Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size

Abstract

For graph G, let  bw(G) denote the branchwidth of G and  gm(G) the largest integer g such that G contains a g×g grid as a minor. We show that  bw(G)≤3 gm(G) for every planar graph G. This is an improvement over the bound  bw(G)≤4 gm(G) due to Robertson, Seymour and Thomas. Our proof is constructive and implies quadratic time constant-factor approximation algorithms for planar graphs for both problems of finding a largest grid minor and of finding an optimal branch-decomposition: (3+ϵ)-approximation for the former and (2+ϵ)-approximation for the latter, where ϵ is an arbitrary positive constant. We also study the tightness of the above bound. We show that for any constant c<2, the bound of ${\mathop {\mathrm {bw}}}(G)\leq c\; {\mathop {\mathrm {gm}}}(G) + o({\mathop {\mathrm {gm}}}(G))$ does not hold in general for a planar graph G.

1 Introduction

The notion of branchwidth of a graph is introduced by Robertson and Seymour [18] in their graph minor theory. It is similar and has close relationship to the more celebrated notion of treewidth [17]. In particular, the linear bound of ${\mathop {\mathrm {bw}}}(G) \leq {\mathop {\mathrm {tw}}}(G) + 1 \leq\frac{3}{2}{\mathop {\mathrm {bw}}}(G)$ holds for every graph G with more than one edge, where  bw(G) denotes the branchwidth of G and  tw(G) the treewidth of G. Both treewidth and branchwidth play essential roles in the graph minor theory. Informally, a branch-decomposition of G is a system of vertex cuts of G represented as edges of a tree whose leaves are the edges of G. The width of a branch-decomposition is the maximum cardinality of the vertex cuts in the system and the branchwidth of G is the minimum width of all possible branch-decompositions of G. See Sect. 2 for more formal definitions.

Grid minors also play an important role in the graph minor theory. Let G and H be graphs. H is a minor of G if H can be obtained from some subgraph of G through a possibly empty sequence of edge contractions. A k×h grid is a graph on vertex set {(i,j)∣0≤i<k,0≤j<h,i,j: integer}, such that vertices (i,j) and (i′,j′) are adjacent with each other if and only if |i−i′|+|j−j′|=1. We denote by  gm(G) the largest integer g such that G contains a g×g grid as a minor. Since the branchwidth of a g×g grid is g and  bw(G)≥ bw(H) holds if H is a minor of G,  gm(G) is a lower bound on  bw(G).

Robertson and Seymour [17] showed that there is a function f such that, for every graph G,  tw(G)≤f( gm(G)). This function f, which was enormous in this result, was improved by Robertson, Seymour, and Thomas [20] to $f(k) = 20^{2(k + 1)^{5}}$. In fact, this bound is shown via a bound on the branchwidth, namely ${\mathop {\mathrm {bw}}}(G) \leq20^{2{\mathop {\mathrm {gm}}}(G)({\mathop {\mathrm {gm}}}(G)+1)^{4}}$. In the same paper, these three authors also show a much better bound for planar graphs: every planar G satisfies  bw(G)≤4 gm(G).¹

This linear bound of the form

$${\mathop {\mathrm {bw}}}(G) \leq c\;{\mathop {\mathrm {gm}}}(G) + o\bigl({\mathop {\mathrm {gm}}}(G)\bigr) $$

(1)

has been extended to bounded genus graphs [7] and to graphs excluding a fixed minor [6], providing a combinatorial basis of the bidimensionality theory [5, 6, 8, 10]. Informally, a graph parameter f(G) is bidimensional, if the value of f for g×g grid grows as g increases and f(H)≤f(G) if H is a minor of G. The bidimensionality theory provides a uniform framework for designing subexponential fixed parameter algorithms for computing bidimensional graph parameters. To decide if f(G)>k, we first determine or estimate  bw(G). If  bw(G) is sufficiently large then, owing to the linear bound, we may conclude that f(G)>k, since G contains a large grid minor. Otherwise, the small-width branch-decomposition of G is used in a dynamic programming algorithm to exactly compute f(G), which typically runs in time polynomial in the size of G but exponential in  bw(G). Combined with the techniques of kernelization (see [11] for introduction), this approach often leads to subexponential fixed parameter algorithms.

Since the coefficient c in the linear bound (1) appears in the exponent of the running time of those algorithms derived from the bidimensionality framework, reducing this coefficient is of extreme importance. The derivations of the bound (1) for bounded genius graphs [7] and for fixed-minor free graphs [6] both depend on the bound for planar graphs and the coefficient c for these classes of graphs, as given by those derivations, depends linearly on the coefficient for planar graphs. Therefore, it is natural to ask for the best possible coefficient in the bound (1) for planar graphs.

Our main results in this paper are the following. We improve the coefficient 4 given by $[20]$ to 3.

Theorem 1.1

For every planar graph G, we have  bw(G)≤3 gm(G).

We also show that the coefficient cannot be smaller than 2.

Theorem 1.2

There is a family of planar graphs {G _h }, h=2,3,… , such that  gm(G _h )=h and  bw(G _h )=2h. Therefore, the coefficient c in the linear bound  bw(G)≤c gm(G)+o( gm(G)) for planar graphs must satisfy c≥2.

The family {G _h } in Theorem 1.2 is explicitly defined as follows. Let k≥3 and h≥1 be integers. A k×h cylinder, denoted by C _k,h , is a graph on vertex set {(i,j)∣0≤i<k,0≤j<h,i,j: integer}, such that vertices (i,j) and (i′,j′) are adjacent if and only if i′≡(i±1)mod k and j′=j or i′=i and |j−j′|=1. In other words, C _k,h is the Cartesian product of a cycle on k vertices and a path on h vertices.

We show that  bw(C _k,h )=min{k,2h} and:

Theorem 1.3

For every integer h≥2, we have  gm(C _2h,h )=h.

With G _h =C _2h,h , Theorem 1.2 follows from this theorem.

The proof of Theorem 1.1 uses an extension of some known upper bounds on the branchwidth of planar graphs and hypergraphs. These upper bounds are based on the “radius” of planar graphs [16], which roughly corresponds to the outerplanarity [1], and are first observed for the treewidth of planar graphs [2, 16] and later for the branchwidth of planar graphs [4, 22] and of planar hypergraphs [22]. Although our results are on planar graphs, our proof of Theorem 1.1 involves hypergraphs and requires a non-trivial extension of the bound of [22], which is embodied by Theorem 3.1 proved in Sect. 3. This extension may be of an independent interest.

The bound of the form (1) on the branchwidth of planar graphs implies similar bounds on the treewidth through the linear relation mentioned above. In particular, the bound  bw(G)≤4 gm(G) of [20] implies  tw(G)≤6 gm(G). Thomas [23] and Grigoriev [12] independently improve the constant in this bound to 5. Theorem 1.1 gives a better constant 4.5. Grigoriev et al. [13] study the tightness of these bounds for treewidth and conjecture that the best constant in this bound is 2. Theorem 1.3 verifies one side of their conjecture that the constant in the bound cannot be smaller than 2, since  tw(C _2h,h )=2h=2 gm(C _2h,h ).

In algorithms derived in the bidimensionality framework mentioned above, we need to compute a large grid minor of the given graph, if we want not only to decide whether the parameter value exceeds k but also to obtain the evidence of a positive answer. It is not known whether a largest ( gm(G)× gm(G)) grid minor of a planar graph can be found in polynomial time. The linear bound of [20] implies a 4-approximation algorithm for this largest grid minor problem on planar graphs. Bodlaender, Grigoriev and Koster give an O(n ²logn) time algorithm with the same approximation ratio for this problem [3], where n is the number of vertices of G. Our proof for Theorem 1.1 is constructive and implies an O(n ²) time (3+ϵ)-approximation algorithm, where ϵ is an arbitrary positive constant.

Theorem 1.1 is a consequence of a slightly more general result.

Theorem 1.4

Let G be a planar graph and k,h be integers with k≥3 and h≥1. Then G has either a minor isomorphic to a k×h cylinder or a branch-decomposition of width at most k+2h−3.

Setting h=k in this theorem gives Theorem 1.1 since a k×h grid is a subgraph of a k×h cylinder. Another interesting case is when h=⌈k/2⌉. As we show in Sect. 6, the branchwidth of C _k,⌈k/2⌉ is k. This motivates us to define another lower bound on the branchwidth:  cm(G) is the largest k such that G contains C _k,⌈k/2⌉ as a minor. Then, our theorem implies that  bw(G)≤2 cm(G) for planar graph G. Thus,  cm(G) is a better lower bound on  bw(G) than  gm(G) in the sense that it is provably tight within a factor of two for planar graphs.

In a related paper [15], based on the construction in Theorem 1.4, we develop efficient approximation algorithms for the largest grid minor and the optimal branch-decomposition of planar graphs. These algorithms give a trade-off between the running time and the approximation ratio. At one end, they run in O(n ²) time and give a (3+ϵ)-approximation for the largest grid-minor and a (2+ϵ)-approximation for the optimal branch-decomposition. At the other end, they run in O(n ^1+ϵ) time and give constant-factor approximations where the constant is, roughly speaking, inversely proportional to ϵ. Although this approximation algorithm for branch-decomposition is less important than that for grid-minor construction, as a polynomial time algorithm for exact optimal decomposition is known for planar graphs [21], (see also [14] for an improvement on the running time), it is still significantly faster than the exact algorithm of $[14]$, which runs in O(n ³) time.

The organization of this paper is as follows. In the next section, we give some preliminaries to what follows. In Sect. 3, we prove Theorem 3.1, the radius-based upper bound. In Sect. 4, based on this bound, we prove Theorem 1.4. In Sect. 5, we show that the upper bound of Theorem 3.1 is tight. In Sect. 6, we determine the branchwidth of a cylinder and prove Theorem 1.3.

2 Preliminaries

Although our results are on graphs, we need to work with hypergraphs, in the inductive proof of Theorem 1.4. To obtain a branch-decomposition of given graph G, we find an appropriate separation of G, construct the branch-decompositions of the two separated parts, and then combine them into a branch-decomposition of G. As will be clear, in each of these “parts” we need to regard the other part as a hyperedge in order for this approach to work.

A hypergraph Γ is a pair of V(Γ), the vertex set of Γ, and E(Γ), the edge set of Γ, together with an incidence relation between V(Γ) and E(Γ). For each e∈E(Γ), we denote by V _Γ(e) the set of vertices in V(Γ) that are incident with e. For A⊆E(Γ), we denote by V _Γ(A) the set of vertices incident with edges in A: V _Γ(A)=⋃ _e∈A V _Γ(e). We denote by ∂ _Γ(A)=V _Γ(A)∩V _Γ(E(Γ)∖A) the set of vertices incident with edges in A and with edges in E(Γ)∖A. The order of an edge e in Γ is the number |V _Γ(e)| of vertices incident with e. A hypergraph Γ is a graph if every edge of Γ has order 2. Note that our definition of a graph allows parallel edges but not self-loops. A hypergraph Δ is a subhypergraph of a hypergraph Γ if V(Δ)⊆V(Γ), E(Δ)⊆E(Γ), and the incidence relation of Δ is a restriction of that of Γ on V(Δ)×E(Δ). Thus, for each e∈E(Δ), V _Δ(e)=V _Γ(e)∩V(Δ).

A walk on graph G is a sequence v ₀, e ₁, v ₁, …, e _k , v _k , which alternates between vertices and edges such that vertex v _i is incident with edge e _i for 1≤i≤k and with e _i+1 for 0≤i<k. We say the above walk is from v ₀ to v _k . For each walk ω on G, we denote by V(ω) and E(ω) the set of vertices and the set of edges, respectively, appearing in ω. The length of a walk is the number of edge occurrences in the walk. We call a walk a path if it does not contain duplicate occurrences of a vertex in the sequence. A walk is closed if it starts and ends with the same vertex. A closed walk is a cycle if no vertex occurs in the walk more than once, except for the first/last vertex. If ω ₁ is a walk to some vertex v and ω ₂ is a walk from v, then we denote by ω ₁+ω ₂ the concatenated walk, ω ₁ followed by ω ₂. It may happen that ω ₁+ω ₂ is a closed walk.

A separation, in this paper, of hypergraph Γ is an ordered pair (A,B) of subsets of edges of Γ, such that A∪B=E(Γ) and A∩B=∅. The order of separation (A,B) is the cardinality of V _Γ(A)∩V _Γ(B)=∂ _Γ(A)=∂ _Γ(B), the set of vertices shared by both sides of the separation.

A branch-decomposition of hypergraph Γ is a pair (T,ϕ), where T is a tree each internal node of which has degree 3 and ϕ is a bijection from the set of leaves of T to E(Γ). Consider an edge a of T and let L ₁ and L ₂ denote the sets of leaves of T in the two respective subtrees of T obtained by removing a. We say that the separation (ϕ(L ₁),ϕ(L ₂)) is associated with this edge a of T and also that this separation belongs to this branch-decomposition. We define the width of the branch-decomposition (T,ϕ) to be the largest order of the separations associated with edges of T or 0 if |E(Γ)|≤1 and hence T does not have any edge. The branchwidth of Γ, denoted by  bw(Γ), is the minimum integer w such that there is a branch-decomposition of Γ with width w. In the rest of this paper, we identify a branch-decomposition (T,ϕ) with the tree T, leaving the bijection implicit and regarding each leaf of T as an edge of Γ.

Let Γ be a hypergraph and (A,B) a separation of Γ. We denote by Γ|A the hypergraph with vertex set V _Γ(B) and edge set B∪{e _A }, where e _A is an edge not in E(Γ), such that each edge e∈E(Γ|A)∖{e _A } is incident with v∈V(Γ|A) if and only if e is incident with v in Γ and e _A is incident with each vertex in ∂ _Γ(A). Thus in Γ|A, all edges in A are replaced by a single edge e _A and each incidence of a vertex in V _Γ(B) with an edge in A is kept as an incidence with e _A . The following lemma is straightforward from the definition of branch-decompositions and is implicit in the literature [18, 21]. It is the use of this lemma that makes us work on hypergraphs even though our main results are on graphs.

Lemma 2.1

Let Γ be a hypergraph and (A,B) a separation of Γ. Let T _A be a branch-decomposition of Γ|A of width w _A and T _B a branch-decomposition of Γ|B of width w _B . Let e _A and e _B be the edges replacing A and B, respectively, in Γ|A and Γ|B. Let tree T be the result of concatenating T _A and T _B by identifying the leaf e _A of T _A and the leaf e _B of T _B and then ignoring the identified node to join the two incident edges into one. Then, T is a branch-decomposition of Γ of width max{w _A ,w _B }.

Corollary 2.1

Let Γ be a hypergraph and ${{\mathcal{A}}}= \{A_{1}, \ldots, A_{m}\}$ a non-empty collection of mutually disjoint subsets of E(Γ). For each 1≤i≤m, let B _i =E(Γ)∖A _i and T _i a branch-decomposition of Γ|B _i and let w _i be the width of this branch-decomposition. Let Γ₀=Γ and, for each 1≤i≤m, let Γ _i =Γ _i−1|A _i . Let T ₀ be a branch-decomposition of Γ _m and let w ₀ be the width of this branch-decomposition. For each 1≤i≤m, let e _i be the edge in T ₀ replacing A _i and $e_{i}'$ the edge in T _i replacing B _i . Construct tree T from T ₀, T ₁, …, T _m by identifying, for 1≤i≤m, leaf e _i of T ₀ with leaf $e_{i}'$ of T _i and then ignoring this node to merge the two incident edges into one. Then, T is a branch-decomposition of Γ and the width of T is max_0≤i≤m w _m .

Let Σ be a fixed sphere. A plane graph is the standard representation of graphs on Σ: a vertex is a point on Σ distinct from any other vertex, an edge is a curve segment on Σ between two vertices such that its interior does not contain any intersection with other edges. A graph is planar if it is isomorphic, as a graph, to some plane graph.

We view walks on a plane graph as a curve on Σ that threads the edges in the walk. Formally, a curve on Σ is a continuous function α:[0,1]→Σ. We say a curve α is simple if α(s)≠α(t) for every pair s,t with 0≤s<t<1. We say a curve α is closed if α(0)=α(1); otherwise it is non-closed. We refer to simple non-closed curves as segments. For each curve α on the sphere, we denote by  rev(α) its reversal:  rev(α)(t)=α(1−t) for 0≤t≤1. A disc on Σ is a set of points of Σ that is homeomorphic to the unit closed disc {(x,y)∣x ²+y ²≤1} on the plane. For a disc d on the sphere, the interior  int(d) is the open region whose closure is d. The boundary of d, denoted by  bd(d) is d∖ int(d). We also denote by ${\mathop {\overrightarrow {\mathrm {bd}}}}(d)$ the simple closed curve whose image is  bd(d) and goes clockwise around d. For each simple closed curve α, we denote by R(α) the closed disc d such that ${\mathop {\overrightarrow {\mathrm {bd}}}}(d) = \alpha$.

We say that a closed curve α on Σ is non-crossing if it has an infinitesimal perturbation that is simple. Let α be a non-crossing closed curve and α′ an infinitesimal perturbation of α that is simple. When we perturb α′ back to α,  int(R(α′)) transforms into a set of open discs. We denote by  discs(α) the set of the closures of these open discs. We denote by R(α) the union α∪⋃ _{d∈ discs(α)} d of α, as a point set, with those discs in  discs(α). Note that this notation is consistent with the special case where α is simple. We say that non-crossing closed curve α is tree-shaped if Σ∖R(α) is connected.

For hypergraphs, we use slightly non-standard embeddings: vertices, edges, and faces are all represented by discs. A tiling is a finite set D of discs on Σ such that ⋃ _d∈D d=Σ and, for each distinct pair d ₁,d ₂∈D, d ₁∩d ₂, is either empty or a single closed segment. In the latter case, we say that the two discs d ₁ and d ₂ are incident with each other. A tiling D is tripartite, if D has a tripartition (D ₁,D ₂,D ₃) such that, for each i=1,2,3 and for each distinct discs d ₁,d ₂∈D _i , d ₁∩d ₂=∅. In this situation, we say that (D ₁,D ₂,D ₃) is a tripartite tiling. See Fig. 1.

Fig. 1

(a) A tripartite tiling with 4 dark, 3 light, and 3 white disks. A white disk in the sphere is drawn as an infinite area in the plane. (b) Not a tiling since a light and a dark disc intersect each other in two segments

A plane hypergraph, in this paper, is a pair Γ=(V(Γ),E(Γ)), such that (V(Γ),E(Γ),F(Γ)) is a tripartite tiling for some set F(Γ) of discs. Note that, given a plane hypergraph Γ, the set F(Γ) such that (V(Γ),E(Γ),F(Γ)) is a tripartite tiling is unique. We call each disc in F(Γ) a face of the plane hypergraph. A plane hypergraph is a hypergraph with the natural incidence relation: vertex v∈V(Γ) and edge e∈E(Γ) are incident with each other if v∩e≠∅. Although not all hypergraphs that are legitimately considered planar can be embedded in the sphere as a plane hypergraph in our definition (for example, if we try to embed a hypergraph that is disconnected or has a cut vertex, there would be a face that is not a closed disc), the restricted definition suffices for our purposes.

A plane hypergraph is trivial if it has exactly one vertex and exactly one edge; otherwise it is non-trivial.

Proposition 2.1

Each edge of a non-trivial plane hypergraph has order at least two.

Proof

Suppose a plane hypergraph Γ has an edge e of order one and let V _Γ(e)={v}. Then, since  bd(e)∩ bd(v) is a single segment (as (V(Γ),E(Γ),F(Γ)) is a tripartite tiling),  bd(e)∖ bd(v) belongs to the boundary  bd(r) of some single face r. Since  bd(v)∩ bd(r) is a single segment with endpoints on  bd(e), {{e},{v},{r}} is a tiling and hence Γ is trivial. □

Let Γ be a plane hypergraph. For each disc d∈V(Γ)∪E(Γ)∪F(Γ), we denote by V _Γ(d), E _Γ(d), and F _Γ(d) the set of discs in V(Γ), E(Γ), and F(Γ), respectively, that are incident with d. We extend this notation to sets of discs: V _Γ(D)=⋃ _d∈D V _Γ(d), and so on.

We define the radial graph [19] of a plane hypergraph, that represents the incidences between vertices and faces. Let Γ be a plane hypergraph and G a plane graph. We say that G is the radial graph of Γ, if there is a bijection ϕ:V(Γ)∪F(Γ)→V(G) such that:

• 1. ϕ(x)∈ int(x) for each x∈V(Γ)∪F(Γ), and

• 2. for each v∈V(Γ) and r∈F(Γ), there is an edge between ϕ(v) and ϕ(r) in ${\mathcal{R}}_{\Gamma}$ if and only if v and r are incident with each other in Γ, and

• 3. for each v∈V(Γ) and r∈F(Γ) incident with each other, the edge of ${\mathcal{R}}_{\Gamma}$ between ϕ(v) and ϕ(r) is contained in  int(v)∪ int(r)∪( bd(v)∩ bd(r)) and intersects  bd(v)∩ bd(r) at exactly one point.

Although the radial graph of Γ is not unique, it is unique up to homeomorphism. We assume a fixed representative radial graph for each plane hypergraph Γ and denote it by ${\mathcal{R}}_{\Gamma}$. We often view ${\mathcal{R}}_{\Gamma}$ as a graph on V(Γ)∪F(Γ) embedded as plane graph ${\mathcal{R}}_{\Gamma}$ in the sphere. In particular, we not only view a walk ω in ${\mathcal{R}}_{\Gamma}$ as a curve on the sphere, we also view it as a walk through V(Γ)∪F(Γ), saying that x∈V(Γ)∪F(Γ) is on ω if the corresponding vertex ϕ(x) is on ω. We also say that the walk is from x to y, where x,y∈V(Γ)∪F(Γ), if it is, more precisely, from ϕ(x) to ϕ(y). For each walk ω on ${\mathcal{R}}_{\Gamma}$, we denote by  length_Γ(ω) the length of this walk, that is, the number of edge occurrences it contains.

Let Γ be a plane hypergraph. For x,y∈V(Γ)∪F(Γ), the Γ-distance between x and y, denoted by  dist_Γ(x,y), is the length of the shortest path in ${\mathcal{R}}_{\Gamma}$ connecting x and y. We extend this notion of distance to sets of vertices/faces: for X,Y⊆V(Γ)∪F(Γ),  dist_Γ(X,Y)=min _x∈X,y∈Y  dist_Γ(x,y). We also write  dist_Γ(x,Y) for  dist_Γ({x},Y) and  dist_Γ(X,y) for  dist_Γ(X,{y}).

Let Γ be a plane hypergraph. For our purposes, a Γ-noose is an oriented cycle of the radial graph of ${\mathcal{R}}_{\Gamma}$ (see Fig. 2).

Fig. 2

A plane hypergraph Γ, its radial graph, and a simple Γ-noose ν: (a) Edges, vertices, and faces are drawn in dark gray, light gray, and white respectively. The solid and dashed segments are the edges of ${\mathcal{R}}_{\Gamma}$. Γ-noose ν consists of dashed edges and is oriented clockwise. (b) The disc R(ν) is shown in dark gray, as a new edge. This entire picture of (b) shows the hypergraph Γ|ν

Let μ be an oriented path or cycle of ${\mathcal{R}}_{\Gamma}$. For x,y∈V(Γ)∪F(Γ) on μ, we denote by μ[x,y] the oriented subpath of μ from x to y. In particular, μ[x,x] is a subpath of length zero consisting of the single vertex of ${\mathcal{R}}_{\Gamma}$ corresponding to x. If μ is an oriented path and y precedes x on μ then μ[x,y] is undefined.

Let ν be a Γ-noose. We say that a separation (A,B) of Γ is induced by ν, if every edge in A is contained in R(ν) and every edge in B is contained in R( rev(ν)). Note that if separation (A,B) is induced by a Γ-noose ν, then V _Γ(A)∩V _Γ(B) is the set of vertices of Γ on ν. Therefore, the order of the separation induced by ν is  length_Γ(ν)/2. Let e be an arbitrary edge of Γ and ν a closed walk of ${\mathcal{R}}_{\Gamma}$ threading the vertices and faces incident with e clockwise around e. Because of our definition of plane hypergraphs based on tiling, each face or vertex incident with e appears exactly once on ν and therefore ν is a Γ-noose that induces separation ({e},E(Γ)∖{e}).

A branch-decomposition of Γ is called a sphere-cut decomposition [9] if each separation that belongs to this decomposition is induced by a Γ-noose.

Let Γ be a plane hypergraph and ν a Γ-noose. We denote by Γ|ν a plane hypergraph with vertex set $\{v \setminus {\mathop {\mathrm {int}}}(R(\nu)) \mid v \in V(\Gamma), v\not\subseteq R(\nu)\}$ and edge set {e∈E(Γ)∣e∩R(ν)=∅}∪{R(ν)}. Note that, in Γ∣ν, we replace all the edges of Γ contained in R(ν) by one edge R(ν) (see Fig. 2(b)). Thus, Γ∣ν is isomorphic to Γ∣A as a hypergraph, where A is the set of edges of Γ contained in R(ν).

We need to slightly generalize the notion of Γ-nooses and related concepts. We call a closed walk ν of ${\mathcal{R}}_{\Gamma}$ a pseudo-Γ-noose if it is non-crossing and tree-shaped as a closed curve on Σ. Let ν be a pseudo-Γ-noose. We denote by Γ|ν the plane hypergraph obtained from Γ by making each disc d∈ discs(ν) into a single edge. More formally, Γ|ν has vertex set $V(\Gamma|\nu) =\{v \setminus\bigcup_{d \in {\mathop {\mathrm {discs}}}(\nu)} {\mathop {\mathrm {int}}}(d)\mid v \in V(\Gamma),v \not\subseteq\bigcup_{d \in {\mathop {\mathrm {discs}}}(\nu)} d\}$ and edge set E(Γ|ν)={e∈E(Γ)∣e∩⋃ _{d∈ discs(ν)} d=∅}∪ discs(ν).

A k×h cylinder, denoted by C _k,h , where k≥3 and h>0 are integers, is the Cartesian product of a cycle on k vertices and a path on h vertices. Its vertex set is V(C _k,h )={(i,j)∣0≤i<k,0≤j<h} and its vertices (i,j) and (i′,j′) are adjacent to each other if and only if i=i′ and (j′−j)≡±1(mod k) or j=j′ and (i′−i)=±1. For 0≤j<h, we call the cycle of C _k,h induced by the vertex set {(i,j)∣0≤i<k} the j-th row, or row j, of C _k,h . For 0≤i<k, we call the path of C _k,h induced by the vertex set {(i,j)∣0≤j<h} the i-th column, or column i, of C _k,h . A k×h grid, denoted by G _k,h , where k,h>0 are integers, is similarly defined as a Cartesian product of a path on k vertices and a path on h vertices. The rows and columns of a grid are defined similarly.

3 A Radius-based Bound on the Branchwidth of Plane Hypergraphs

The goal of this section is to prove an upper bound on the branchwidth of a plane hypergraph based on the radius of its radial graph, which serves as the base case of the inductive proof of Theorem 1.4. This type of radius-based upper bounds are first observed for the treewidth of planar graphs [2, 16], and later for the branchwidth [4, 22]. Tamaki [22] generalizes the bound to plane hypergraphs and gives a linear time algorithm for constructing a branch-decomposition achieving the bound.

We prove and use the following upper bound on the branchwidth of plane hypergraphs which extends the bound in [22]. While the previous bound is with respect to the radius of the radial graph with the center corresponding to a face of the plane hypergraph, our bound is, roughly speaking, with respect to the radius with the center being an edge of the plane hypergraph.

Theorem 3.1

Let k≥2 and d≥1 be integers. Let Γ be a plane hypergraph of maximum edge order at most k. Suppose there is an edge e ₀∈E(Γ) such that  dist_Γ(v,V _Γ(e ₀))≤d for every vertex v∈V(Γ). Then, we have  bw(Γ)≤k+d.

We need some lemmas, all of which involve the notion of d-compact nooses defined below. For a plane hypergraph Γ, a connected subset R of Σ, and elements x,y∈V(Γ)∪F(Γ) such that the vertices of ${\mathcal{R}}_{\Gamma}$ corresponding to x and y are contained in R, we denote by  dist_Γ,R (x,y) the length of the shortest path of ${\mathcal{R}}_{\Gamma}$ from x to y contained in R;  dist_Γ,R (x,y) is undefined if such a path does not exist. This notation is extended to  dist_Γ,R (x,Y),  dist_Γ,R (X,y), and  dist_Γ,R (X,Y), where X and Y are subsets of V(Γ)∪F(Γ).

Let d be a positive integer, Γ a plane hypergraph, ν a pseudo-Γ-noose, and X either a set of vertices or a set of faces of Γ. We say that ν is d-compact for Γ with center X, if the following conditions are satisfied.

• 1. The vertices or the faces in X appear consecutively on ν, that is, there is a subwalk ν′ of ν such that X is the set of vertices on ν′ or the set of faces on ν′.

• 2. For every vertex or face y of Γ that intersects R(ν), we have  dist_Γ,R(ν)(y,X)≤d.

Lemma 3.1

Let ν be a pseudo-Γ-noose and suppose ν is d-compact with center X for some d and X. Then, each Γ-noose μ such that R(μ)∈ discs(ν) is d-compact for some Y, where Y⊆X or |Y|=1.

Proof

Let T be a breadth-first forest, with roots in X, of the subgraph of ${\mathcal{R}}_{\Gamma}$ contained in R(ν). Since ν is d-compact with center X, the depth of this forest is at most d. Let T _μ be the subgraph of T consisting of the vertices and edges in R(μ). Let Y be the set of elements of X that are intersected by R(μ). Since ν is tree-shaped, if Y is non-empty then T _μ is a sub-forest of T with roots in Y and therefore μ is d-compact with center Y. Otherwise, T _μ is a subtree of some tree of T. We redefine Y to be the singleton set consisting of the root of this subtree. Then, μ is d-compact with center Y. □

Lemma 3.2

Let k,d be positive integers. Let Γ be a plane hypergraph with maximum edge order k. Suppose there is a Γ-noose ν that is (d+1)-compact for Γ with some center X with |X|≤k/2+1 and has length  length_Γ(ν)≤2d+2k, then Γ| rev(ν) has a branch-decomposition of width at most d+k.

Proof

Let k, d, Γ, ν, and X be as in the statement of the lemma. We prove by induction on the number of edges of Γ| rev(ν) that Γ| rev(ν) has a branch-decomposition of width at most d+k.

If |E(Γ| rev(ν))|≤2, then the statement is trivial. So, assume that |E(Γ| rev(ν))|≥3.

Let V ₀ denote the set of vertices of Γ that have corresponding vertices in Γ| rev(ν): V ₀={v∈V(Γ)∣v∩R(ν)≠∅}. Similarly define F ₀ to be the set of faces of Γ that have corresponding faces in Γ| rev(ν).

We construct a breadth-first forest T in ${\mathcal{R}}_{\Gamma}$ contained in R(ν) and spans the vertices of ${\mathcal{R}}_{\Gamma}$ corresponding to vertices and faces in V ₀∪F ₀, where the roots are the vertices of ${\mathcal{R}}_{\Gamma}$ corresponding to the elements of X. We view this forest as a forest on V ₀∪F ₀, referring to the vertices in X as the roots of T. For each y∈(V ₀∪F ₀)∖X, let π(y) denote the parent of y in T. For each y∈V ₀∪F ₀, let ρ(y) denote the root of the tree in T to which y belongs and let δ _y denote the oriented path in T from y to ρ(y). Since T is breadth-first, the length of δ _y equals  dist_Γ,R(ν)(y,ρ(y))= dist_Γ,R(ν)(y,X).

Let y be a vertex or face on ν but not in X. We say that δ _y is separating, if neither δ _y nor its reversal is a subpath of ν.

(Case 1) There is some vertex or face y such that δ _y is separating (see Fig. 3). Let y ₀ be a vertex or face on ν but not in X such that $\delta_{y_{0}}$ is separating and is the shortest subject to this condition. Then, π(y ₀) is not on ν; otherwise, $\delta_{\pi (y_{0})}$ would be separating and shorter. Let z be the first vertex or face on $\delta_{y_{0}}$, after y ₀, that is on ν. Then, since $\delta_{z} = \delta_{y_{0}}[z, \rho (y_{0})]$ is not separating, it is either a subpath of ν or  rev(ν). Thus, the subpath $\delta_{y_{0}}[y_{0}, z]$ divides R(ν) into two closed discs R ₁ and R ₂. Let ν _i , for i=1,2, be the Γ-noose such that R(ν _i )=R _i . We define X _i for i=1,2 as follows. If no element of X is on ν _i then X _i ={z}; otherwise, X _i is the set of vertices and faces of X on ν _i .

Fig. 3

Separating paths in Case 1: Drawing conventions are the same as in the previous figure. The Γ-noose ν is shown by a dashed curve and is oriented clockwise. The edges of the breadth-first forest T are shown by solid lines. Where an edge of T is drawn along ν, it must be considered as identical to the corresponding segment of ν. For each face or vertex x, the corresponding vertex in ${\mathcal{R}}_{\Gamma}$ is shown by a small black circle or square, the latter being the case when x∈X. The paths δ _y and δ _y′ in ${\mathcal{R}}_{\Gamma}$, for face y and vertex y′, are both separating. The first point at which δ _y intersects ν after y is in face z≠ρ(y). The first point at which δ _y′ intersects ν after y′ is in the root ρ(y′) of y′

We claim that ν _i is (d+1)-compact for Γ with center X _i for i=1,2. To see this, simply observe that, if y∈V(Γ)∪F(Γ) intersects R _i , i=1 or 2, either δ _y is contained in R _i or z is on δ _y and δ _y [y,z] is contained in R _i .

Fix i=1 or 2. We have confirmed above that ν _i is (d+1)-compact with center X _i . Moreover, we have  length_Γ(ν _i )≤ length_Γ(ν), because ν _i is obtained from ν by replacing a subpath of ν with a chordal path $\delta_{y_{0}}[y_{0},z]$ not longer than the replaced subpath.

Observe furthermore that R(ν _3−i ) contains at least one edge. This is because π(y ₀) is not on ν and, for each vertex or face on ν that is adjacent to y ₀ on ν, there must be an edge to make it distinct from π(y ₀). Therefore, we may apply the induction hypothesis to ν _i and obtain a branch-decomposition of Γ| rev(ν _i ) with width at most d+k.

The hypergraph ((Γ| rev(ν)|ν ₁)|ν ₂, consisting of three edges, has a unique branch-decomposition of width at most d+k. Therefore, applying Corollary 2.1, we obtain from these decompositions a branch-decomposition of Γ| rev(ν) of width at most d+k.

(Case 2) There is no vertex or face y on ν such that δ _y is separating. Thus, for every vertex or face y∈V(Γ)∪F(Γ) that is on ν, δ _y is a subpath of ν or  rev(ν). Let x ₁ and x ₂ be the two elements of X such that all the elements of X are on ν[x ₂,x ₁]. If X is all the vertices on ν or all the faces on ν then such a pair of x ₁ and x ₂ is not unique but an arbitrary choice suffices.

For each y∈V(Γ)∪F(Γ) on ν[x ₁,x ₂], since δ _y is a subpath of ν or  rev(ν) and ends in an element of X, either ρ(y)=x ₁ or ρ(y)=x ₂. Therefore, there are y ₁,y ₂∈V(Γ)∪F(Γ) that appear consecutively on ν[x ₁,x ₂] in this order, such that $\delta_{y_{1}} = {\mathop {\mathrm {rev}}}(\nu)[y_{1}, x_{1}]$ and $\delta_{y_{2}} = \nu[y_{2}, x_{2}]$. Since one of y ₁ and y ₂ is a vertex and the other is a face, it is impossible for both to be at distance d+1 from the roots in T. Therefore, without loss of generality, we may assume that

$${ {\mathop {\mathrm {length}}}_\Gamma }(\delta_{y_1}) \leq d + 1 $$

(2)

and

$${ {\mathop {\mathrm {length}}}_\Gamma }(\delta_{y_2}) \leq d $$

(3)

Let f denote the unique edge of Γ contained in R(ν) that is incident with both y ₁ and y ₂. Let α be the Γ-noose that induces separation (E(Γ)∖{f},{f}). Note that the length of α is twice the order of edge f and is at most 2k.

Let y ₃ be a vertex or face of Γ on α, such that

$${ {\mathop {\mathrm {length}}}_\Gamma }\bigl(\alpha[y_2, y_3]\bigr) = { {\mathop {\mathrm {length}}}_\Gamma }\bigl(\alpha[y_3, y_2]\bigr) \leq k . $$

(4)

This implies that

$${ {\mathop {\mathrm {length}}}_\Gamma }\bigl(\alpha[y_1, y_3]\bigr) \leq k - 1. $$

(5)

Let z be the first point on $\delta_{y_{3}}$ that is also on ν. Since $\delta_{z} = \delta_{y_{3}}[z,\rho (y_{3})]$ is not separating, it must be either a subpath of ν or  rev(ν) (see Fig. 4).

Fig. 4

Case 2, no separating paths from elements on ν : Drawing conventions are the same as in the previous figures. The deviations of the drawings of the pseudo-Γ-nooses ν ₁ and ν ₂ from the corresponding segments of ν and from the corresponding edges of T are for the sake of visibility

Let z ₁=z if z is on ν[ρ(y ₃),y ₁] and z ₁=ρ(y ₃) otherwise. Let z ₂=z if z is on ν[y ₂,ρ(y ₃)] and z ₂=ρ(y ₃) otherwise. Define closed walks ν ₁ and ν ₂ of ${\mathcal{R}}_{\Gamma}$ by

$$\nu_1 = \nu[z_1, y_1] +\alpha[y_1, y_3] + \delta_{y_3}[y_3,z_1],$$

and

$$\nu_2 = \nu[y_2, z_2] + {\mathop {\mathrm {rev}}}{\delta_{y_3}[y_3, z_2]} + \alpha[y_3,y_2].$$

Then, ν _i for i=1,2 is a pseudo-Γ-noose. Note that the discs in  discs(ν ₁), those in  discs(ν ₂), R( rev(ν)), and R( rev(α)) have mutually disjoint interiors and together cover the entire sphere Σ.

For i=1,2, let X _i be the set of elements of X on ν _i if there are such elements; otherwise let X _i ={z _i }. Then, for i=1,2, ν _i is d-compact for Γ with center X _i , since for each vertex or face y that intersects R(ν _i ), δ _y stays in R(ν _i ) until it reaches ρ(y) or z _i .

We first bound the length of ν _i for each i=1,2. Using (2) and (5), we have

where we have used |X ₁|≤|X|≤k/2+1. Since the length of a noose is even, the bound reduces to 2d+2k. Similarly, using (3) and (4), we have

and hence  length_Γ(ν ₂)≤2d+2k.

For each disc g∈ discs(ν ₁)∪ discs(ν ₂), let μ _g denote the Γ-noose such that R(μ _g )=g. Since ν _i for i=1,2 is (d+1)-compact for Γ with center X _i , |X _i |≤|X|, μ _g is (d+1)-compact for Γ with some center X _g , |X _g |≤|X|≤k/2+1, by Lemma 3.1. Therefore, we may apply the induction hypothesis to μ _g and obtain a branch-decomposition T _g of Γ| rev(μ _g ) of width at most d+k.

Let Γ₀=((Γ| rev(ν))|ν ₁)|ν ₂ be the plane hypergraph obtained from Γ| rev(ν) by replacing the edges contained in disc g by a single edge g, for every g∈ discs(ν ₁)∪ discs(ν ₂). Consider separations (A _i ,B _i ), 1≤i≤6, where

• 1. A ₁={R( rev(ν))}, B ₁= discs(ν ₁)∪ discs(ν ₂)∪{f},

• 2. A ₂= discs(ν ₁), B ₂={R( rev(ν))}∪ discs(ν ₂)∪{f},

• 3. A ₃= discs(ν ₂), B ₃={R( rev(ν))}∪ discs(ν ₁)∪{f},

• 4. A ₄={f}, B ₄={R( rev(ν))}∪ discs(ν ₁)∪ discs(ν ₂),

• 5. A ₅= discs(ν ₁)∪{f}, B ₅={R( rev(ν))}∪ discs(ν ₂), and

• 6. A ₆= discs(ν ₂)∪{f}, B ₆={R( rev(ν))}∪ discs(ν ₁).

Separation (A ₁,B ₁) is induced by  rev(ν) and hence its order is at most d+k. Separation (A ₂,B ₂) is induced by the pseudo-Γ-noose ν ₁ and hence its order is at most d+k. Separation (A ₃,B ₃) is induced by the pseudo-Γ-noose ν ₂ and hence its order is at most d+k. The order of separation (A ₄,B ₄) is at most |V _Γ(f)|≤k.

Let $\nu_{1}'$ and $\nu_{2}'$ denote the pseudo-Γ-nooses that induce separations (A ₅,B ₅) and (A ₆,B ₆), respectively. By an analysis similar to the one to bound the length of ν ₁ and ν ₂, we have ${ {\mathop {\mathrm {length}}}_{\Gamma}}(\nu_{i}') \leq { {\mathop {\mathrm {length}}}_{\Gamma}}(\nu_{i}) + 2$, for i=1,2. The bound of  length_Γ(ν _i )≤2d+2k is tight for at most one of i=1,2, because we cannot have |X ₁|=k/2+1 and |X ₂|=k/2+1 at the same time. Therefore, we have ${ {\mathop {\mathrm {length}}}_{\Gamma}}(\nu_{i}') \leq2d + 2k$ for either i=1 or 2 and either separation (A ₅,B ₅) or separation (A ₆,B ₆) has order at most d+k. Assume without loss of generality that (A ₅,B ₅) has order at most d+k.

Let T ₀ be a branch-decomposition of Γ₀ that consists of separations (A _i ,B _i ), 1≤i≤5, together with other separations induced by pseudo-Γ-nooses that are subwalks of ν ₁ or ν ₂. Each of those other separations has order at most d+k, since  length_Γ(ν _i )≤2(d+k) for i=1,2, and therefore the width of T ₀ is at most d+k.

Applying Corollary 2.1 to branch-decompositions T ₀ and T _g , g∈ discs(ν ₁)∪ discs(ν ₂), we obtain a branch-decomposition of Γ| rev(ν) of width at most d+k. □

For the next lemma, we need some additional structure in the radial graph. Let Γ be a plane hypergraph, v a vertex of Γ and r a face of Γ. Because V(Γ)∪E(Γ)∪F(Γ) is a tiling by the definition of plane hypergraphs, v∩r is either empty or a single closed segment. In the latter case, we call the segment the b-arc demarcating v and r. Note that the endpoints of a b-arc are on the boundary of edges of Γ. The graph whose vertices are the edges of Γ and whose edges are b-arcs is essentially the medial graph of Γ introduced in [21].

Lemma 3.3

Let d,k be positive integers, with k≥2. Let Γ be a non-trivial plane hypergraph, with maximum edge order k. Suppose there is an edge e ₀∈E(Γ) such that  dist_Γ(v,V _Γ(e ₀))≤d for every vertex v∈V(Γ). Then, there are t pseudo-Γ-nooses ν _j , 1≤j≤t, where t=2 or 3, t sets X _j , 1≤j≤t, each of which is either a set of vertices or set of faces of Γ, and an edge f∈E(Γ) such that the following conditions are satisfied.

• 1. The interiors of the discs in ⋃_1≤j≤t  discs(ν _j ) are mutually disjoint.

• 2. For 1≤j≤t, e ₀∩R(ν _j )=∅ and f∩R(ν _j )=∅.

• 3. E(Γ)=⋃_1≤j≤t (E _j )∪{e ₀,f}, where E _j is the set of edges of Γ contained in R(ν _j ).

• 4. For 1≤j≤t, the elements of X _j appear consecutively on ν _j .

• 5. For 1≤j≤t, ν _j is (d+1)-compact for Γ with center X _j .

• 6. For 1≤j≤t, |X _j |≤k/2+1.

• 7. For 1≤j≤t,  length_Γ(ν _j )≤2(d+k).

• 8. Let Γ _j , 0≤j≤t, be defined by Γ₀=Γ and Γ _j =Γ _j−1|ν _j for 0<j≤t. Then, Γ _t , which has edges e ₀, f, and those in  discs(ν _j ), 1≤j≤t, has a branch-decomposition of width at most d+k.

Proof

Let β denote the Γ-noose that induces separation (E(Γ)∖{e ₀},{e ₀}) of Γ.

We construct a breadth-first spanning forest T of ${\mathcal{R}}_{\Gamma}$ where the roots are the vertices of ${\mathcal{R}}_{\Gamma}$ corresponding to vertices of Γ incident with e ₀. As we did in the proof of Lemma 3.2, we view this forest as a forest on V(Γ)∪F(Γ) with roots in V _Γ(e ₀). For each vertex or face y∈V(Γ)∪F(Γ), we denote by ρ(y) the root of the tree in this forest y belongs to, and δ _y the oriented path in this forest from y to ρ(y). If $y \not\in V_{\Gamma}(e_{0})$, we furthermore denote by π(y) the parent of y in this forest. The choice of this breadth-first forest is arbitrary except for the following rule: for each face r incident with e ₀, π(r) is the vertex on β immediately after r.

For b-arc a of Γ, we denote by  edges(a) the set of two edges of Γ to which the endpoints of a belong. Since Γ is non-trivial, we have | edges(a)|=2 for each b-arc a of Γ.

Let T ^∗ be the graph whose vertex set is E(Γ) and whose edges are the b-arcs of Γ that are not intersected by edges of T. Let T ^† denote the subgraph of T ^∗ induced by E(Γ)∖{e ₀}. Since Γ is non-trivial, the order of e ₀ is at least two by Proposition 2.1 and E(Γ)∖{e ₀} is non-empty. We claim that T ^† is a tree. Suppose for contradiction that T ^† is not connected. Then, there is a cycle C of ${\mathcal{R}}_{\Gamma}$ each edge of which belongs either to T or to β. Since T is a forest, C must contain an edge that belongs to β. Let p be a maximal subpath of C that does not contain an edge of β. Then p is between some x,y⊆V _Γ(e ₀)∪F _Γ(e ₀). Since p is a path in T, x and y must belong to the same tree in T. But this is possible only if one of x and y is the root of the tree and the other is an immediate child of the root, a contradiction because then the edge between x and y together with p would form a cycle in T. Therefore, T ^† is connected. Suppose for contradiction that T ^† contains a cycle. Then, the b-arcs forming this cycle, together with the edges of Γ on this cycle, separate Σ into two regions both containing vertices or faces, a contradiction since there must be an edge of T connecting these two regions. Therefore, T ^† does not have a cycle and hence is a tree.

For each b-arc a of T ^† and each edge e∈ edges(a), let A(a,e) denote the set of edges e′∈E(Γ)∖{e ₀} such that the path between e and e′ in T ^† contains a. Thus, if  edges(a)={e ₁,e ₂} then (A(a,e ₁),A(a,e ₂)) is a bipartition of E(Γ)∖{e ₀}, with e ₁∈A(a,e ₂) and e ₂∈A(a,e ₁).

We now orient each b-arc of T ^∗. Each b-arc with one end on  bd(e ₀) is oriented away from e ₀. Other b-arcs, that is, those in T ^†, are oriented as follows. For each A⊆E(Γ)∖{e ₀}, let  weight(A) denote the number of b-arcs between A and e ₀ that are in T ^∗. Note that  weight(E(Γ)∖{e ₀}) is the order of e ₀, as each vertex of Γ incident with e ₀ contributes exactly one b-arc in T ^∗ incident with e ₀. Let  edges(a)={e ₁,e ₂}. We orient a from e ₁ to e ₂ if  weight(A(a,e ₁))> weight(A(a,e ₂)) or  weight(A(a,e ₁))= weight(A(a,e ₂)) and |A(a,e ₁)|>|A(a,e ₂)|, from e ₂ to e ₁ if  weight(A(a,e ₂))> weight(A(a,e ₁)) or  weight(A(a,e ₂))= weight(A(a,e ₁)) and |A(a,e ₂)|>|A(a,e ₁)|, and arbitrarily if  weight(A(a,e ₁))= weight(A(a,e ₂)) and |A(a,e ₁)|=|A(a,e ₂)|. Since T ^† is a tree, the resulting directed graph has a sink f.

We say that a face r incident with f is bad if  dist_Γ(r,V _Γ(e ₀))=d+1 and π(r) is not incident with f. If there is a bad face r, we eliminate it by redefining π(r) to be one of the two vertices incident with r and f, modifying T and hence T ^∗ and T ^†, provided that f remains a sink of the orientated version of T ^∗. Let v ₁ and v ₂ be two vertices incident with both r and f. Let a _i , for i=1,2, the b-arc demarcating r and v _i . Then, f ceases to be a sink after the above modification only if  weight(a ₁,f)+ weight(a ₂,f)≥|V _Γ(e ₀)|/2. Since this weight condition may hold for at most one face incident with f, if there are more than one bad face, then we can eliminate at least one of them. Thus, we assume in the following that there is at most one bad face.

Let α be the Γ-noose that induces separation (E(Γ)∖{f},{f}). Let a ₁,…,a _m be the b-arcs of Γ incident with f that are in T ^∗, listed in the counterclockwise order around f. For convenience, we denote a _m also by a ₀. We claim that m≥2. If f is the only edge of Γ other than e ₀, this is the case since the order of e ₀ is at least two. Otherwise, f has an incoming b-arc in T ^∗, say a, and because of the orientation of this b-arc, we have that  weight(A(a,e))>0, where e is the other end of a, and hence f has at least one more b-arc of T ^∗ incident with it.

For each 0≤i≤m, let ${x_{i}^{+}}$ and ${x_{i}^{-}}$ denote the vertex and the face (not respectively) that a _i demarcates, with ${x_{i}^{+}}$ on the left and ${x_{i}^{-}}$ on the right as we look from f toward the other end of a _i (see Fig. 5).

Fig. 5

Forest T and tree T ^†: The edges of the breadth-first forest T are shown with thin solid lines. The oriented b-arcs of T ^∗ are shown with thick arrows: those in T ^† are black and others are gray. The b-arcs of T ^† around the sink f and the vertices/faces incident with them are shown with symbols to denote them

For each pair of distinct indices i,j, 1≤i,j≤m, we define a pseudo-Γ-noose μ _ij as follows. Fix i,j, 1≤i,j≤m, i≠j. If $\rho ({x_{i}^{-}}) \neq \rho ({x_{j-1}^{+}})$ then we let

$$\mu_{ij} = \delta_{{x_{j-1}^{+}}} + \beta\bigl[\rho ({x_{j-1}^{+}}), \rho ({x_i^{-}})\bigr] + {\mathop {\mathrm {rev}}}(\delta_{x_i^{-}}) + \alpha[{x_i^{-}},{x_{j-1}^{+}}]. $$

(6)

Note that μ _ij may not be a Γ-noose because $\beta[\rho ({x_{j-1}^{+}}), \rho ({x_{i}^{-}})]$ and $\alpha[{x_{i}^{-}}, {x_{j-1}^{+}}]$ may share vertices. If $\rho ({x_{i}^{-}}) = \rho ({x_{j-1}^{+}})$ then we let z _ij be the first vertex or face that is on both $\delta_{{x_{i}^{-}}}$ and $\delta_{{x_{j-1}^{+}}}$ and let

$$\mu_{ij} = \delta_{{x_{j-1}^{+}}}[{x_{j-1}^{+}}, z_{ij}] + {\mathop {\mathrm {rev}}}\bigl(\delta_{x_i^{-}}[z_{ij}, {x_i^{-}}]\bigr) + \alpha[{x_i^{-}},{x_{j-1}^{+}}]. $$

(7)

Note that the orientation of μ _ij specified here, in either of the above two cases, is such that f lies outside of R(μ _ij ) (see Fig. 6). If e ₀ also lies outside of R(μ _ij ) then we say that the ordered pair (i,j) is legal; otherwise it is illegal.

Fig. 6

Pseudo-Γ-nooses μ _ij : For the same instance as in the previous figure, only μ ₅₁=μ ₀₁ and μ ₃₄ are shown. The pseudo-Γ-nooses in this example are Γ-nooses

Let $X'_{ij}$ be the set of vertices on μ _ij that are incident with e ₀. If $X'_{ij} \neq\emptyset$ then we let $X_{ij} = X'_{ij}$; otherwise, if pair (i,j) is legal then we let X _ij ={z _ij }; otherwise we let X _ij =V _Γ(e ₀)∪F _Γ(e ₀). The last case is for technical convenience utilized later.

We claim that if μ _ij is legal then it is (d+1)-compact with center X _ij . To see this, first observe that, for every vertex v of Γ that intersects R(μ _ij ), the path in T from v to ρ(v) stays in R(μ _ij ) until it reaches either z _ij (when μ _ij is defined by (7) or a vertex or face incident with e ₀. Moreover, if μ _ij is defined by (6) then we have ρ(v)∈X _ij . If μ _ij is defined by (7) then we have either ρ(v)∈X _ij or the face that is on δ _v immediately before ρ(v) is incident with another root of T in X _ij . Therefore, we have ${\mathop {\mathrm {dist}}}_{\Gamma|{\mathop {\mathrm {rev}}}(\mu_{ij})}(v, X_{ij}) \leq d$ for every vertex v and ${\mathop {\mathrm {dist}}}_{\Gamma|{\mathop {\mathrm {rev}}}(\mu_{ij})}(r, X_{ij}) \leq d + 1$ for every face r.

To bound the length of μ _ij , let l _ij denote the number of vertices of Γ on $\mu_{ij}[{x_{i}^{-}}, {x_{j-1}^{+}}]$. From the assumption in the lemma, ${ {\mathop {\mathrm {length}}}_{\Gamma}}(\delta_{{x_{i}^{-}}}) \leq d$ if ${x_{i}^{-}}$ is a vertex and ${ {\mathop {\mathrm {length}}}_{\Gamma}}(\delta_{{x_{i}^{-}}}) \leq d + 1$ if ${x_{i}^{-}}$ is a face. In either case, at most d/2 vertices of Γ are on $\delta_{{x_{i}^{-}}}$, excluding $\rho ({x_{i}^{-}})$. Similarly, at most d/2 vertices of Γ are on $\delta_{{x_{j-1}^{+}}}$, excluding $\rho ({x_{j-1}^{+}})$. Therefore, the number of vertices of Γ on $\mu_{ij}[{x_{j-1}^{+}}, {x_{i}^{-}}]$ excluding those in $\{{x_{i}^{-}}, {x_{j-1}^{+}}\} \cup X_{ij}$ is at most d. Therefore, we have

$${ {\mathop {\mathrm {length}}}_\Gamma }(\mu_{ij}) \leq2\bigl(d + l_{ij} + |X_{ij}|\bigr) $$

(8)

We say that an ordered pair (i,j), 1≤i,j≤m, is good if l _ij ≤(k−1)/2 and |X _ij |≤|V _Γ(e ₀)|/2+1. Note that from the technical definition of X _ij for illegal pair (i,j), an illegal pair can never be good.

We say that a t-tuple of indices (i ₁,i ₂,…,i _t =i ₀) with 1≤i ₁<i ₂<…<i _t ≤m is cyclically good, if, for each 1≤j≤t, pair (i _j−1,i _j ) is good. We claim that there is a cyclically good t-tuple with t=2 or 3. To show this, we first show that the m-tuple (1,2,…,m) is cyclically good. Fix 0≤i<m. We show that pair (i,i+1) is good. If $\rho ({x_{i}^{-}}) = \rho ({x_{i}^{+}})$ and hence X _i(i+1)={z _i(i+1)} then certainly |X _i(i+1)|≤|V _Γ(e ₀)|/2+1. Otherwise, each vertex of X _i(i+1) has a b-arc in T ^∗ on its boundary that directly connects an edge in A(a _i ,f) and e ₀. Therefore, we have |X _i(i+1)|≤ weight(A(a _i ,f)). From the definition of the orientation of the b-arcs in T ^† and the choice of f as a sink, we have  weight(A(a _i ,f))≤|V _Γ(e ₀)|/2. Therefore, we have |X _i(i+1)|≤|V _Γ(e ₀)|/2. We also have l _i(i+1)=0, since ${x_{i}^{-}}$ and ${x_{i}^{+}}$ are adjacent around f and, though one of them is a vertex, it is not counted in l _i(i+1). Therefore, pair (i,i+1) is good for every 0≤i<m and hence (1,2,…,m) is cyclically good.

We now show that, if there is a cyclically good t-tuple and t>3, then there is a cyclically good (t−1)-tuple. Together with the existence of a cyclically good m-tuple we proved above, this shows that there is a cyclically good t-tuple with t≤3.

Let (i ₁,i ₂,…,i _t =i ₀) be a cyclically good t-tuple, with t>3.

Since ${\mathop {\mathrm {int}}}(R(\mu_{i_{0}i_{2}}))$ and ${\mathop {\mathrm {int}}}(R(\mu_{i_{2} i_{4}}))$ are disjoint, $X_{i_{0}i_{2}}$ and $X_{i_{2} i_{4}}$ may have at most two elements in common. Therefore, we have $|X_{i_{0} i_{2}}| + |X_{i_{2} i_{4}}|\leq|V_{\Gamma}(e_{0})| + 2$, and either $|X_{i_{0} i_{2}}| \leq|V_{\Gamma}(e_{0})|/2 + 1$ or $|X_{i_{2} i_{4}}| \leq|V_{\Gamma}(e_{0})|/2 + 1$. Similarly, we have either $|X_{i_{1} i_{3}}| \leq |V_{\Gamma}(e_{0})|/2 + 1$ or $|X_{i_{3} i_{5}}| \leq |V_{\Gamma}(e_{0})|/2 + 1$ if t≥5 and either $|X_{i_{1} i_{3}}| \leq |V_{\Gamma}(e_{0})|/2 + 1$ or $|X_{i_{3} i_{1}}| \leq |V_{\Gamma}(e_{0})|/2 + 1$ if t=4. With possible renumbering, we need to consider only two cases: (Case 1) t≥5, $|X_{i_{0} i_{2}}| \leq|V_{\Gamma}(e_{0})|/2 + 1$, and $|X_{i_{3} i_{5}}| \leq|V_{\Gamma}(e_{0})|/2 + 1$; (Case 2) $|X_{i_{0} i_{2}}| \leq|V_{\Gamma}(e_{0})|/2 + 1$, and $|X_{i_{1} i_{3}}| \leq|V_{\Gamma}(e_{0})|/2 + 1$.

(Case 1) First observe that $l_{i_{0} i_{2}} + l_{i_{3} i_{5}}\leq|V_{\Gamma}(f)| - 1 \leq k - 1$, since at least one vertex of V _Γ(f) is between $x^{+}_{i_{2} - 1}$ inclusive and $x^{-}_{i_{3}}$, and is not counted in $l_{i_{0} i_{2}} + l_{i_{3} i_{5}}$. Therefore, we have either $l_{i_{0} i_{2}} \leq(k - 1)/ 2$ or $l_{i_{3} i_{5}} \leq(k - 1)/ 2$ and hence either pair (i ₀,i ₂) or pair (i ₃,i ₅) is good. It follows that either (t−1)-tuple (i ₂,i ₃,…,i _t ) or (t−1) tuple (i ₁,i ₂,i ₃,i ₅,…,i _t ) is cyclically good.

(Case 2) We claim that, for each i with i ₁≤i<i ₂, either $l_{i_{0}(i + 1)} \leq(k - 1) / 2$ or $l_{i i_{3}} \leq(k - 1) / 2$. Since $l_{i_{0} i_{1}} \leq(k - 1) / 2$ and $l_{i_{2} i_{3}} \leq(k - 1) / 2$, this claim implies that if we let i ^∗ be the largest i in the range i ₁≤i≤i ₂ such that $l_{i_{0} i} \leq(k - 1) / 2$ then $l_{i_{0} i^{*}} \leq(k - 1) / 2$ and $l_{i^{*} i_{3}} \leq(k - 1) / 2$. Therefore, both pairs (i ₀,i ^∗) and (i ^∗,i ₃) are good and hence (i ^∗,i ₃,…,i _t ) is a cyclically good (t−1)-tuple.

To verify the claim, let i be arbitrary with i ₁≤i<i ₂. No vertices in V _Γ(f) are counted both in $l_{i_{0}(i + 1)}$ and in $l_{i i_{3}}$, since if $x^{-}_{i}$ is a vertex then it is not counted in $l_{i i_{3}}$ and if $x^{+}_{i}$ is a vertex then it is not counted in $l_{i_{0} (i + 1)}$. Moreover, from $x^{+}_{i_{3} - 1}$ to $x^{-}_{i_{0}}$ in counterclockwise order around f, there must be at least one vertex since t≥4 and hence i ₃≠i ₀, which is not counted either in $l_{i_{0} (i + 1)}$ or in $l_{i i_{3}}$. Therefore we have $l_{i_{0} (i + 1)} + l_{i i_{3}} \leq k - 1$, and hence either $l_{i_{0} (i + 1)} \leq(k - 1) / 2$ or $l_{i i_{3}} \leq(k - 1)/2$. This verifies the claim.

We have verified that there is a cyclically good t-tuple with t≤3. Let (i ₁,…,i _t =i ₀) be a cyclically good t-tuple with t≤3. It is clear that t cannot be 1, so t=2 or 3. We set $\nu_{j}=\mu_{i_{j-1} i_{j}}$ and $X_{j}=X_{i_{j-1} i_{j}}$ for 1≤j≤t.

We now verify that the conditions in the lemma are satisfied.

• 1. Let g ₁∈ discs(ν _j ) and g ₂∈ discs(ν _j′) be two distinct discs, where 1≤j,j′≤t. If j=j′ then  int(g ₁)∩ int(g ₂)=∅ by the definition of pseudo-Γ-nooses. If j≠j′, then  int(g ₁)∩ int(g ₂)=∅ because the subtree of T ^† enclosed by ν _j is disjoint from the subtree enclosed by ν _j′.

• 2. For 1≤j≤t, we have f∩R(ν _j )=∅ because of the definition of ν _j and e ₀∩R(ν _j )=∅ because the pair i _j−1,i _j is chosen to be legal.

• 3. E(Γ)=⋃_1≤j≤t (E _j )∪{e ₀,f}, where E _j is the set of edges of Γ contained in R(ν _j ), because ⋃_1≤i≤m A(a _i ,f)=E(Γ)∖{e ₀,f} and, for each 1≤i≤m, A(a _i ,f) is contained in R(ν _j ) for some 1≤j≤t.

• 4. For 1≤j≤t, elements of X _j appear consecutively on ν _j , by the definition of X _j .

• 5. For 1≤j≤t, ν _j is (d+1)-compact for Γ with center X _j . This is a consequence of the more general statement that μ _ij is (d+1)-compact for X _ij for every legal pair of indices 1≤i,j≤m, which we have already verified.

• 6. For 1≤j≤t, we have |X _j |≤|V _Γ(e ₀)|/2+1≤k/2+1 because $|X_{i_{j-1} i_{j}}|\leq|V_{\Gamma}(e_{0})|/2 + 1$ from the definition of a good pair.

• 7. For 1≤j≤t,  length_Γ(ν _j )≤2(d+k). This follows from the fact that the pair (i _j−1,i _j ) is good: we have $l_{i_{j-1} i_{j}}\leq(k-1)/2$, $|X_{i_{j-1} i_{j}}| \leq k/2+1$, and hence $l_{i_{j-1} i_{j}}+|X_{i_{j-1} i_{j}}| \leq k+1/2$. But since this is the number of vertices and possibly faces, it must be integral and we may remove 1/2 from the bound. Our required bound follows from inequality (8).

It remains to verify the last condition that Γ _t has branchwidth at most d+k. We show this for t=3; the case t=2 is similar and simpler. For indices j=1,2,3, we use a cyclic arithmetic where j+1=1 if j=3 and j−1=3 if j=1.

Recall that the edges of Γ₃ are e ₀, f, and the discs in  discs(ν _j ), j=1,2,3. For each subset I of {1,2,3}, let S _I denote the separation (A,B) of Γ₃, with A={f}∪⋃ _j∈I  discs(ν _j ) and $B = \{e_{0}\} \cup\bigcup_{j \not\in I} {\mathop {\mathrm {discs}}}(\nu_{j})$. Our goal is to find two distinct indices j ₁,j ₂∈{1,2,3} such that two separations $S_{\{j_{1}\}}$ and $S_{\{j_{1}, j_{2}\}}$ have order at most k+d. These separations essentially determine a branch-decomposition of Γ₃. This branch-decomposition is completed by adding a set of separations, for each j=1,2,3, used to decompose  discs(ν _j ) further. Such a set of separations can be chosen so that each separation in the set is induced by a pseudo-Γ-noose that is a subwalk of ν _j and hence has order at most d+k. Therefore, finding indices j ₁ and j ₂ as above will finish the proof.

For j=1,2,3, let α _j denote the common subpath of α and ν _j ; similarly, β _j is the common subpath of β and ν _j . If ν _j and β do not have any edge or vertex in common, we say that β _j is empty.

Suppose first that β _j is empty for some j, say j=1. If furthermore β ₂ is empty, then index pair (i ₃,i ₁) used to define β ₃ would be illegal, so β ₂ is not empty. Similarly, β ₃ is not empty.

Separation S _{1}=({f}∪ discs(ν ₁),{e ₀}∪ discs(ν ₂)∪ discs(ν ₃)) is induced by a Γ-noose consisting of  rev(α ₂+α ₃) and a path in T of length at most 2d. The length of this Γ-noose is at most 2d+2k and therefore the order of separation S _{1} is at most d+k. Separation S _{1,2}=({f}∪ discs(ν ₁)∪ discs(ν ₂),{e ₀}∪ discs(ν ₃)) is induced by a Γ-noose obtained from  rev(ν ₃) by replacing its subpath  rev(β ₃) by β ₂. The length of this Γ-noose is at most 2d+2k, by a calculation similar to the one that bounded the length of ν ₃. Therefore, the order of separation S _{1,2} is at most d+k and we are done in this case.

So suppose β _j is non-empty for j=1,2,3. For j=1,2,3, let ν _j be decomposed as

$$\nu_j = \alpha_j + \gamma_j +\beta_j + {\mathop {\mathrm {rev}}}(\gamma_{j - 1}),$$

where γ _j is a path of T (see Fig. 7). Because of the radius condition, we have  length_Γ(γ _j )≤d+1 for j=1,2,3 and, because there is at most one bad face, we have  length_Γ(γ _j )=d+1 for at most one value of j.

Fig. 7

α _j , β _j , and γ _j

First suppose  length_Γ(α)≤ length_Γ(β). Then, for at least one j∈{1,2,3},  length_Γ(α _j )+ length_Γ(α _j+1)≤ length_Γ(β _j )+ length_Γ(β _j+1) holds. Without loss of generality, we assume  length_Γ(α ₂)+ length_Γ(α ₃)≤ length_Γ(β ₂)+ length_Γ(β ₃). Define a closed walk ω ₁ in ${\mathcal{R}}_{\Gamma}$ by

$$\omega_1 = {\mathop {\mathrm {rev}}}(\alpha_2 + \alpha_3) +\gamma_1 + \beta_1 + {\mathop {\mathrm {rev}}}(\gamma_3).$$

Then ω ₁ is a Γ-noose that induces separation S _{1}. Its length is bounded by

Therefore, the order of separation S _{1} is at most d+k. Since  length_Γ(α ₂)+ length_Γ(α ₃)≤ length_Γ(β ₂)+ length_Γ(β ₃), we have either  length_Γ(α ₂)≤ length_Γ(β ₂) or  length_Γ(α ₃)≤ length_Γ(β ₃). We assume the latter is the case; the other case is similar. Define a closed walk ω ₂ in ${\mathcal{R}}_{\Gamma}$ by

$$\omega_2 = {\mathop {\mathrm {rev}}}(\alpha_3) + \gamma_2 +\beta_2 + \beta_1 + {\mathop {\mathrm {rev}}}(\gamma_3).$$

Then ω ₂ is a Γ-noose that induces separation S _{1,2}. Its length is bounded by

Therefore the order of S _{2,3} is at most d+k and we are done in this case as well.

Finally suppose  length_Γ(α)> length_Γ(β). Similarly as above, we find two distinct indices j ₁ and j ₂ such that ${ {\mathop {\mathrm {length}}}_{\Gamma}}(\beta_{j_{1}}) < { {\mathop {\mathrm {length}}}_{\Gamma}}(\alpha_{j_{1}})$ and ${ {\mathop {\mathrm {length}}}_{\Gamma}}(\beta_{j_{1}}) + { {\mathop {\mathrm {length}}}_{\Gamma}}(\beta_{j_{2}}) <{ {\mathop {\mathrm {length}}}_{\Gamma}}(\alpha_{j_{1}}) + { {\mathop {\mathrm {length}}}_{\Gamma}}(\alpha_{j_{2}})$. We assume below that j ₁=1 and j ₂=2. Other cases are similar. Define Γ-nooses ω ₁ and ω ₂ as above. Then,

and

Therefore, the orders of both separations S _{1} and S _{1,2} are at most d+k and we are done. □

We are now ready to prove Theorem 3.1.

Let k≥2 and d≥1 be integers. Let Γ be a plane hypergraph of maximum edge order at most k and suppose there is an edge e ₀∈E(Γ) such that  dist_Γ(v,V _Γ(e ₀))≤d for every vertex v∈V(Γ). We construct a branch-decomposition of Γ with width at most d+k as follows. We first apply Lemma 3.3 and obtain an edge f, t pseudo-Γ-nooses ν _j , 1≤j≤t, where t=2 or 3, and t sets of vertices or faces X _j , 1≤j≤t, such that the conditions of Lemma 3.3 are satisfied. We construct Γ _j , 1≤j≤t, as in the condition in the lemma, and a branch-decomposition T _t of Γ _t with width at most d+k.

Let D=⋃_1≤j≤t  discs(ν _j ). For each disc g∈D, let ω _g be the Γ-noose such that R(ω _g )=g. Since ν _j for j=1,2,3 is (d+1)-compact for Γ with center X _j with |X _j |≤k/2+1, ω _g for each g∈D is (d+1)-compact for Γ with center Y _g for some Y _g with |Y _g |≤k/2+1, by Lemma 3.1. We apply Lemma 3.2 to ω _g for each g∈D, and obtain a branch-decomposition T _g of Γ| rev(ω _g ) with width at most d+k. Applying Corollary 2.1 to Γ∣ rev(ω _g ), g∈D, and Γ _t , we combine the branch-decompositions T _g , g∈D, and T _t into a branch-decomposition of Γ of width at most d+k. This completes the proof of Theorem 3.1.

4 Upper Bound

The goal of this section is to prove Theorem 1.4, which is restated below.

Theorem 1.4

Let G be a planar graph and k,h be integers with k≥3 and h≥1. Then G has either a minor isomorphic to a k×h cylinder or a branch-decomposition of width at most k+2h−3.

We first sketch the proof ideas. The proof is by induction and at each induction step, we work on a plane hypergraph Γ, which is almost a graph, in the sense that all but one of its edges, say e ₀, are of order 2. The order of e ₀ is smaller than k. We construct contours each of which consists of vertices of Γ that are equidistant from F _Γ(e ₀) in the radial graph of Γ. If the nest of contours is shallow then we may apply Theorem 3.1 and obtain a branch-decomposition of small width. If the nest is deep, then we look for promising separations of small order that separate those deep contours from e ₀ and commit to those separations, hoping that they will lead to a branch-decomposition of small width. We may fail to find a separation of small order between e ₀ and some deep contour, but then there must be a large enough number of vertex-disjoint paths from V _Γ(e ₀) to the vertices on the contour, which will form a large cylinder minor together with the nested contours.

We need some definitions and lemmas to formalize these ideas.

Let Γ be a plane hypergraph that is a graph, that is, every edge of Γ has order two, and r ₀ a face of Γ. (We could alternatively say that Γ is a biconnected plane graph, but we would then need to repeat many definitions already done for plane hypergraphs to adapt them to plane graphs.) Let ${\mathcal{R}}_{\Gamma}(r_{0}, i)$ denote the subgraph of the radial graph ${\mathcal{R}}_{\Gamma}$ induced by the set of vertices corresponding to the set of vertices and faces of Γ that are at distance 2i+2 or greater from r ₀ in ${\mathcal{R}}_{\Gamma}$. Then, each connected component of ${\mathcal{R}}_{\Gamma}(r_{0}, i)$ is bounded by a cycle of Γ consisting of vertices at distance 2i+1 from r ₀. We call such a cycle a contour at depth i from r ₀. Let c be a contour at depth i from r ₀. We say that a vertex or face x of Γ is inside of contour c, if the vertex of ${\mathcal{R}}_{\Gamma}$ corresponding to x belongs to the connected component of ${\mathcal{R}}_{\Gamma}(r_{0}, i)$ bounded by c; x is outside of c if it is not inside of c and is not a vertex on c. We say that Γ-noose ν separates contour c from r ₀, if R(ν) contains all edges of c and R( rev(ν)) contains r ₀. We say ν crosses c if some face of Γ that lies inside of c is on ν.

We need the following version of the Menger’s theorem.

Lemma 4.1

Let Γ be a plane hypergraph that is a graph, c ₁ and c ₂ two vertex-disjoint cycles of Γ, and k a positive integer. We say that a Γ-noose ν separates c ₁ from c ₂ if R(ν) contains all the edges of c ₁ and R( rev(ν)) contains all the edges of c ₂. Then, there are k pairwise vertex-disjoint paths from V(c ₁) to V(c ₂), if and only if there is no Γ-noose ν of length smaller than 2k that separates c ₁ from c ₂.

Proof

One direction is straightforward: if ν is a Γ-noose that separates c ₁ from c ₂, then every path of Γ from V(c ₁) to V(c ₂) must intersect ν. Therefore, k pairwise vertex-disjoint paths between c ₁ and c ₂ imply  length_Γ(ν)≥2k.

To prove the other direction, we suppose that P is a largest set of pairwise vertex-disjoint paths from V(c ₁) to V(c ₂) and show that there is a Γ-noose of length 2|P| that separates c ₁ from c ₂. Recall that paths in our definition based on walks are oriented. The proof is an adaptation of the standard augmenting path argument for the maxflow-mincut theorem. Let V(P)=⋃ _p∈P V(p) and E(P)=⋃ _p∈P E(p). Let q be an arbitrary path in Γ. We say that an edge e in E(q)∩E(P) is a backward edge of q, if the orientation of e in q is the opposite from its orientation in a path in P. We say that a path q of Γ is a partial augmenting path if it is from a vertex on c ₁, each edge e in E(q)∩E(P) is a backward edge of q, and each vertex in V(q)∩V(P) except for the last vertex of q is incident with a backward edge of q. Let S denote the set of vertices of Γ such that there is a partial augmenting path ending at v. Then, S does not contain any vertex on c ₂, since otherwise there is a partial augmenting path that ends in a vertex on c ₂, which would imply that there are |P|+1 vertex-disjoint paths from V(c ₁) to V(c ₂), a contradiction. Note that V(c ₁)⊆S, since the path of length zero from each vertex on c ₁ is a partial augmenting path by definition. For each path p∈P, let v _p denote the last vertex on p that belongs to S and e _p the edge on p that immediately follows v _p . By the definition of partial augmenting paths, all vertices of p up to v _p belong to S.

Let u and v be an arbitrary pair of vertices adjacent in Γ such that u∈S and $v \not\in S$. Then there is a partial augmenting path ending with u and this partial augmenting path cannot be extended to v, u and v consecutively appear in some path p∈P in this order. This means that u=v _p and the edge between u and v is e _p . Therefore, every path from V(c ₁) to V(c ₂) contains v _p and e _p for some p∈P.

Let V _P ={v _p ∣p∈P}, E _p ={e _p ∣p∈P}, and F _P the set of faces of Γ incident with some edge in E _p . Let H be the subgraph of the radial graph ${\mathcal{R}}_{\Gamma}$ induced by the set of vertices corresponding to the elements of V _P ∪F _P . Then, every path of Γ from V(c ₁) to V(c ₂) intersects H since it contains v _p and e _p for some p∈P. Therefore, H contains a cycle ν that separates V(c ₁) from V(c ₂). Clearly, we have  length_Γ(ν)≤2|V _P |=2|P|. Since every path in P intersects ν, the equality must hold. □

Lemma 4.2

Let Γ be a plane graph that is a graph and k≥3, h≥2 integers. Let r ₁ and r ₂ be two faces of Γ such that the following conditions are satisfied.

• 1. There is no Γ-noose ν of length smaller than 2k such that r ₁⊆R(ν) and r ₂⊆R( rev(ν)).

• 2.  dist_Γ(r ₁,r ₂)≥2h.

Then, Γ contains a k×h cylinder as a minor.

Proof

Since the vertex of ${\mathcal{R}}_{\Gamma}$ corresponding to r ₂ belongs to ${\mathcal{R}}_{\Gamma}(r_{1}, h - 1)$, there is a unique contour c _i at depth i from r ₁ that contains r ₂ in its inside, for each 0≤i<h. By assumption 1, there is no Γ-noose of length smaller than 2k that separates c _h−1 from c ₀. Therefore, by Lemma 4.1, there are k vertex-disjoint paths between V(c ₀) and V(c _h−1). These k vertex-disjoint paths, together with the h contours, constitute a minor of Γ isomorphic to a k×h cylinder. □

The induction would be straightforward if this lemma held for plane hypergraphs that are not graphs. This unfortunately is not the case and we need a more elaborate structure of induction in which we work on hypergraphs that are almost graphs. An appropriate generalization of Theorem 1.4 for an induction proof turns out to be as follows.

Lemma 4.3

Let Γ be a plane hypergraph with at least one edge and k≥3 and h≥2 be integers. Suppose that every edge of Γ except for one edge e ₀ is of order 2 and e ₀ is of order smaller than k. Then either Γ has a branch-decomposition of width at most k+2h−3 or the subhypergraph Γ′ of Γ with vertex set V(Γ) and edge set E(Γ)∖{e ₀}, which is a graph, has a k×h cylinder as a minor.

To prove this lemma, we need the following technical lemma.

Lemma 4.4

Let Γ be a plane hypergraph that is a graph, r ₀ a face of Γ, and c ₁, and c ₂ be two distinct contours of Γ at depth d from r ₀, where d is a positive integer. For i=1,2, let ν _i be a Γ-noose such that

• 1. ν _i separates c _i from r ₀,

• 2.  length_Γ(ν _i ) is the smallest subject to condition 1, and

• 3. R(ν _i ) is minimal subject to conditions 1 and 2.

Then, either ν ₁ separates both c ₁ and c ₂ from r ₀, ν ₂ separates both c ₁ and c ₂ from r ₀, or  int(R(ν ₁))∩ int(R(ν ₂))=∅.

Proof

We first show that ν ₁ does not cross c ₂. Suppose otherwise. For each maximal subpath μ of ν ₁ that does not intersect any vertex on or inside of c ₂, let μ′ denote the Γ-noose consisting of μ and a path of ${\mathcal{R}}_{\Gamma}$ around c ₂ such that R( rev(μ′)) contains all the edges of c ₂. Then, since c ₁ and c ₂ are edge-disjoint, there is one such μ such that R(μ′) contains all the edges of c ₁. Let μ denote this particular path and let μ=ν ₁[v ₁,v ₂], where v ₁,v ₂∈V(c ₂).

For each vertex v∈V(c ₁), let δ _v be a path in ${\mathcal{R}}_{\Gamma}$ of length d from v to r ₀. We assume without loss of generality that, for any pair of distinct vertices v,w∈V(c ₁), δ _v and δ _w do not cross each other. For each v∈V(c ₁), δ _v intersects ν ₁ since ν ₁ separates c ₁ from r ₀. Let x _v denote the vertex or face of Γ in which δ _v and ν ₁ intersect each other; we choose the one closest to v on δ _v if there are multiple intersections. Note that, since every vertex on δ _v except for v is at distance smaller than 2d+1 from r ₀, no vertex on c ₂ is on δ _v . Therefore, x _v is on the subpath μ of ν ₁. For i=1,2, let w _i ∈V(c ₁) be the vertex such that $x_{w_{i}}$ is the closest to v _i on μ. We claim that w ₁ and w ₂ are adjacent on c ₀. For, otherwise there are two vertices u ₁,u ₂∈V(Γ) such that u ₁, w ₁, u ₂, w ₂ appear in this order on c ₁ while $x_{u_{1}}$ and $x_{u_{2}}$ appear between $x_{v_{1}}$ and $x_{v_{2}}$ on μ. This contradicts our assumption since $\delta_{u_{i}}$ and $\delta_{w_{j}}$ must cross each other for at least one pair (i,j)∈{1,2}×{1,2}.

Since  dist_Γ(w _i ,r ₀)= dist_Γ(v _i ,r ₀)=2d+1 for i=1,2, we have ${ {\mathop {\mathrm {length}}}_{\Gamma}}(\mu[v_{1},\allowbreak x_{w_{1}}]) \geq { {\mathop {\mathrm {length}}}_{\Gamma}}(\delta_{w_{1}}[w_{1}, x_{w_{1}}])$ and ${ {\mathop {\mathrm {length}}}_{\Gamma}}(\mu[x_{w_{2}}, v_{2}]) \geq { {\mathop {\mathrm {length}}}_{\Gamma}}(\delta_{w_{2}}[w_{2}, x_{w_{2}}])$.

Let ν be a Γ-noose obtained by closing the path $\delta_{w_{1}}[w_{1}, x_{w_{1}}]+\mu[x_{w_{1}}, x_{w_{2}}]+ {\mathop {\mathrm {rev}}}(\delta_{w_{2}}[w_{2}, x_{w_{2}}])$ by a path of length two from w ₂ to w ₁ through a face outside of c ₁. Then  length_Γ(ν)≤ length_Γ(ν ₁), all the edges of c ₁ are contained in R(ν), and r ₀ is contained in R( rev(ν)). Since R(ν) is a proper subset of R(ν ₁), this is a contradiction to the minimality of ν ₁. This proves the claim that ν ₁ does not cross c ₂. Similarly, ν ₂ does not cross c ₁.

We are now to prove that either ν ₁ separates c ₂ from r ₀, ν ₂ separates c ₁ from r ₀, or  int(R(ν ₁))∩ int(R(ν ₂))=∅. Suppose that none of these three conditions hold. Since ν _3−i does not cross c _i , for i=1,2, the assumption that ν _3−i does not separate c _i from r ₀ implies that no edge of c _i is contained in R(ν _3−i ). For the same reason, all the edges of c _i are contained in one connected component of R(ν _i )∖R(ν _3−i ), which we call R _i . Let $\nu'_{i}$ be the Γ-noose such that $R(\nu'_{i}) = R_{i}$, for i=1,2. Then, $\nu'_{i}$ consists of subpaths of ν _i and subpaths of  rev(ν _3−i ). But if an oriented edge of ν _i appears in $\nu'_{i}$, then it is not contained in R(ν _3−i ) and hence its reversal does not appear in $\nu'_{3 - i}$. Therefore we have ${ {\mathop {\mathrm {length}}}_{\Gamma}}(\nu'_{1}) + { {\mathop {\mathrm {length}}}_{\Gamma}}(\nu'_{2}) \leq { {\mathop {\mathrm {length}}}_{\Gamma}}(\nu_{1}) +{ {\mathop {\mathrm {length}}}_{\Gamma}}(\nu_{2})$ and hence ${ {\mathop {\mathrm {length}}}_{\Gamma}}(\nu'_{i}) \leq { {\mathop {\mathrm {length}}}_{\Gamma}}(\nu_{i})$ for i=1 or 2. Since we are assuming  int(R(ν ₁))∩ int(R(ν ₂))≠∅ and hence $R(\nu'_{i})$ is a proper subset of R(ν _i ), this contradicts the minimality conditions for ν _i . □

We are now ready to prove Lemma 4.3. The proof is by induction on the number of edges of Γ. The base case |E(Γ)|≤2 is trivial. So, suppose |E(Γ)|>2.

Suppose first that for every face r of Γ, we have  dist_Γ(r,F _Γ(e ₀))≤2h−2. Then, we apply the “dual” version of Theorem 3.1, in which the roles of vertices and faces are interchanged, with d replaced by 2h−2 and k replaced by k−1, and obtain a branch-decomposition of Γ of width at most (k−1)+2h−2=k+2h−3.

Therefore, suppose that there is some face r of Γ such that  dist_Γ(r,F _Γ(e ₀))≥2h. Let Γ′ be a plane hypergraph with vertex set V(Γ) and edge set E(Γ)∖{e ₀}. Then F(Γ′)=(F(Γ)∖F _Γ(e ₀))∪{r ₀}, where $r_{0} = e_{0} \cup\bigcup_{r \in F_{\Gamma}(e_{0})} r$. Since every edge of Γ except for e ₀ is of order two, Γ′ is a graph. Because there is a face r of Γ′ with  dist_Γ′(r,r ₀)≥2h, Γ′ has some contours at depth h−1 from r ₀. Let m>0 be the number of those contours and c ₁, …, c _m the list of those contours.

For each 1≤i≤m, let ν _i be a shortest Γ′-noose that separates c _i from r ₀. Suppose first that  length_Γ(ν _i )≥2k for some 1≤i≤m. Let $\Gamma'_{i}$ be the plane hypergraph obtained from Γ′ by removing all vertices and edges that lie inside of c _i and thus making the inside of c _i into a single face. Applying Lemma 4.2 to $\Gamma'_{i}$ with faces r ₀ and r _i , we obtain a minor of $\Gamma'_{i}$ isomorphic to a k×h cylinder and are done.

So, suppose  length_Γ(ν _i )<2k for every 1≤i≤m. Let ${\mathcal{N}}= \{\nu_{i} \mid1 \leq i \leq m\}$. By Lemma 4.4, we may choose these Γ′-nooses in ${\mathcal{N}}$ so that, for each pair 1≤i<j≤m,  int(R(ν _i ))∩ int(R(ν _j ))=∅ unless either ν _i separates c _j from r ₀ or ν _j separates c _i from r ₀. If ν _i separates c _j from r ₀ then we remove ν _j from the collection; if ν _j separates c _i from r ₀ then we remove ν _i from the collection. In this manner, we obtain a subcollection ${\mathcal{N}}'$ of Γ′-nooses such that

• 1. for each $\nu\in {\mathcal{N}}'$,  length_Γ(ν)<2k,

• 2. for each 1≤i≤m, there is some $\nu\in {\mathcal{N}}'$ that separates c _i from r ₀, and

• 3. for each pair of distinct elements $\nu, \nu' \in {\mathcal{N}}'$, we have  int(R(ν))∩ int(R(ν′))=∅.

Let ${\mathcal{N}}' = \{\nu'_{1}, \ldots, \nu'_{m'}\}$. Noting that these Γ′-nooses are also Γ-nooses, let Γ₀=Γ and, for 1≤i≤m′, $\Gamma_{i} = \Gamma_{i - 1} | \nu'_{i}$.

Since the faces in F _Γ(e ₀) are contained in face r ₀ of Γ′, none of them is contained in R(ν) for any $\nu\in {\mathcal{N}}'$. Therefore, each of these faces are also a face of Γ _m′. By construction, each face r of Γ _m′ has ${\mathop {\mathrm {dist}}}_{\Gamma_{m'}}(r, F_{\Gamma}(e_{0})) \leq2(h - 1)$. Therefore, the Γ _m′ satisfies the dual version of Theorem 3.1, with d replaced by 2h−2 and k by k−1, and hence has a branch-decomposition of width ≤k+2h−3.

Since $R(\nu'_{i})$ does not contain edges of Γ incident with e ₀, the number of edges in $\Gamma| {\mathop {\mathrm {rev}}}(\nu'_{i})$ is strictly smaller than the number of edges in Γ, for 1≤i≤m′. Therefore, we may apply the induction hypothesis to $\Gamma| {\mathop {\mathrm {rev}}}(\nu'_{i})$ and obtain either a k×h cylinder as a minor or a branch-decomposition of width at most k+2h−3. If all of $\Gamma| {\mathop {\mathrm {rev}}}(\nu'_{i})$, 1≤i≤m′, have such a branch-decomposition, we may apply Corollary 2.1 and combine those branch-decompositions and the branch-decomposition of Γ _m′ into a branch-decomposition of Γ of width at most k+2h−3. This completes the proof of Lemma 4.3.

In proving Theorem 1.4, we may assume that the given planar graph G is biconnected, since otherwise we may work separately on biconnected components. Then, the statement of the theorem immediately follows from the application of Lemma 4.3 to a plane hypergraph Γ that is a graph and is isomorphic to G. This completes the proof of Theorem 1.4.

5 Tightness of Theorem 3.1

In this section, we show that the bound of Theorem 3.1 is tight. That is, for every integer k≥2 and every even integer d≥2, there is a plane hypergraph satisfying the conditions of the theorem and has branchwidth exactly d+k.

A (k,h,w)-capped cylinder, denoted by CC _k,h,w where k≥2, h≥1, w≥2 are integers, is a hypergraph that consists of the following:

• 1. a kw×h cylinder C _kw,h ,

• 2. an edge e ₀, where the set V(e ₀) of vertices incident with e ₀ is {v _j ∣0≤j<k} and is disjoint from V(C _kw,h ),

• 3. an edge between vertex v _j of e ₀ and vertex (wj+l,0) of C _kw,h , for 0≤j<k and 0≤l<w,

• 4. an edge e ₁ that is incident with vertices (wj,h−1) for 0≤j<k.

It is not difficult to verify that CC _k,h,w has an essentially unique plane embedding: draw C _kw,h on the side of a geometric cylinder in the standard manner, draw e ₀ in the top of this cylinder, and draw e ₁ in the bottom. In the following, we will refer to this plane hypergraph as CC _k,h,w .

The maximum edge order of CC _k,h,w is k and for every vertex v of CC _k,h,w we have ${\mathop {\mathrm {dist}}}_{{CC}_{k, h, w}}(v, V(e_{0})) \leq2h$. Therefore by Theorem 3.1, we have a branch-decomposition of CC _k,h,w with width k+2h. The following theorem shows that Theorem 3.1 is tight.

Theorem 5.1

Let k≥2, h≥1, w≥3h be integers. Then the branchwidth of CC _k,h,w is at least k+2h.

To prove this theorem, we need the following notion and lemma. Let Γ be a plane hypergraph. A noose-tangle for Γ of order k, where k>0 is an integer, is a collection ${\mathcal{T}}$ of separations of Γ of order strictly smaller than k that are induced by Γ-nooses, such that the following three axioms are satisfied.

• 1. For each separation (A,B) of order strictly smaller than k induced by some Γ-noose, exactly one of (A,B) and (B,A) is in ${\mathcal{T}}$.

• 2. If $(A_{1}, B_{1}), (A_{2}, B_{2}), (A_{3}, B_{3}) \in {\mathcal{T}}$ then A ₁∪A ₂∪A ₃≠E(Γ).

• 3. If $(A, B) \in {\mathcal{T}}$, then V _Γ(A)≠V(Γ).

If we change the first axiom and require that exactly one of (A,B) and (B,A) to belong to ${\mathcal{T}}$ for every separation (A,B) of order smaller than k, then we get the definition of standard tangles [18]. Robertson and Seymour [18] show that a hypergraph Γ has a branch-decomposition of width k if and only if there is no tangle of Γ of order k.

For plane hypergraphs, we have the following parallel.

Lemma 5.1

Let Γ be a plane hypergraph and let k>0 be an integer. Then, Γ has a branch-decomposition of width k if and only if there is no noose-tangle for Γ of order k+1.

Proof

Since a tangle of order k+1 implies a noose-tangle of order k+1, “if” part follows immediately from the corresponding direction of the tangle characterization of branchwidth. For “only if” part, suppose Γ has a branch-decomposition of width k. The result of Seymour and Thomas on bond carvings (5.1 in [21]) implies that Γ has a sphere-cut decomposition of width k. With an argument almost identical to the one showing that a tangle of order k+1 precludes a branch-decomposition of width k [18], we can show that a noose-tangle of order k+1 precludes a sphere-cut decomposition of width k. Therefore, Γ cannot have such a noose-tangle. □

We are now ready to prove Theorem 5.1. Let Γ=CC _k,h,w , k≥2, h≥1, and w≥3h. Let ν be an arbitrary Γ-noose of length strictly smaller than 2(k+2h). We claim that if ν separates e ₀ from e ₁ then

• 1. ν intersects either some vertex of e ₀ or some vertex of e ₁, and

• 2. if ν intersects some vertex of e ₀ and e ₀⊆R(ν) then R(ν) does not contain any edge in row h−1 of the cylinder in Γ.

The first claim holds, since if ν separates e ₀ from e ₁ and intersects only the vertices of the cylinder then it must intersect at least one vertex in each of the kw≥3kh≥k+2h columns. For the second claim, suppose ν intersects some vertex of e ₀ and e ₀⊆R(ν). If ν does not intersect some vertex v _j , 0≤j<k, then it must intersect at least one vertex in each of the w columns indexed by wj through wj+w−1. This is impossible since w≥3h and therefore ν intersects all the k vertices of e ₀. To have an edge of in row h−1, ν must intersect at least 2 vertices in each of the h rows of the cylinder, which is impossible.

Our noose-tangle ${\mathcal{T}}$ for Γ of order k+2h is defined as follows. Let (A,B) be an arbitrary separation of Γ induced by some Γ-noose, say ν, whose order is strictly smaller than k+2h. If e ₀,e ₁∈B then we put (A,B) in ${\mathcal{T}}$; if e ₀,e ₁∈A then we put (B,A) in ${\mathcal{T}}$; if e ₀∈A,e ₁∈B then we put (A,B) in ${\mathcal{T}}$ if ν intersects vertices of e ₀ and put (B,A) in ${\mathcal{T}}$ otherwise.

${\mathcal{T}}$ satisfies the first axiom for noose-tangles due to its construction.

To verify the second axiom, suppose for contradiction that ⋃_1≤i≤3 A _i =E(Γ) where $(A_{i}, B_{i}) \in {\mathcal{T}}$ for 1≤i≤3. Since it is impossible for any A _i to have both e ₀ and e ₁, at least one of them, say A ₁ contains e ₀ and at least one of the others, say A ₂ contains e ₁. By the second claim above, A ₁ does not contain any edge of row h−1 of the cylinder. Let F be the set of edges in row h−1 contained in A ₂. Then the simple Γ-noose ν ₂ that induces separation (A ₂,B ₂) must intersect at least one vertex in each column of the cylinder that contains a vertex incident to some edge in F. Therefore, we have |F|≤k+2h−1. Similarly, the number of edges of row h−1 belonging to A ₃ is at most k+2h−1. Therefore, A ₁∪A ₂∪A ₃ contains at most 2k+4h−2<kw vertices, a contradiction.

For the last axiom, let (A,B) be an arbitrary member of ${\mathcal{T}}$. If this separation separates e ₀ from e ₁ then V _Γ(A) cannot contain both vertices of e ₀ and e ₁, by our second claim above. If e ₀,e ₁∈B, on the other hand, V _Γ(A) may contain at most k+2h vertices of any single row of the cylinder. In either case, we have V _Γ(A)≠V(Γ).

Therefore, ${\mathcal{T}}$ is indeed a noose-tangle for Γ of order k+2h and by Lemma 5.1, we have the result. This completes the proof of Theorem 5.1.

Remark 5.1

The proof would work, with finer calculations, for somewhat smaller values of w.

6 The Branchwidth and Grid-Minor Size of a Cylinder

In this section, we study the branchwidth and the grid-minor size of the cylinder.

Theorem 6.1

For arbitrary integers k≥3 and h≥1, we have  bw(C _k,h )=min{k,2h}.

Proof

The upper bound can be exhibited by simple “linear” branch-decompositions as follows.

Consider first a linear branch-decomposition of C _k,h induced by the “row-major” total order <_row defined as follows. We first define the row major order <_row on V(C _k,h ) to be a lexicographic total order: (i ₁,j ₁)<_row(i ₂,j ₂) if and only if i ₁<i ₂ or i ₁=i ₂ and j ₁<j ₂. We extend this order to a total order on E(C _k,h ) as follows. For e ₁,e ₂∈E(C _k,h ), let e ₁={u ₁,v ₁} with u ₁<_row v ₁ and let e ₂={u ₂,v ₂} with u ₂<_row v ₂. Then e ₁<_row e ₂ if and only if u ₁<_row u ₂ or u ₁=u ₂ and v ₁<_row v ₂.

For each separation (A,B) in this branch-decomposition, vertices in V(A)∩V(B) are contained in one row and therefore the width of this branch-decomposition is k.

Next consider another linear branch-decomposition of C _k,h induced by the “column-major” order defined similarly but with a slight modification so that the edges between column 0 and column k−1 are the largest in this ordering.

For each separation (A,B) in this branch-decomposition, vertices in V(A)∩V(B) are contained in two columns (column 0 and another) and therefore the width of this branch-decomposition is 2h. These two types of branch-decompositions prove that  bw(C _k,h )≤min{k,2h}.

To prove the inequality in the other direction, we use noose-tangles of Sect. 5.

We assume an arbitrary plane hypergraph Γ that is a graph and is isomorphic to C _k,h as a graph and construct a noose-tangle ${\mathcal{T}}$ of order k with respect to this embedding. Let D ₁ be the set of edges in row 0 of C _k,h and D ₂ the set of edges in row h−1. Let (A,B) be an arbitrary noose-induced separation of Γ of order smaller than min{k,2h}, and let ν denote the noose that induces this separation. Since there are k vertex-disjoint paths between row 0 and row h−1 of C _k,h , by Menger’s theorem ν cannot separate D ₁ and D ₂. Moreover, ν cannot separate D ₁ into more than one non-empty subsets and at the same time separate D ₂ into more than one non-empty subset, since if it did then its length would be at least 2h. To summarize, exactly one of A and B contains D ₁ or D ₂. If A does then we put (B,A) in ${\mathcal{T}}$ and if B does then we put (A,B) in ${\mathcal{T}}$. As a result, for each $(A, B) \in {\mathcal{T}}$, we have either A∩D ₁=∅ and D ₂∖A≠∅ or A∩D ₂=∅ and D ₁∖A≠∅.

The first axiom of noose-tangles holds because of the construction.

The second axiom holds, since if $(A_{1}, B_{1}), (A_{2}, B_{2}), (A_{3}, B_{3}) \in {\mathcal{T}}$ then it is impossible for A ₁∪A ₂∪A ₃ to contain all the edges in D ₁ and D ₂; at least two of A ₁, A ₂, A ₃ must be dedicated to cover D ₁ and the same holds for D ₂.

The third axiom holds, since for each $(A, B) \in {\mathcal{T}}$, V _Γ(A) does not contain either the vertices in row 0 or the vertices in row h−1 and hence V _Γ(A)≠V(Γ).

Therefore, G has a noose-tangle of order k and the branchwidth of G is at least k. □

Next we prove Theorem 1.3 that states  gm(C _2h,h )=h. One direction,  gm(C _2h,h )≥h, is trivial as a h×h grid is clearly a subgraph of C _2h,h . The other direction,  gm(C _2h,h )≤h, may be somewhat surprisingly the most difficult result to prove in this paper.

We first review a few basic facts about the branch-decomposition of grids.

It is well-known that  bw(G _k,h )=min{k,h}. This can be shown similarly to the proof of Theorem 6.1 which determines the branchwidth of cylinders. We need a slightly stronger observation. We call a vertex set W⊆V(G _k,h ) semi-vertical, if W={(i ₁,j ₁),(i ₂,j ₂),…,(i _m ,j _m )} with j ₁<j ₂<…<j _m and |i _a+1−i _a |≤j _a+1−j _a for 1≤a<m.

Lemma 6.1

Let k≥h≥1 be integers. Let W⊆V(G _k,h ) be an arbitrary non-empty semi-vertical set of vertices and Λ a hypergraph with V(Λ)=V(G _k,h ) and E(Λ)=E(G _k,h )∪{e ₀}, where V _Λ(e ₀)=W. Then  bw(Λ)=h.

Proof

Since  bw(Λ)≥ bw(G _k,h )≥min{k,h}=h, it suffices to exhibit a branch-decomposition of Λ of width h. First observe that each semi-vertical set W can be extended into a semi-vertical set W′⊇W with |W′|=h. Moreover, the width of a branch-decomposition of a hypergraph does not increase when we replace an edge e of that hypergraph with some edge e′ with V(e′)⊆V(e). Therefore, it suffices to give a construction for the case |W|=h. So let W={(i ₀,0),(i ₁,1),…,(i _h−1,h−1)}. Let W ₁={(i,j)∈V(G _k,h )∣(i<i _j )} and W ₂={(i,j)∈V(G _k,h )∣(i>i _j )}. Note that W∪W ₁∪W ₂=V(G _k,h ) and that W, W ₁, and W ₂ are pairwise disjoint. Let $E_{1} = \{e \in E(G_{k, h}) \mid V_{G_{k, h}}(e) \cap W_{1} \neq \emptyset\}$ be the set of edges incident with a vertex in W ₁, $E_{2} = \{e \in E(G_{k, h}) \mid V_{G_{k, h}}(e) \cap W_{2} \neq\emptyset \}$ the set of edges incident with a vertex in W ₂, and E ₃=E(G _k,h )∖(E ₁∪E ₂). Note that E ₁∩E ₂=∅ and therefore E ₁, E ₂, and E ₃ partitions E(G _k,h ).

Let <_col be the column major total order on V(G _k,h ) (=V(C _k,h )) as defined in the proof of Theorem 6.1. We define a total order < on |E(Λ)| as follows.

• 1. For each e ₁∈E ₁, each e ₂∈E ₂ and each e ₃∈E ₃, e ₁<e ₃<e ₀<e ₂.

• 2. For each pair e ₁,e ₂∈E(G _k,h ) for which the order is not specified by the above, let e ₁=(u ₁,v ₁) with u ₁<_col v ₁ and e ₂=(u ₂,v ₂) with u ₂<_col v ₂. Then, e ₁<e ₂ if and only if u ₁<_col u ₂ or u ₁=u ₂ and v ₁<_col v ₂.

Let (A,B) be an arbitrary separation of E(Λ) such that e ₁<e ₂ for each e ₁∈A and each e ₂∈B. We claim that V(A)∩V(B) contains at most one vertex from each row. To verify this claim, suppose for a contradiction that (i,j ₁),(i,j ₂)∈V(A)∩V(B) for some row i, where j ₁<j ₂. Let e ₁∈B be an edge incident to (i,j ₁) and e ₂∈A an edge incident to (i,j ₂). From the assumption on the separation (A,B), we must have e ₂<e ₁. Since (i,j ₁)∈e ₁ and (i,j ₂)∈e ₂ with j ₁<j ₂ implies either e ₁=e ₂ or e ₁<_col e ₂, this order e ₂<e ₁ must be determined by the first case of the definition of < and therefore we have either e ₂∈E ₁ and e ₁∈E ₃∪{e ₀}∪E ₂, e ₂∈E ₃ and e ₁∈{e ₀}∪E ₂, or e ₂=e ₀ and e ₁∈E ₂. In each of these cases, we must have j ₂≤j ₁, a contradiction. This verifies the claim.

Therefore, the width of the branch-decomposition of Λ induced by this total order < is h. □

Since C _2h,h contains G _h,h as a subgraph and hence as a minor, that  gm(C _2h,h )≥h clearly holds. Therefore, it suffices to show that C _2h,h does not contain G _h+1,h+1 as a minor in order to prove Theorem 1.3.

We fix h≥2 and, for a contradiction, assume that C _2h,h contains a minor isomorphic to G _h+1,h+1. To deal with C _2h,h , we often need arithmetic on column indices, which are naturally modulo 2h. We use arithmetic symbols ⊕ and ⊖ for this purpose: a⊕b is integer c with 0≤c<2h such that a+b≡c(mod2h) and a⊖b is integer c with 0≤c<2h such that a−b≡c(mod2h).

Let M be a minor of C _2h,h isomorphic to G _h+1,h+1 and let H be a minimal subgraph of C _2h,h that is contracted into M. Let ψ:V(H)→V(M) denote the mapping such that ψ(v) is the vertex of M into which v is contracted. By the minimality of H, the following holds.

• 1. For each v∈V(M), the subgraph of H induced by the vertex set ψ ⁻¹(v) is a tree. We denote this tree by T _v .

• 2. For each pair of vertices u,v adjacent in M, there is exactly one edge between ψ ⁻¹(u) and ψ ⁻¹(v).

We embed C _2h,h as a plane hypergraph Γ, H as a plane hypergraph Δ such that V(Δ)⊆V(Γ) and E(Δ)⊆E(Γ). The embedding of M is a plane hypergraph Ω obtained from Δ as follows. For each v∈V(M), we replace the tree in Δ representing T _v in H by a single vertex that is the union of all the vertices and edges in the tree. For each 0≤i<2h and 0≤j<h, we denote by c _ij the vertex of Γ representing vertex (i,j) of C _2h,h .

Proposition 6.1

We have E(Ω)⊆E(Γ) and, for every e∈E(Γ)∖E(Ω), e⊆x for some x∈V(Ω)∪F(Ω). For every v∈V(Γ), we have v⊆x for some x∈V(Ω)∪F(Ω). For every r∈F(Γ), we have r⊆r′ for some r′∈F(Ω).

From now on, we identify the graphs H and M with their embeddings Δ and Ω, respectively. In particular, we view the mapping ψ:V(H)→V(M) as a mapping from V(Δ) to V(Ω). Thus, we have v⊆ψ(v) for each v∈V(Δ).

We call the face of Γ bounded by the cycle in row 0 the top face and the one bounded by the cycle in row h−1 the bottom face of Γ.

Lemma 6.2

Suppose Ω has a face that contains both the top and bottom faces of Γ. Then, a k×h grid G _k,h for sufficiently large k contains a minor isomorphic to Ω.

Proof

Let r be a face of Ω that contains both the top and bottom faces of Γ. Choose an arbitrary path μ of Γ contained in r that connects the top and the bottom faces of Γ. Let Γ′ be the subhypergraph of Γ with E(Γ′)={e∈E(Γ)∣e∩μ=∅} and V(Γ′)=V(Γ). Then, k×h grid G _k,h for sufficiently large k contains a subgraph G isomorphic to Γ′: the isomorphism from Γ′ to G is obtained by viewing the sphere as a geometrical cylinder whose top/bottom faces are the top/bottom faces of Γ and cutting the side of this cylinder along μ into a strip. Since Γ′, as a graph, has Ω as a minor, G _k,h has a minor isomorphic to Ω. □

We call the unique face of Ω bounded by 4h edges the large face of Ω. We call the four vertices of degree 2 on the larges face of Ω the corner vertices of Ω. Let S ₀, S ₁, S ₂, and S ₃ be oriented paths between adjacent corner vertices such that their cyclic concatenation in this order is a directed cycle S that bounds the large face of Ω.

We define some more symbols B, B ₀, B ₁, B ₂, B ₃, R _out and R _in to be used throughout the proof as follows.

B is an oriented cycle of Δ, hence of Γ, such that ψ(B)=S. For each i, 0≤i<4, B _i is the maximal subpath of B such that ψ(B _i )=S _i . Of the two discs of Σ bounded by B, R _out is the one that contains the large face of Ω and R _in is the other.

We assume the following throughout the proof.

A1 :

In the embedding Γ of the cylinder C _2h,h , the columns are arranged clockwise around the bottom face under the ascending order of the column indices.

A2 :

When we proceed along the oriented cycle B of Γ, we see R _in to the right.

Let ω be an arbitrary walk on ${\mathcal{R}}_{\Gamma}$. By Proposition 6.1, ω does not intersect any edge of Ω and defines a walk on V(Ω)∪F(Ω). Let ω′ be the walk on ${\mathcal{R}}_{\Omega}$ that defines the same walk on V(Ω)∪F(Ω) as ω. We call ω′ the walk in ${\mathcal{R}}_{\Omega}$ corresponding to ω. Clearly,  length_Ω(ω′)≤ length_Γ(ω).

We say that vertices u,v∈V(B) are antipodal to each other, if u∈V(B _i ) and v∈(B _(i+2)mod 4) for some 0≤i<4. We call a path μ of ${\mathcal{R}}_{\Gamma}$ an antipodal chord if μ is between an antipodal pair of vertices and avoids R _out.

Lemma 6.3

Every antipodal chord has length 2h or greater in Γ.

Proof

Let u and v be arbitrary vertices of B that are antipodal and μ an antipodal chord between u and v. Let μ′ be the path in ${\mathcal{R}}_{\Omega}$ corresponding to μ. Then, since μ′ is between ψ(u) and ψ(v) which belong to two opposite sides of the (h+1)×(h+1) grid, we have  length_Γ(μ)≥ length_Ω(μ′)≥2h. □

A slit is a path in ${\mathcal{R}}_{\Gamma}$ that connects distinct vertices u,v∈V(B), intersects each row of Γ at most once, and stays in R _in. Let σ be a slit connecting u,v∈V(B). Since  length_Γ(σ)<2h, the pair u, v is not antipodal by Lemma 6.3. It follows that, of the two parts of R _in separated by σ, one is incident with B _i for three or more indices i and the other is incident with B _i for two or fewer indices i. We call the former the major side and the latter the minor side of slit σ.

Lemma 6.4

Let σ be a slit. Then the major side of σ contains either the top face or the bottom face of Γ.

Proof

Let σ be a slit and assume σ connects vertices b ₁ and b ₂ in V(B). Let σ′ be the path in ${\mathcal{R}}_{\Omega}$ corresponding to σ and let W be a set of vertices of Γ on σ such that, for each vertex v on σ′, exactly one vertex in ψ ⁻¹(v) is contained in W.

Let R ₁ and R ₂ be the major and minor sides of this slit respectively. Let Ω _i , i=1,2, be the subhypergraph of Ω such that E(Ω _i ) consists of the edges of Ω contained in R _i and V(Ω _i ) consists of vertices of Ω that are intersected by R _i . Note that V(Ω₁)∩V(Ω₂)=ψ(W) and (E(Ω₁),E(Ω₂)) is a separation of Ω with order |ψ(W)|.

Suppose R ₁ contains neither the top nor the bottom face of Γ. We argue similarly to the proof of Lemma 6.2. Choose an arbitrary curve ν that connects the top and the bottom faces of Γ avoiding R ₁. Let Γ′ be a subhypergraph of Γ induced by the edge set {e∈E(Γ)∣e∩ν=∅}. Let ϕ be a natural isomorphism, obtained similarly to the one in the proof of Lemma 6.2, from Γ′ to a subgraph of k×h grid G _k,h , where k is large enough. Let M ₂ be a graph isomorphic to Ω₂ with an isomorphism χ:V(Ω₂)→V(M ₂). Let J be a graph obtained from G _k,h by attaching M ₂ by identifying, for each w∈W, the vertex ϕ(w) of G _k,h and the vertex χ(ψ(w)) of M ₂. In this construction of J we disregard the embeddings of G _k,h and M ₂, dealing with them simply as graphs.

We claim that  bw(J)=h. This gives us the desired contradiction, because J contains G _h+1,h+1 as a minor and hence  bw(J)≥h+1. Since σ is a slit, ϕ(W) is semi-vertical. Therefore, the hypergraph Λ with V(Λ)=G _k,h and E(Λ)=E(G _k,h )∪{e ₀}, where e ₀ is incident with each vertex in ϕ(W), has branchwidth  bw(Λ)=h, by Lemma 6.1. Moreover, the hypergraph $M'_{2}$ with $V(M'_{2}) = V(M_{2})$ and $E(M'_{2}) = E(M_{2}) \cup\{e_{0}\}$ has branchwidth ${\mathop {\mathrm {bw}}}(M'_{2}) \leq h$ since M ₂ is a subgraph of the (h+1)×(h+1) grid M drawn on the minor side of the slit. Therefore, by Lemma 2.1, we have  bw(J)=h. This verifies the claim and completes the proof of the lemma. □

Lemma 6.5

There does not exist a pair of slits σ ₁ and σ ₂ that satisfies the following conditions.

• 1. σ ₁ and σ ₂ do not intersect each other,

• 2. the major side of σ ₁ contains the minor side of σ ₂ (and hence vice versa), and

• 3. the minor side of σ ₁ contains the bottom face of Γ, and the minor side of σ ₂ contains the top face or Γ.

Proof

Since the proof is similar to the proof of Lemma 6.4, we only give a sketch. Suppose there is such a pair of slits σ ₁ and σ ₂ satisfying the above conditions. Let Ω _i , for i=1,2, be the subhypergraph of Ω consisting of the vertices and faces of Ω contained in or intersecting the minor side of σ _i . We attach a graph isomorphic to Ω₁ and a graph isomorphic to Ω₂ to a grid G _k,h for sufficiently large k in an appropriate manner to construct a graph of branchwidth h that has a minor isomorphic to G _h+1,h+1, deriving a contradiction. □

Lemma 6.6

Region R _in must contain at least one of the top and bottom faces of Γ.

Proof

Suppose that R _in does not contain either the top or bottom face of Γ. Then, the large face of Ω contains both the top and bottom face of Γ. It follows from Lemma 6.2 that G _k,h for sufficiently large k contains G _h+1,h+1 as a minor, a contradiction since  bw(G _k,h )=h for k≥h and  bw(G _h+1,h+1)=h+1. □

From now on, we add the following to our list of assumptions.

A3 :

R _in contains the bottom face of Γ.

We also need some more global symbols: L, L ₀, L ₁, L ₂, and L ₃. We say that vertex v=c _ij ∈V(B) of the cylinder Γ is extremal if j≥j′ for every j′ such that c _ij′∈V(B). Then, L is the set of all extremal vertices in V(B) and L _i =L∩V(B _i ) for 0≤i<4.

The following lemma is straightforward.

Lemma 6.7

When we arrange the elements of L cyclically according to the ascending order of column indices, the elements of L _i for each i appear consecutively, and as a whole, they appear in the cyclic order of L ₀, L ₁, L ₂, L ₃, skipping empty sets if any.

We fix a point b ₀ in the bottom face. For each vertex v=c _ij of Γ contained in the closure of R _in, let $\beta_{v}^{+}$ and $\beta_{v}^{-}$ be fixed paths in ${\mathcal{R}}_{\Gamma}$ chosen as follows. Path $\beta_{v}^{+}$ starts from v=c _ij , threads vertices c _(i⊕a)(j+a), for a=1,…,h−j−1, in this order and ends in b ₀. Similarly path $\beta_{v}^{+}$ starts from v=c _ij , threads vertices c _(i⊖a)(j+a), for a=1,…,h−j−1, in this order and ends in b ₀.

Let v=c _ij ∈V(B). We say that vertex u=c _i′j′∈V(B) obstructs $\beta_{v}^{+}$ if i′=i⊕a and j′>j+a for some 0≤a<h−j; that u obstructs $\beta_{v}^{-}$ if i′=i⊖a and j′>j+a for some 0≤a<h−j. We also say that u obstructs v if it obstructs either $\beta_{v}^{+}$ or $\beta_{v}^{-}$.

We say that v∈V(B) is exposed if both $\beta_{v}^{+}$ and $\beta_{v}^{-}$ stays in R _in. Note that v is exposed if and only if no u∈V(B) obstructs it. It is also true that v is exposed if and only if no exposed vertex u∈V(B) obstructs it.

Let X be the set of exposed vertices of V(B) and X _i =V(B _i )∩X for 0≤i<4.

Since X _i ⊆L _i for each 0≤i≤4, the following lemma follows from Lemma 6.7.

Lemma 6.8

When we arrange the elements of X cyclically according to the ascending order of column indices, the elements of X _i for each i appear consecutively, and as a whole, they appear in the cyclic order of X ₀, X ₁, X ₂, X ₃, skipping empty sets if any.

Lemma 6.9

At least one of X ₀ and X ₂ is empty and at least one of X ₁ and X ₃ is empty.

Proof

We prove the first part of the statement; the proof of the second part is similar. For contradiction, suppose X ₀ and X ₂ are both non-empty. If there are $v_{0} = c_{i_{0}j_{0}} \in X_{0}$ and $v_{2} = c_{i_{2}j_{2}} \in X_{2}$ with i ₂⊖i ₀<h, let μ be the concatenation of $\beta_{v_{0}}^{+}$ and $\beta_{v_{2}}^{-}$. Then, μ is an antipodal chord of length strictly smaller than 2h, a contradiction to Lemma 6.3. If there are $v_{0} = c_{i_{0}j_{0}} \in X_{0}$ and $v_{2} = c_{i_{2}j_{2}} \in X_{2}$ with i ₀⊖i ₂<h, we have a similar contradiction.

Therefore, the only possibility is that X ₀ has only one element $v_{0} = c_{i_{0}j_{0}}$, X ₂ has only one element $v_{2} = c_{i_{2}j_{2}}$, and i ₂⊖i ₀=h. If j ₀+j ₂≥h, a path in ${\mathcal{R}}_{\Gamma}$ between v ₀ and v ₂ going through the bottom face is an antipodal chord of length strictly smaller than 2h, a contradiction to Lemma 6.3. So suppose j ₀+j ₂<h. Then, vertices in column i ₀⊕a for j ₀≤a≤h−j ₂ are not obstructed by v ₀ or v ₂. Therefore, there must be some vertex $v_{1} = c_{i_{1}j_{1}} \in X$ where 0<i ₁⊖i ₀<h. Since X ₀={v ₀} and X ₂={v ₂}, by Lemma 6.8, v ₁ must belong to X ₁. Similarly, there is a vertex $v_{3} = c_{i_{3}j_{3}} \in X_{3}$, where 0<i ₃⊖i ₂<h. By an argument similar to the one in the beginning of this proof, X ₁={v ₁}, X ₃={v ₃}, and i ₃⊖i ₁=h. Assume without loss of generality that j ₀ is the largest among j _q , 0≤q<4. It is clear that j ₀>0, since otherwise none of v ₀,v ₁,v ₂,v ₃ obstructs any other vertices of B and hence there must be other vertices in X. Since v ₀ is the vertex of B with the largest row index in column i ₀, either $c_{(i_{0} \oplus1)j_{0}}$ or $c_{(i_{0} \ominus1)j_{0}}$ must be in V(B). We assume the former: the other case is similar. Then, $c_{(i_{0} \oplus1)j_{0}}$ is not obstructed by any vertex of X and hence must be exposed. This means that i ₁=i ₀⊕1 and i ₃=i ₂⊕1. Let i=i ₀⊕1⊕j ₀. Then, vertices in column i are not obstructed by v ₀ or v ₁ since j ₀=i⊖(i ₀+1); they are not obstructed by v ₂ since j ₀+j ₂<h and hence

It is impossible for v ₃ to obstruct vertices in column i, since it must obstruct either v ₀ or v ₂ in order to do so, but v ₀ and v ₂ are exposed. Therefore, there must be some exposed vertex in column i, a contradiction since only columns i _q for 0≤q<4 have exposed vertices. □

Lemma 6.10

Either L ₀ or L ₂ is empty and either L ₁ or L ₃ is empty.

Proof

We first assume that both L ₀ and L ₂ are non-empty and show that both L ₁ and L ₃ are non-empty. To show that L ₁ is non-empty, we suppose L ₁ is empty and derive a contradiction. Let i ₀ and i ₂ be column indices such that column i ₀ contains an element of L ₀, column i ₂ contains an element of L ₂, and column i ₀⊕a for 0<a<i ₂⊖i ₀ does not contain any element of L ₀ or of L ₂. If i ₂=i ₁⊕1, there is trivially an antipodal chord of length strictly smaller than 2h between v ₀ and v ₂, a contradiction to Lemma 6.3. So, it must be that i ₂⊖i ₀>1, that is, there are columns after i ₀ and before i ₂ and they are free of vertices of B. Along the directed subpath of B from v ₀ to v ₂, the next vertex of v ₀ must be $c_{i_{0}(j_{0} - 1)}$, which we call $v_{0}'$. Since L ₁=∅, $v_{0} \not\in V(B_{1})$ and therefore $v_{0}' \in V(B_{0})$ ($v_{0}'$ may also be in V(B ₁) but that is irrelevant). Then, there is an antipodal chord between $v_{0}'$ and v ₂ of length j ₀+j ₂ through the top face and one of length 2h−j ₀−j ₂−1 through the bottom face. At least one of them has length smaller than 2h, a contradiction. Therefore, L ₁ is non-empty. Similarly, L ₃ is non-empty under the assumption that both L ₀ and L ₂ are non-empty.

If we assume that L ₁ and L ₃ are both non-empty, then we have both L ₂ and L ₄ non-empty by an argument similar to the above. Therefore, if the conclusion of the Lemma does not hold, then all of L ₀, L ₁, L ₂, and L ₃ are non-empty; from this we derive a contradiction below.

Owing to Lemma 6.9 and using symmetry, we have to consider only two cases.

(Case 1) X ₀ and X ₁ are non-empty but X ₂ and X ₃ are empty. Recall Lemma 6.8 and let $x_{0} = c_{i_{0}j_{0}} \in X_{0}$ and $x_{1} = c_{i_{1}j_{1}} \in X_{1}$ such that column i ₁⊕a for 0<a<i ₀⊖i ₁ does not contain any element of X. Then, each element of L ₂∪L ₃ must be obstructed by either x ₀ or x ₁; otherwise it would be exposed and contradicts the assumption X ₂=X ₃=∅. Recall Lemma 6.7 and let i ^∗ be the index of the last column, under the standard cyclic order, whose extremal vertex $v = c_{i^{*}j^{*}}$ is in L ₂. Either x ₀ or x ₁ obstructs v and, because of the relative column positions of x ₀, x ₁ and v, x ₀ obstructs $\beta_{v}^{+}$ in the first case and x ₁ obstructs $\beta_{v}^{-}$ in the second case. Suppose first that x ₀ obstructs $\beta_{v}^{+}$. Let $v' = c_{i^{*}j'}$ be the vertex in column i ^∗ such that $\beta_{v'}^{+}$ goes through x ₀. Since x ₀ obstructs $\beta_{v}^{+}$, we have j′>j ^∗. Let μ be a path in ${\mathcal{R}}_{\Gamma}$ from v to x ₀, that first threads vertices in column i ^∗ between v and v′ and then follow $\beta_{v'}^{+}$ up to x ₀. Then μ is an antipodal chord of length <2h, a contradiction. Suppose next that x ₀ does not obstruct $\beta_{v}^{+}$ and therefore x ₁ obstructs $\beta_{v}^{-}$. Let u be the extremal vertex in column i ^∗⊕1 and let j″ be the row index of u. By the choice of i ^∗, u∈L ₃. If $\beta_{u}^{-}$ is obstructed by x ₁, we have a contradiction similar to the above. So, suppose that $\beta_{u}^{-}$ is not obstructed by x ₁ and therefore $\beta_{u}^{+}$ is obstructed by x ₀. To summarize current assumptions, x ₁ obstructs $\beta_{v}^{-}$ but not $\beta_{u}^{-}$ and x ₀ obstructs $\beta_{u}^{+}$ but not $\beta_{v}^{+}$. This is possible only if j ^∗=j″, $\beta_{v}^{+}$ goes through x ₀, and $\beta_{u}^{-}$ goes through x ₁. The part of $\beta_{v}^{+}$ connecting v and x ₀ provides an antipodal chord of length smaller than 2h, a contradiction.

(Case 2) X ₀ is non-empty but X ₁, X ₂, X ₃ are all empty. Since some element of L ₂ is obstructed by some element of X ₀, we have a contradiction similarly to Case 1. □

An ordered pair of columns of Γ, with indices i ₁ and i ₂, is a band if it satisfies the following conditions.

• 1. Both column i ₁ and column i ₂ of Γ have a vertex of B.

• 2. For each i=i ₁⊕a, 0<a<i ₂⊖i ₁, each edge e of Γ between a vertex in column i⊕a and a vertex in column i⊕a⊕1 is drawn in R _in, i.e., e⊆R _in.

Lemma 6.11

There is a unique band.

Proof

Since B is a directed cycle of Γ, R _out is connected and hence there is at most one band. Suppose, for contradiction, that there is no band. By Lemma 6.10, we may assume without loss of generality that L ₀≠∅ and L ₂=L ₃=∅. Since there is no band, for some column i, the extremal vertex $v_{1} = c_{i j_{1}}$ of column i is in L ₁∪L ₀, the extremal vertex $v_{2} = c_{(i \oplus1) j_{2}}$ of column i⊕1 is in L ₀ and the directed subpath from v ₁ to v ₂ of B contains B ₂ and B ₃. Then there is a slit σ connecting v ₁ and v ₂: σ goes through the vertices $c_{i (j_{1} + 1)}$, …, $c_{i (j_{2} - 1)}$ if (j ₁+1)<j ₂ and other cases are similar. Since the minor side of σ contains the bottom face, by Lemma 6.4, the major side of σ must contain the top face.

Now, we view the cylinder Γ upside down and use the same argument to find another slit σ′ that contains the top face in its minor side. This pair σ, σ′ of slits satisfies the conditions of Lemma 6.5 and therefore we have a contradiction. □

We are ready to finish the proof of Theorem 1.3. Let columns i ₁ and i ₂ form a band. Let v ₂ (v ₃, resp.) be the vertex of V(B) in column i ₁ (i ₂, reps.) with the smallest row index and v ₁ (v ₀, resp.) be the vertex of V(B) in column i ₁ (i ₂, resp.) with the largest row index. Note that v ₀, v ₁, v ₂, v ₃ appear in the directed cycle B in this cyclic order. Let P _i , 0≤i≤3, denote the subpath of B from v _i to v _i+1mod 4.

Assume without loss of generality that v ₀∈V(B ₀) and suppose P ₀ intersects B _i for 0≤i≤k (and only for 0≤i≤k), where 0≤k≤3. If k=3, then similarly to the proof of Lemma 6.11, we find a slit whose minor side contains the bottom face. But, because of the band, the minor side then contains the top face as well, contradicting Lemma 6.4. Since v ₀∈V(B ₀) is extremal and hence L ₀≠∅, we must have L ₂=∅ by Lemma 6.10 and hence $v_{1} \not\in V(B_{2})$, or k≠2. Therefore, we have k≤1. Similarly, P ₂ intersects B _i for at most two values of i.

We also observe that P ₃ intersects B _i for at most two values of i, since otherwise there is a slit between v ₃ and v ₀ whose minor side contains both the top and the bottom faces. Similarly, P ₁ intersects B _i for at most two values of i.

We have shown that for every j, 0≤j≤3, P _j intersects B _i for at most two values of i. If any P _j intersects B _i for only one value of i then this cannot hold. Therefore, we conclude that for every j, 0≤j≤3, P _j intersects B _i for exactly two values of i. From this conclusion and the assumption v ₀∈V(B ₀), we have v _i ∈V(B _i ) for 0≤i≤3. Let $v_{1}'$ be the vertex of B immediately after v ₁ and $v_{2}'$ the vertex of B immediately before v ₂. Because of the band assumption, v ₁, $v_{1}'$, $v_{2}'$, v ₂ are all in column i ₁. If v ₁ and v ₂ are not adjacent, then either $v_{1}' \in V(B_{1})$ or $v_{2}' \in V(B_{2})$. If v ₁ and v ₂ are adjacent, then either v ₁ or v ₂ belongs to V(B ₁)∩V(B ₂) and therefore we also have either $v_{1}' \in V(B_{1})$ or $v_{2}' \in V(B_{2})$. Suppose $v_{1}' \in V(B_{1})$. The case $v_{2}' \in V(B_{2})$ is similar. Let μ ₁ be an antipodal chord from $v_{1}'$ to v ₃ first going through vertices in column i ₁ to the top face and then through vertices in column i ₂. Let μ ₂ be a similar path in ${\mathcal{R}}_{\Gamma}$ from v ₁ to v ₃ through the bottom face. Since the sum of the lengths of these two antipodal chords is 4h−2, at least one of them has length smaller than 2h, a contradiction. This completes the proof of Theorem 1.3.