DeepAI

# A tight Erdős-Pósa function for planar minors

Let H be a planar graph. By a classical result of Robertson and Seymour, there is a function f:N→R such that for all k ∈N and all graphs G, either G contains k vertex-disjoint subgraphs each containing H as a minor, or there is a subset X of at most f(k) vertices such that G-X has no H-minor. We prove that this remains true with f(k) = c k k for some constant c=c(H). This bound is best possible, up to the value of c, and improves upon a recent result of Chekuri and Chuzhoy [STOC 2013], who established this with f(k) = c k ^d k for some universal constant d. The proof is constructive and yields a polynomial-time O(OPT)-approximation algorithm for packing subgraphs containing an H-minor.

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## 1. Introduction

In 1965, Erdős and Pósa [15] proved that there is a function such that for every graph and every , either contains vertex-disjoint cycles, or there is a set of at most vertices such that is a forest. Many variants and generalizations of this theorem have been developed over the years, such as for cycles satisfying various constraints [33, 7, 34, 49, 31, 4, 17, 6, 35, 40, 41, 50, 51, 32, 29], directed cycles [44, 28], matroid circuits [24], and immersions [36, 25]; see [43, 42] for surveys.

In this paper, the objects of interest are graph minor models. A graph is a minor of a graph if can be obtained from a subgraph of by contracting edges. If is not a minor of , then is said to be -minor free. For every graph , an -model in a graph is a collection of vertex-disjoint connected subgraphs of such that and are linked by an edge in for every edge . We define and the size of as . Two -models and are disjoint if . It is easy to see that is a minor of if and only if there is an -model in .

Let be the maximum number of pairwise disjoint -models in . Let be the minimum size of a subset such that has no -model. Clearly, . We say that the Erdős-Pósa property holds for -models if there exists a bounding function such that

 τH(G)≤f(νH(G))

holds for every graph .

By a classical result of Robertson and Seymour [46], the Erdős-Pósa property holds for -models if and only if is planar; the fact that it does hold when is planar is a consequence of their Grid Minor Theorem. The original bounding function obtained by Robertson and Seymour for planar was exponential. In 2013, Chekuri and Chuzhoy [10] proved that one can take as bounding function for some universal constant , and some constant .

No explicit value for the constant is given in [10] but a quick analysis of their proof suggests that it is at least a double-digit integer. Our main result is that we can take , which is best possible.

###### Theorem 1.1 (Main theorem).

For each planar graph , there exists a constant such that the Erdős-Pósa property holds for -models with bounding function .

A bounding function is best possible for the following reason. For an lower bound on bounding functions was already established by Erdős and Pósa [15]. This lower bound holds more generally when is any planar graph containing a cycle, as can be seen by considering -vertex graphs with treewidth and girth (as constructed in [38], for instance). Then (because -minor free graphs have treewidth when is planar) and (because each -model contains a cycle). We also note that if, on the other hand, is a forest, then the lower bound does not apply, and it is in fact already known that there is a bounding function [19].

Before pursuing further, let us emphasize that the constant we obtain in the proof of Theorem 1.1 is enormous, in fact it is not even known to be computable. On the other hand, depends polynomially on in the bounding function established by Chekuri and Chuzhoy [10] (this follows from [10] combined with their celebrated Polynomial Grid Minor Theorem [11, 12]). Thus, our main result can be seen as a trade-off, where we decrease the value of to an optimal at the price of a much bigger constant factor . Computability of our constant in Theorem 1.1 does not follow from our proofs because we use a result of Fomin, Lokshtanov, Misra, and Saurabh [20] about minor-minimal graphs with , whose proof is non-constructive (see Section 5). Finding good bounds on as a function of is left as an open problem, in particular it would be interesting to determine whether could depend polynomially on .

Prior to this paper, when is planar but not a forest, a bounding function was known to hold for -models if is a triangle [15], a cycle [17, 39], a multigraph consisting of two vertices linked by parallel edges [9], and more generally if is any minor of a wheel [1]. The authors of [1] developed general tools to tackle arbitrary planar graphs , together with some techniques that are specific to wheels. In this paper we build on their approach. Our main technical contribution is a series of lemmas which allowed us to develop the ‘right’ generalization of the objects used in [1]. An overview of the proof will be given shortly but first let us mention some combinatorial and algorithmic consequences of our result.

## 2. Consequences of our results

We describe in this section several consequences of our results. Their proofs are given in Sections 9 and 10.

### Approximation algorithms for packing and covering models

Our proof of Theorem 1.1 is constructive, in the sense that it can be turned into a polynomial-time algorithm computing both a collection of disjoint -models in the input graph , and a subset of at most vertices such that has no -model, for some constant depending on the constant in Theorem 1.1 and for some . Note that , together witness the fact that (1) is within a factor of (since ), and (2) is within a factor of (since ). Thus, we get -approximation algorithms for both the packing and covering problems associated to planar -models.

###### Corollary .

For each fixed planar graph , there is a polynomial-time -approximation algorithm both for computing and .

The result for covering is already known. In fact, for every planar graph , there is even a constant factor approximation algorithm for computing . Indeed, a randomized constant factor approximation was first developed by Fomin, Lokshtanov, Misra, and Saurabh [21], and very recently a deterministic one was obtained by Gupta, Lee, Li, Manurangsi, and Włodarczyk [27].

On the other hand, the result for packing is new. It is also close to best possible in the following sense: When the packing problem corresponds to the well-studied problem of packing cycles, which is known to be quasi-NP-hard to approximate to within a ratio of  [23]. We note also that when is a forest, can be approximated to within a constant factor [19].

### Large treewidth graph decompositions

A second consequence of our main theorem is the following partitioning corollary.

###### Corollary .

There is a function such that for all integers , every graph of treewidth at least

 s???(r)⋅klog(k+1)

has vertex-disjoint subgraphs , each of treewidth at least .

In particular, the treewidth of every graph not containing disjoint copies of a fixed planar graph as a minor is , where the hidden constant depends on . This is best possible when contains a cycle (see the paragraph following Theorem 1.1). A similar result with an bound for some universal constant was obtained by Chekuri and Chuzhoy [10, Theorem 1.1]. Again, we remark that, while the poly-logarithmic dependency on in their bound is not optimal, their theorem has the extra advantage that can be taken as a polynomial, which is not the case in our proof of Section 2.

### Computing minor-closed bidimensional parameters.

Let be a graph parameter, that is, a function mapping graphs to integers and that is constant within each isomorphism class. We say that is minor-closed if for every minor of every graph . In [10], Chekuri and Chuzhoy gave algorithms to compute graph parameters satisfying certain conditions.

###### Theorem 2.1 ([10, Theorem 5.3]).

Let be a minor-closed parameter that is positive on all graphs with treewidth at least , is at least the sum over the components of a disconnected graph, and can be computed in time given a tree-decomposition of width of the graph.

Then there is a constant and an algorithm that, given an -vertex graph and an integer , decides whether in time

 (2O(p2k⋅logdpk)+h(O(p2k⋅logdpk)))nO(1).

Note that the requirements of Theorem 2.1 are satisfied by several well-studied parameters such as feedback vertex set, vertex cover, and more generally any packing or covering problem of models of a fixed planar graph (as described at the beginning of the section). By plugging the improved bounds of our partitionning result Section 2 in the proof of Theorem 2.1 from [10], we obtain the following result.

###### Corollary .

Let be a minor-closed parameter that is positive on all graphs with treewidth at least , is at least the sum over the components of a disconnected graph, and can be computed in time given a tree-decomposition of width of the graph.

Then there is an algorithm that, given an -vertex graph and an integer , decides whether in time

 (2O(s???(p)klogk)+h(O(s???(p)klogk)))nO(1).

Observe that Section 2 improves the dependence on of the algorithm from Theorem 2.1, at the cost of a worse dependence on . However, in the natural setting where is fixed and we want to check for various pairs , is a constant so its contribution is less relevant. As noted in [10], the requirements on can also be stated as follows.

###### Corollary .

Let be a minor-closed parameter that is positive on some -vertex planar graph , is at least the sum over the components of a disconnected graph, and can be computed in time given a tree-decomposition of width of the graph.

Then there is a function and an algorithm that, given an -vertex graph and an integer , decides whether in time

 (2s???(t)klogk+h(s???(t)klogk))nO(1).

### Erdős-Pósa property in minor-closed classes

For a graph and a class of graphs, we say that the Erdős-Pósa property holds for -models in if there exists a bounding function such that holds for every graph . Restricting the class sometimes yields improved bounding functions. For instance, while the bounding function in the classic Erdős-Pósa theorem is , it can be improved to when restricted to planar graphs [3]. In fact, this is true more generally for -models for any fixed planar graph when restricted to any proper minor-closed class , as shown by Fomin, Saurabh, and Thilikos [22].

###### Theorem 2.2 (Fomin, Saurabh, and Thilikos [22]).

Let be a proper minor-closed graph class and let be a planar graph. Then there exists a constant such that the Erdős-Pósa property holds for -models in with bounding function .

As it turns out this theorem also follows directly from our main technical theorem (stated in the next section).

### Packing cycles with modularity constraints

In 1988, Thomassen obtained the following modularity-constrained variant of the Erdős-Pósa theorem:

###### Theorem 2.3 (Thomassen [49]).

For every there is a function such that, for every and every graph , either contains vertex-disjoint cycles of length 0 modulo , or there is a subset of at most vertices such that has no such cycle.

Wollan [51] obtained a similar statement for cycles with non-zero length modulo , when

is odd. As proved by Dejter and Neumann-Lara

[14], the same statement does not hold in general for cycles of length modulo , when . Thomassen’s upper-bound (for fixed ) has later been improved to for some by Chekuri and Chuzhoy [10], who used a partitioning theorem similar to our Section 2. As a consequence of our main theorem, we obtain a bounding function for cycles of length 0 modulo , which is the same as in the original Erdős-Pósa Theorem.

###### Corollary .

For every positive integer there is a constant such that, for every and every graph , either contains vertex-disjoint cycles of length 0 modulo , or there is a subset of at most vertices such that has no such cycle.

Extremal graphs showing that this bound is tight (up to the value of ) can be obtained from extremal graphs for the original Erdős-Pósa Theorem by subdividing every edge times. We actually prove a stronger statement about modularity-constrained subdivisions of planar subcubic graphs, whose proof we postpone to Section 10.

## 3. Overview of the proof

In this paper, all logarithms are binary. Unless otherwise specified, the graphs we consider are finite, simple, and undirected. In particular, when contracting edges of a graph, we subsequently delete resulting loops and parallel edges. Let be a graph. We use and as shorthand for and , respectively.

A separation of a graph is a pair of subsets of such that and has no edge from to . Observe that our definition allows or to be empty. The order of the separation is .

The heart of our proof is the following technical theorem.

###### Theorem 3.1 (Main technical theorem).

For every , every planar graph , and every non-decreasing function with , there is a constant such that for every graph , at least one of the following holds.

1. [label = ()]

2. contains an -model of size at most ;

3. contains a -model of size at most ;

4. has a separation of order at most such that does not contain as a minor and .

Theorem 1.1 follows quickly from Theorem 3.1 using previous results. We give the derivation in Section 5. Thus, it only remains to prove Theorem 3.1.

To give a high-level idea of our proof strategy for Theorem 3.1, we sketch it for the case . Note that every cycle in our graph is a -model. First, we consider a maximum-size collection of paths of length , for some large enough constant . Assume for simplicity that these paths cover all vertices of . If one of the paths in is not induced, we find a cycle of length at most . Similarly, if two of these paths are connected by at least two edges, we get a cycle of length at most . In both cases, we find a -model of size at most for a suitable choice of the constant , and 1 is satisfied. Thus we may assume this does not happen.

Then, we consider the auxiliary graph on vertex set where two vertices are adjacent if the corresponding paths are connected by an edge in . If has large enough minimum degree (as a function of ), then a known result (see Theorem 4.4 in the next section) yields a -model of size in , which translates into a -model of size in , which is outcome 2.

Hence, we may assume that has a vertex of degree bounded by some function of . Then the corresponding path has neighbors in only a few other paths of . By letting and letting be the rest of the graph plus the vertices of with a neighbor in , we obtain outcome 3 (assuming has been chosen large enough).

While the arguments leading to outcomes 2 and 3 above work for all planar graphs , this approach fails in general as the existence of many edges between two paths of does not always yield a small -model (outcome 1).

The aforementioned result of [1] for the case where is a wheel avoids this difficulty by packing paths and cycles instead of just paths. However, this technique breaks down when trying to pack subgraphs having a vertex of degree at least .

In our proof, we addressed this difficulty by introducing a family of objects called orchards and considering orchard packings as a counterpart to the family of paths/cycles. Roughly speaking, orchards have the property that two disjoint orchards connected by many edges either can be combined into more desirable structures (in the same sense that two paths connected by two edges induce a cycle in the proof sketch above), or the orchards can be separated in a ‘clean way’ from each other using a small set of vertices. This allows us to conclude similarly as above. However, the proof is more involved.

The rest of the paper is organized as follows. The next section contains the general definitions and results we use. In Section 5 we prove Theorem 1.1 assuming Theorem 3.1. Orchards and orchard packings are introduced in Section 6 and Section 7, along with some key separation lemmas. Using these results we finally prove Theorem 3.1 in Section 8. The proofs of the algorithmic and combinatorial consequences of our results stated in Section 2 are given in Sections 9 and 10, respectively.

## 4. Preliminaries

A tree-decomposition of a graph is a tree together with subsets of for each satisfying

• ,

• for each , there exists such that , and

• for each , the set of all such that induces a subtree of .

The width of the tree-decomposition is . The treewidth of , denoted , is the minimum width taken over all tree-decompositions of .

###### Theorem 4.1 (Robertson and Seymour [46]).

There exists a function such that for every , every graph of treewidth at least contains every -vertex planar graph as a minor.

By the results of Chekuri and Chuzhoy [11, 12], can be bounded from above by a polynomial function.

We do not directly use tree-decompositions in this paper. Instead, we use the following dual notion. A bramble in a graph is a collection of vertex sets of connected subgraphs of , called bramble sets of , such that for all , or there is an edge between and . The order of is the minimum size of a set such that intersects all bramble sets.

###### Theorem 4.2 (Seymour and Thomas [47]).

Let be an integer. A graph has treewidth at least if and only if it contains a bramble of order at least .

We also require the following two theorems.

###### Theorem 4.3 (Erdős-Szekeres Theorem [16]).

Let . Every sequence of at least distinct integers contains an increasing subsequence of length or a decreasing subsequence of length .

###### Theorem 4.4 (Fiorini, Joret, Theis, and Wood [18], see also [37, 48]).

There is a function such that, for every , if an -vertex graph has average degree at least , then it contains a -model on vertices.

## 5. From the main technical theorem to the main theorem

In this section, we show how Theorem 1.1 can be deduced from Theorem 3.1. We follow the same line of proof as in [1] by considering a minimal counterexample and showing that the outcomes of Theorem 3.1 contradict its minimality. By minor-minimal we mean minimal with respect to the minor ordering. We rely on the following results.

###### Theorem 5.1 (Fomin, Lokshtanov, Misra, and Saurabh [20, Corollary 1]).

For every planar graph , there is a polynomial such that for every , every graph with and minor-minimal with this property satisfies .

Let us emphasize that the polynomial in Theorem 5.1 depends (non-constructively) on .

###### Theorem 5.2 (Fiorini, Joret, and Wood [19]).

For every connected planar graph , there is a computable and non-decreasing function such that, for every graph , if is a separation of where is -minor free and , then there exists a graph such that

 τH(G′)=τH(G),νH(G′)=νH(G),and|G′|<|G|.

Theorem 5.2 as originally stated in [19] does not guarantee that is non-decreasing. We can however easily obtain this property by defining , with the function given in [19], and clearly then has the properties claimed in Theorem 5.2.

###### Lemma (Aboulker, Fiorini, Huynh, Joret, Raymond, and Sau [1, Lemma 2.7], reworded).

Let be a planar graph and let be a bounding function for -models. Then, for each minor of , there is a bounding function for -models with .

We are now ready to prove Theorem 1.1, assuming Theorem 3.1.

###### Proof of Theorem 1.1.

Let us first assume that is connected. We explain at the end of the proof how the result extends to disconnected graphs.

Let and be positive integers such that for every integer , we have , where is the function of Theorem 5.1 for . Such numbers exist as this function is a polynomial.

Let be the function of Theorem 5.2 for the graph . Clearly we can assume . Let be the constant of Theorem 3.1 for and for the function . We prove Theorem 1.1 for , where is a positive integer such that .

Towards a contradiction, suppose for some graph . Among all such graphs, we choose such that the tuple ( is lexicographically minimum. Let .

We apply Theorem 3.1 on with and . According to Theorem 5.2, the outcome 3 of Theorem 3.1 implies the existence of a graph such that

 τH(G′)=τH(G),νH(G′)=νH(G),and|G′|<|G|.

This would however contradict the minimality of . Therefore we may now assume that one of the first two outcomes of Theorem 3.1 holds. Which of the two outcomes holds is not important for the rest of the proof, as we will only use the fact that contains a model of of size at most , which is true in both cases. Using properties of we will show that . Once this is established, using that the graph is not a counterexample to Theorem 3.1, we will conclude that cannot be a counterexample either.

The definition of implies that it is minor-minimal with the property . Thus, if is a proper minor of , then (since ). In particular, is minor-minimal with the property . Now, observe that for any vertex (simply add to an optimal hitting set for ). Hence, and . Therefore, we can apply Theorem 5.1.

 |G| ≤ατH(G)β (\autoref{thm:fominkernel}) =α(f(k)+1)β ≤α(c(k+1)2)β.

Then

 |V(M)| ≤σ⋅log|G| (definition of M) ≤σ(logα+βlogc+2β)log(k+1) ≤clog(k+1) (definition of c).

Let us consider the graph . Observe that

 νH(G)≥νH(G′)+1andτH(G)≤τH(G′)+clog(k+1).

By minimality of , we have . Then

 τH(G) ≤f(νH(G′))+clog(k+1) ≤f(k−1)+clog(k+1) (f is non-decreasing) ≤f(k).

Therefore, is not a counterexample, a contradiction.

We now consider the case where is not connected. Let be a planar connected graph with and . Such a graph can be obtained from planar drawings of the components of by adding edges between their external faces in a planar way. As shown in the first part of the proof, there is a bounding function for -models, for some constant depending on only. By applying Section 5 to , , and , we obtain a bounding function for -models which is of the same order of magnitude as , as desired. ∎

## 6. Orchards

We prove in this section a series of lemmas about bramble-like objects that we call orchards. Given positive integers , an -orchard in is a collection of pairwise vertex-disjoint paths, called horizontal paths, and a collection of pairwise vertex-disjoint trees, called vertical trees, such that

• is non-empty and connected (and thus a path) for each and , and

• each leaf of is on some horizontal path, for each .

With a slight abuse of notation we also write for the subgraph formed by the union of the horizontal paths and vertical trees of . It should be clear from the context whether means the orchard itself or the corresponding subgraph of .

Orchards are similar to brambles in the sense that they can serve as certificates for large treewidth. In fact, every large enough orchard contains a bramble of large order (see the proof of Section 6). However, they are more structured, which makes them easier to handle. We note that grids are particular examples of orchards. Thus, in this sense orchards lie somewhere in between grids and brambles. We note that a concept similar to orchards is that of grid-like minors, introduced by Reed and Wood [45]. Grid-like minors are collections of paths whose intersection graphs are bipartite and contain a large clique minor. While orchards and grid-like minors have common features (note that the intersection graph of the horizontal paths and vertical trees of an orchard is a complete bipartite graph), in general they are incomparable objects.

The main result of this section is a separation lemma for orchards, Section 6, which will be used in the proof of Theorem 3.1.

###### Lemma .

If a graph contains an -orchard, then contains every -vertex planar graph as a minor.

###### Proof.

Let be an -orchard with a collection of horizontal paths and a collection of vertical trees . Consider the bramble in . Since the vertical trees are vertex-disjoint, the horizontal paths are vertex-disjoint, and , it follows that the order of is at least . By Theorem 4.2, has treewidth at least , and therefore by Theorem 4.1, contains every -vertex planar graph as a minor. ∎

Let be an -orchard with horizontal paths and vertical trees . Let and . We say that is a horizontal section if for some or if is a component of . Note that the set of all horizontal sections is a collection of vertex-disjoint paths whose union covers all vertices of . Let be the set of vertices such that for some , and . We say that is a vertical section if is a vertex in (seen as a single-vertex path) or if is a component of . We say that is a section if is a horizontal or a vertical section. Note that the set of all sections is a collection of vertex-disjoint paths whose union covers all vertices of . In the proofs of Lemmas 6 and 6 below, we will use several times that if has horizontal paths, then each of its vertical trees defines at most vertical sections.111 Proof. We proceed by induction on . If , then every vertical tree has one vertical section. Let . Let be an orchard. In the following, whenever we speak of a neighbor, it is with respect to viewed as a graph. Given a vertical tree , let be a horizontal path such that at most one vertex of has a neighbor in . Let be the tree obtained from by iteratively deleting the unique leaf that is not on any horizontal path other than . Let be the sequence of such leaves, and let denote the neighbor of in . Let be the orchard obtained from by deleting and replacing with . Let be the unique section of containing . Every vertical section of in which is not a vertical section of in must be one of the following. (i) the path or (ii) or (iii) one of the at most two components of . (We remark that situations (ii) and (iii) only apply if is a vertical section of and .) By induction, there are at most vertical sections of on . By the discussion above, has at most three more vertical sections.

We define a myriapod to be a tree of maximum degree at most such that all its degree vertices are on a single path , called the spine of . The components of will be called the legs of .

We show that the sections of an orchard can be covered by few myriapods.

###### Lemma .

Let be an -orchard. There is a collection of at most subgraphs of such that:

• every element of is a myriapod whose spine is a horizontal path of and each of whose legs is contained in some vertical tree;

• every section of is contained in some element of .

###### Proof.

For each ordered pair of distinct horizontal paths

in , we take and extend it to a myriapod by adding to it the following legs. For each vertical tree in , we add the (unique) subpath of that has endpoints in respectively but has no vertex of in its interior. By the uniqueness of the paths and because vertical trees are vertex-disjoint, the resulting graph is a myriapod. There are less than ordered pairs of horizontal paths. Since each horizontal section of is contained in some horizontal path and each vertical section is contained in some connecting subpath , it follows that the constructed myriapods together cover all sections of . ∎

Recall that each vertical tree intersects each horizontal path in a subpath and that these subpaths are disjoint. Thus, each horizontal path defines two symmetric total orders on the vertical trees, which are given by the order in which we meet these trees when following from one endpoint to the other.

We say that an -orchard is tame if its vertical trees appear in the same order along every horizontal path. Formally, is tame if there is a permutation of such that for every , we meet the horizontal trees of in the order , or the reverse order, when following from one endpoint to the other.

Given a horizontal section of a horizontal path and a vertical tree in an orchard , we say that is bordered by if does not intersect and, with respect to viewed as a graph, one of the endpoints of has a neighbor which is a vertex of . If additionally (given an ordering of ‘from left to right’) there is such a neighbor to the left (right) of , then we say that is bordered by on its left (right).

An orchard is a suborchard of an orchard if is obtained from by selecting a subset of its horizontal paths and a subset of its vertical trees.

###### Lemma .

There exists a function such that, for every , if is an -orchard, then contains a tame -suborchard .

###### Proof.

We claim that we may take . The proof is by induction on . Note that every -orchard is tame, and , so the claim holds for .

For the inductive step, let be the horizontal paths of and let us consider the orchard obtained from by ignoring and contracting some edges of the vertical trees so that the leaves of each vertical tree lie on . More precisely, from each vertical tree we iteratively delete the leaves that are not in . Observe that . Therefore, by induction, this orchard contains a tame -suborchard .

Let be the vertical trees of , named according to the order in which they intersect . Since is tame, this is also the order in which they intersect for all . Let be the corresponding trees in . Choose one of the two possible orientations for arbitrarily and let be the order in which intersect . By Theorem 4.3, contains an increasing or decreasing subsequence . Let be the vertical trees of corresponding to . By reversing the orientation of if necessary, we obtain a tame suborchard of , as required. ∎

Using Lemmas 6 and 6, we now derive separation lemmas that will be key tools in the main proof. These lemmas and those in Section 7 are all parameterized by some positive integer . In Section 8 we will apply these lemmas with the value .

Given two disjoint subsets of vertices of a graph , we say that sees if there is an edge in linking a vertex of to one of .

###### Lemma .

Let . Suppose that is an -orchard and is an -orchard vertex-disjoint from in a graph , with . Then, for each at least one of the following holds.

• contains pairwise vertex-disjoint -orchards;

• there exists with such that sees at most sections of the orchard .

###### Proof.

Let denote the set of sections of . Consider the auxiliary bipartite graph with vertex partition with the vertices of in one part and the sections of in the other part, such that is an edge in if and only if sees section in .

Let denote the set of vertices of that see more than sections of . Suppose . Then contains a matching of size . Suppose on the other hand that . If sees at most sections of , then we are done. Thus we may suppose that sees more than sections of . By definition of , each vertex in sees at most sections of . Hence, there exists a matching of size between and the sections of .

Thus in both cases, contains a matching of size . From this fact we will derive that contains pairwise vertex-disjoint -orchards.

By Section 6 applied to and by the pigeonhole principle, there is a myriapod in such that:

• the spine of is a horizontal path of and each leg of is a subgraph of a vertical tree of ; and

• at least sections matched by are contained in .

If , then each leg of is a subgraph of a vertical tree of and hence contains at most sections. (If , then has no legs.) It follows that there is a submatching of size at least such that the sections of matched by are

1. [label = ()]

2. all on the spine of , or

3. on distinct legs of .

By 2 we mean that each section matched by is on some leg of and no two such sections are on the same leg of .

We can apply a similar reduction to the vertices of matched by . By Section 6, the vertices of can be covered with at most (recall is the number of horizontal paths of ) myriapods whose spines are horizontal paths of and each of whose legs is a subgraph of a vertical tree of . Thus there is such a myriapod in such that at least vertices of are matched by .

Next, we claim that we can find a submatching of size at least

 √1m6⋅f???(c,m)=2⋅(2m⋅c)2m+1

such that the vertices of matched by are

1. [label = (0)]

2. all on the spine of , or

3. on distinct legs of , or

4. all on a single leg of .

This can be seen as follows. Let . As , the myriapod has at least vertices matched by . A part is the spine or a leg of . If some part of contains at least matched vertices, then 1 or 3 holds, and we are done. Otherwise, strictly more than parts have at least one matched vertex. Since is an integer, there are at least such parts. By possibly discarding the spine, we obtain distinct legs each having a matched vertex, and thus 2 holds.

We now extend the -orchard to an -orchard, as follows. As the -th horizontal path of the new orchard we take (in case 1 and 2) the spine of or (in case 3) the leg of that is matched by . For each edge in we choose an edge in the original graph , which has endpoints and some vertex on section . In case 1 and 3 we call this edge . In case 2, by using the leg of it intersects, we extend this edge to a path with an endpoint on the spine of and all its internal vertices on . We also call this new path . After this, has one endpoint on the chosen -th horizontal path.

Given two subgraphs and of , we write for the graph with vertices and edges .

In case 2, the other endpoint of is on a vertical tree of . We extend to a larger vertical tree . For distinct edges , the vertical trees and are distinct and thus the extended vertical trees and are still vertex-disjoint. In this way we obtain extended vertex-disjoint vertical trees that each intersect our chosen extra horizontal path. Thus we have constructed an -orchard, which contains an -suborchard.

In case 1, we do almost the same. The difference is that the -endpoint of is possibly not on a vertical tree. In that case, in order to appropriately extend the vertical trees, we need to add some subpaths of the spine of . In doing that, we need to take care that the extended vertical trees are still vertex-disjoint. One can do this by ordering the vertices of ‘from left to right’. If intersects a vertical tree, then we extend the tree as before. If does not intersect a vertical tree, then we consider a tree that has the closest intersection point with to the left of . There may exist (at most) one such that has no vertical tree strictly to its left. In that case we drop from . Next, we extend to , where is the smallest subpath of containing both and a vertex of . Since each meets a unique section of the horizontal path and since for every vertical tree there exist at most two horizontal sections of that intersect or are bordered by on their left, this ordering guarantees that at least half of the extended vertical trees remain pairwise vertex-disjoint. We thus obtain an -orchard, which contains a suborchard of the desired size since