pXq be 0. The rank r M pXq is given by the formula |V pXq|´bpXq when GrXs contains a cycle in F, and otherwise by |V pXq|´cpXq´ pXq [1, Lemma 2.4].
Let n be the number of vertices in G and let E :" EpGq. Let n A and n B be the number of vertices in the subgraphs GrA e s and GrB e s, so that n A`nB´n is the number of vertices incident both with edges in A e and edges in B e . First assume that F is non-empty.
Then rpM q " n´bpEq. Assume that both GrA e s and GrB e s contain cycles in F. Any subgraph of a balanced subgraph is itself balanced, and it follows that bpA e q`bpB e q ě bpEq.
Therefore λ M pA e q " r M pA e q`r M pB e q´rpM q " pn A´b pA e qq`pn B´b pB e qq´pn´bpEqq ď |V pA e q X V pB e q| ď w as desired. Now assume that GrA e s contains a cycle in F but that GrB e s does not. In this case bpA e q`cpB e q ě bpEq, so λ M pA e q " pn A´b pA e qq`pn B´c pB e q´ pB e qq´pn´bpEqq ď |V pA e q X V pB e q| ď w.
If neither GrA e s nor GrB e s contains a cycle in F, then since cpA e q`cpB e q ě cpEq ě bpEq, we can again reach the conclusion that λ M pA e q ď w.
Now we assume that F is empty. Therefore rpM q " n´cpEq´ pEq, rpA e q " n A´c pA e q´ pA e q, and rpB e q " n B´c pB e q´ pB e q.
As cpA e q`cpB e q ě cpEq, it follows easily that λ M pA e q ď w`1, and this completes the proof.
We will use the following grid theorem due to Robertson and Seymour. Note that in [18] they proved this theorem in terms of tree-width, but in [19] they established that graphs have small tree-width if and only if they have small branch-width, yielding the following version of the theorem. For a positive integer n, let Pn be the graph obtained from the path on n vertices by adding one loop at each vertex. By comparing circuits, it is easy to observe the following lemma. Proof. We may assume that n ą 2. Let w :" N pn 2 q`2 where N pn 2 q is the integer given in Theorem 5.4.
Let M " M pG, B, L, Fq be a quasi-graphic matroid of branch-width at least w. Assume for a contradiction that M does not have a minor isomorphic to M pF n q.
By Proposition 5.3, G has branch-width at least N pn 2 q. By Theorem 5.4, G has a minor G 1 isomorphic to the n 2ˆn2 grid. We may assume that G 1 is equal to the n 2ˆn2 grid.
As As n 2 is greater than four, it follows that for some α P rn 2 s the graph G 2 rtpi, jq : i P tα, α`1u, j P rn 2 sus is a subgraph of G 2 {c 4 isomorphic to the 2ˆn 2 grid. By contracting the edges in the path pα, 1qpα, 2q¨¨¨pα, n 2 q, we obtain a minor isomorphic to F n 2 . Now n 2 ą n implies that M has a minor isomorphic to M pF n q, a contradiction. Therefore L 1 contains no cycle of length 4.
Consider the subgraph G 1 :" G 1 rtpi, jq : i P t1, 2u, j P rn 2 sus. If G 1 contains n vertexdisjoint cycles of length 4 in F 1 , then M 1 has a minor isomorphic to M pPn , ∅, ∅, Cnq, where Cn is the set of cycles of Pn , contradicting our assumption by Lemma 5.5.
Since the 2ˆn grid contains F n as a minor, by our assumption, any sequence of consecutive balanced cycles of length 4 in G 1 contains at most n´2 such cycles. As G 1 contains no cycles of length 4 in L 1 , and at most n´1 vertex-disjoint cycles of length 4 in F 1 , it follows that G 1 contains at most pn´2qn`2pn´1q " n 2´2 cycles of length 4. This is impossible, as G 1 has at least n 2´1 cycles of length 4.
Now it is routine to combine Proposition 5.6 with Theorem 1.2 to deduce the following result.
Corollary 5.7. For every positive integer n, there is an integer d such that every quasigraphic matroid with branch-depth at least d contains a minor isomorphic to M pF n q.

General matroids.
The following conjecture about branch-width is due to Johnson, Robertson, and Seymour. Proof. Since both the cycle matroid and the bicircular matroid of the nˆn grid are quasi-graphic and both have large branch-depth, this follows from Corollary 5.7.