Polylogarithmic approximation for minimum planarization (almost)
In the minimum planarization problem, given some -vertex graph, the goal is to find a set of vertices of minimum cardinality whose removal leaves a planar graph. This is a fundamental problem in topological graph theory. We present a -approximation algorithm for this problem on general graphs with running time . We also obtain a -approximation with running time for any arbitrarily small constant . Prior to our work, no non-trivial algorithm was known for this problem on general graphs, and the best known result even on graphs of bounded degree was a -approximation [CS13].
As an immediate corollary, we also obtain improved approximation algorithms for the crossing number problem on graphs of bounded degree. Specifically, we obtain -approximation and -approximation algorithms in time and respectively. The previously best-known result was a polynomial-time -approximation algorithm [Chu11].
Our algorithm introduces several new tools including an efficient grid-minor construction for apex graphs, and a new method for computing irrelevant vertices. Analogues of these tools were previously available only for exact algorithms. Our work gives efficient implementations of these ideas in the setting of approximation algorithms, which could be of independent interest.
In the minimum planarization problem, given a graph , the goal is to find a set of vertices of minimum cardinality whose removal leaves a planar graph. This is a fundamental problem in topological graph theory, which been extensively studied over the past 40 years. It generalizes planarity, and has connections to several other problems, such as crossing number and Euler genus. The problem is known to be fixed-parameter tractable [Kaw09, MS12, JLS14], but very little is known about its approximability.
1.1 Our contribution
Prior to our work, no non-trivial approximation algorithm for minimum planarization was known for general graphs. The only prior result was a -approximation for graphs of bounded degree [CS13]. We present the first non-trivial approximation algorithms for this problem on general graphs. Our main results can be summarized as follows:
There exists a -approximation algorithm for the minimum vertex planarization problem with running time .
For any arbitrarily small constant , there exists a -approximation algorithm for the minimum vertex planarization problem with running time .
Applications to crossing number.
The crossing number of a graph , denoted , is the minimum number of crossings in any drawing of into the plane (see [Chu11]). Prior to our work, the best-known approximation for the crossing number of bounded-degree graphs was due to Chuzhoy [Chu11]. Given a bounded-degree graph, her algorithm computes a drawing with crossings, which implies a -approximation. We now explain how our result on minimum planarization implies an improved approximation algorithm for crossing number on bounded-degree graphs. It is easy to show that for any graph , , simply by removing one endpoint of one edge involved in each crossing in some optimal drawing. Thus, using our -approximation algorithm for minimum planarization, we can compute a planarizing set of size at most . Thus, in graphs of maximum degree , we can compute some , with , such that is planar. Chimani and Hlinený [CH17] (see also [CMS11]) have given a polynomial-time algorithm which given some graph and some , such that is planar, computes a drawing of with at most crossings. Combining this with our result we immediately obtain an algorithm with running time , which given a graph of bounded degree, computes a drawing of with at most crossings. Similarly, we obtain an algorithm with running time , which given a graph of bounded degree, computes a drawing of with at most crossings, for any fixed . Combining this with existing approximation algorithms for crossing number of graphs of bounded degree that are based on balanced separators, we obtain the following (see [Chu11] for details).
There exists a -approximation algorithm for the crossing number of graphs of bounded degree, with running time . Furthermore there exists a -approximation algorithm for the crossing number of graphs of bounded degree, with running time , for any fixed .
1.2 Related work
In the -deletion problem, the goal is to compute a minimum vertex set in an input graph such that is -minor-free. Characterizing graph properties for which the corresponding vertex deletion problem can be approximated within a constant factor or a polylogarithmic factor is a long standing open problem in approximation algorithms [Yan94, LY93]. In spite of a long history of research, we are still far from resolving the status of this problem. Constant-factor approximation algorithms for the vertex Cover problem (i.e., ) are known since 1970s [BYE81, NT74].
Yannakakis [Yan79] showed that approximating the minimum vertex set that needs to be deleted in order to obtain a connected graph with some property within factor is NP-hard, for a very broad class of properties (see [Yan79]). There was not much progress on approximability/non-approximability of vertex deletion problems until quite recently. Fomin et al. [FLMS12] showed that for every graph property expressible by a finite set of forbidden minors containing at least one planar graph, the vertex deletion problem for property admits a constant factor approximation algorithm. They explicitly mentioned that the most interesting case is when contains a non-planar graph (they said that perhaps the most interesting case is when ), because there is no poly-logarithmic factor approximation algorithm so far. Indeed, the planar graph case and the non-planar case for the family may be quite different, as the graph minor theory suggests. The main result of this paper almost settles the most interesting case. We believe that our techniques can lead to further results on approximation algorithms for minor-free properties.
2 High level description of the algorithm
We now give a brief overview of our approach and highlight some of the main challenges. Our approximation algorithm is inspired by fixed-parameter algorithms for the minimum planarization problem, where one assumes that the size of the optimum planarizing set is some fixed constant (see [Kaw09, MS12, JLS14]).
2.1 Overview of previous fixed-parameter algorithms.
The known fixed-parameter algorithms for minimum planarization work as follows: If the treewidth of the input graph is large enough (say, , for some constant ), then one can efficiently compute a large grid minor in (that is, a minor of that is isomorphic to some grid). A subgraph of is called flat if it admits some planar drawing with outer face , such that all edges between and have one endpoint in . If some vertex is surrounded by a flat subgrid of of size , then it is irrelevant; this means that by removing , we do not change any optimal solution. Thus, if such an irrelevant vertex exists, we recurse on , and return the optimum solution found. We define the face cover of some set of vertices to be the minimum number of faces of that are needed to cover . If there exists some vertex such that the neighborhood of has face cover of size , then is universal; that is, removing decreases the size of some optimum planarizing set by 1. Thus, if such a universal vertex exists, we recurse on , and return the optimum solution found, together with . If the grid is large enough, then we can always find either an irrelevant or a universal vertex. Thus, by repeatedly removing such vertices, we arrive at a graph of bounded treewidth, where the problem can be solved using standard dynamic programming techniques.
2.2 Obtaining an approximation algorithm.
We now discuss the main challenges towards extending the above approach to the approximate setting. In order to simplify the exposition, we discuss the -approximation algorithm. The -approximation is essentially identical, after changing some parameters.
1. The small treewidth case. In the above fixed-parameter algorithms, the problem is eventually reduced to the bounded-treewidth case. That is, one has to solve the problem on a graph of treewidth , for some function . Since the optimum is assumed to be constant, this can be done in polynomial time (in fact, linear time). However, in our setting, can be as large as , and thus this approach is not applicable. Instead, we try to find some small balanced vertex separator . If the treewidth is at most , then we can find some separator of size . In this case, we recurse on all non-planar connected components of , and we add to the final solution. It can be shown that can be charged to the optimum solution, so that the total increase in the cost of the solution is .
2. The large treewidth case. We say that a graph is -apex if it can be made planar by the removal of at most vertices. Since any planar graph of treewidth contains a grid minor of size , it easily follows that any -apex graph of treewidth , for some universal constant , also contains a grid minor of size . To see that, first delete some planarizing set of size , and then find a grid minor in the resulting planar graph, which has treewidth at least . However, even thought it is trivial to prove the existence of such a large grid minor, computing it in polynomial time when is not fixed turns out to be a significant challenge. We remark that it is known how to compute a grid minor of size when [CS13, KS15], and this is enough to obtain a -approximation algorithm. However, in order to obtain a -approximation, we need to find a grid minor when .
3. Doubly-well-linked sets The first main technical contribution of this work is an algorithm for computing a large grid minor in -apex graphs, when is not fixed. Suppose that the treewidth of is . As a first step, we compute some separation of order (that is, some with and ), and some such that is well-linked in both sides of the separation. Intuitively, for a set to be well-linked in some graph means that does not have any sparse cuts, w.r.t. ; in other words, contracting into results in an “expander-like” graph (see Section 3 for a formal definition). We refer to such a set as doubly-well-linked. We remark that the notion of doubly-well-linked set considered here is similar to, and inspired by, the well-linked bipartitions introduced by Chuzhoy [Chu11] in her work on the crossing number problem. It is well-known that in any graph, such a separation can be found so that is well-linked in at least one of the two sides. However, as we explain below, we need to be well-linked in both sides of the separation.
4. From a doubly-well-linked set to a grid minor. There are several algorithms for computing large grid minors in planar graphs [CKS04, RST94b]. A key ingredient in these algorithms is the duality between cuts and cycles in embedded planar graphs. That is, any cut of a planar graph corresponds to a collection of cycles in its dual. The algorithms for the planar case exploit this duality by first computing a well-linked set , and then finding some disk in the plane, that contains and has a large fraction of on its boundary. Then, one can find two sets of paths and , with endpoints on the boundary of , such that every path in intersects every path in . By planarity, this yields a grid minor.
In our case we cannot apply this idea since we don’t have a planar drawing of the graph (indeed, this is precisely what we want to compute). However, it turns out that, intuitively, any doubly-well-linked set behaves as a Jordan curve. That is, if we remove any optimal solution from (that is, any planarizing set of minimum cardinality), then there exists a planar drawing of such that most of the vertices in are close to the outer face. Since is well-linked in , we can route in , with low congestion, a multicommodity flow that routes a unit demand between every pair of vertices in . We then sample paths from this flow, for some sufficiently large constant . The fact that the congestion is low, can be used to deduce that the resulting paths will avoid all vertices in some optimal solution, with some constant probability. Thus, the union of the sampled paths admits a planar drawing. Furthermore, since is doubly-well-linked, we can show that, with some constant probability, the union of the sampled paths can be drawn so that their endpoints are all in the outer face. We thus use the union of the sampled paths to construct a skeleton graph. Specifically, we find a suitable subgraph of which does not intersect some optimal solution. We then sample more paths from the flow, and partition them into two sets and , depending on the structure of their intersection with the skeleton graph. Conditioned on the event that the skeleton graph does not intersect some optimal solution, we can find sets of paths and such that every path in intersects every path in .
5. Computing a partially triangulated grid minor. Having computed a large grid minor , we wish to use to find either universal or irrelevant vertices. To that end, we need to ensure that there are no edges between different faces of . We first compute some , such that can be contracted into some grid of size , where is obtained by “eliminating” some rows and columns on .
6. Computing a semi-universal set. The next main technical challenge in our algorithm is the computation of universal vertices. In the fixed-parameter algorithms described above, in order to compute a universal vertex, one needs a grid of size at least . However, in our case, we only have a grid of size . Thus, we cannot always find a universal vertex. We overcome this obstacle by introducing the notion of a semi-universal vertex: We say that a set of vertices is semi-universal if deleting from decreases the cost of the optimum by at least . We can prove that if the size of the neighborhood of in is at least , then we can find some that is semi-universal. Intuitively, the algorithm finds some that behaves as an expander: for every , the size of the face cover of is at least .
7. Computing an irrelevant vertex. The next technical difficulty is computing an irrelevant vertex, when the size of the neighborhood of in is at most . As in the computation of universal vertices, this is a fairly easy task when the treewidth is at least . However, here we can only find a grid minor of size . We overcome this difficulty as follows: We first partition the grid minor into subgrids, in a fashion similar to a quad-tree decomposition: There are partitions of , each into subgrids of size , for all . Then, for each subgrid in this collection of partitions, we compute an upper estimate on its planarization number: If the number of vertices of in this subgrid is at least , then we set the estimate to be , and otherwise we recursive approximate its minimum planarization number. Finally, we add to this estimate the number of neighbors of in this subgrid. We say that some subgrid of of size is active if its upper estimate is at least , for some constant . Since the size of the neighborhood of is small, we can show that there exists some that lies outsize all active subgrids; we can then show that must be irrelevant.
8. Embedding into a higher genus surface. Given the above algorithms for computing grid minors, semi-universal vertices, and irrelevant vertices, the algorithm proceeds as follows. We iteratively compute one of the following (1) a small balanced separator, (2) a semi-universal set, or (3) a set of irrelevant vertices. We refer to such a sequence of reductions as a pruning sequence. The graph obtained at the end of some pruning sequence is planar. Let be the set of vertices removed in Cases (1) and (2), throughout the pruning sequence. We have . We say that the cost of the pruning sequence is . However, the number of irrelevant vertices removed can be as large as . We need to add these vertices to the planar drawing of the resulting graph. It turns out that this is not always possible. The reason is that the irrelevant vertices removed are only guaranteed to be irrelevant w.r.t. any optimal planarizing set; in contrast, the set does not form an optimal planarizing set (indeed, we remove vertices). We overcome this obstacle using a technique that was first introduced in [CS13]: When deleting a set of irrelevant vertices, we add a grid of width 3, referred to as a frame, around the “hole” that is created. The point of adding this grid is that we can inductively add the irrelevant vertices back to the graph as follows: If does not intersect the frame, then we can simply add the irrelevant vertices back to the graph without violating planarity. Otherwise, if the frame intersets vertices from , then we can extend the current drawing to the irrelevant vertices corresponding to that frame by adding at most handles or antihandles. This leads to an embedding into a new non-planar surface. Repeating this process over all frames, we obtain an embedding of into some surface of Euler genus .
9. The final alorithm. The last remaining step is to compute a planarizing set for . It turns out that this can be done by exploiting the embedding of into the surface of Euler genus , that was computed above. Using tools from the theory of graphs of surfaces, we show that we can decrease the Euler genus of by one, while deleting at most vertices. Repeating this process times, we obtain a planar graph after deleting a set of at most vertices. The final output of the algorithm is , which is a planarizing set for of size .
The rest of the paper is organized as follows. Section 3 introduces some basic definitions and results that are used throughout the paper. Section 4 presents the main algorithm, by putting together the main ingredients of our approach. Section 5 presents our algorithm for computing a doulby-well-linked set. Section 6 introduces the notion of a pseudogrid, which are used to construct grid minors. Section 7 presents the algorithm for computing a grid minor. Section 8 gives the algorithm for contracting the graph into a partially triangulated grid, with a small number of apices. Section 9 shows how to compute a semi-universal set, given a partially-triangulated grid contraction, such that the apex set has a large neighborhood. Section 10 shows how to compute irrelevant vertices, for the case where the apex set of the partially triangulated grid contraction has a small neighborhood. Section 11 shows how, given an algorithm for computing irrelevant vertices, we can compute a patch. Section 12 combines the above algorithms for computing grid minors, semi-universal vertex sets, and patches, to obtain an algorithm for computing a pruning sequence of low cost. Section 13 presents an algorithm which given given a pruning sequence of low cost, computes an embedding into a surface of low Euler genus. Finally, Section 14 gives the algorithm for planarizing a graph embedded into some non-planar surface.
3 Definitions and preliminaries
This section provides some basic notations needed in this paper.
Let be a graph and . We use to denote the shortest path metric on . For any we write . For any we define . For a simple path and we denote by the subpath of between and . We define the grid to be the Cartesian product , where denote the path with vertices. Let be the -grid. For each we denote by and by the indexes of the row and column of respectively. We denote by the boundary cycle of . Every face in other than is a cycle of length 4. We say that some graph is the partially triangulated -grid if is obtained from by adding for every face of , with , at most one diagonal edge.
For some graph , and some such that is planar, we say that is planarizing (for ). We denote by the minimum vertex planarization number of , i.e.
We remark that deciding whether is precisely the problem of deciding whether is planar, which can be solved in linear time [HT74].
Let be a graph and let be a collection of pairwise vertex-disjoint subgraphs of . Then .
Let with and such that is planar. For each let . Then for any , we have that is a subgraph of , and thus it is planar. Thus . It follows that . ∎
Minors and contractions.
A graph obtained via a sequence of zero or more edge contractions on a graph is called a contraction of . Unless stated otherwise, we will replace a set of parallel edges in a contraction of a graph by a single edge. The induced mapping is called the contraction mapping (w.r.t. and ). Similarly, we say that is a minor of if it can be obtained via a sequence of zero or more edge contractions, edge deletions, and vertex deletions, performed on . The induced mapping is called the minor mapping (w.r.t. and ). For any we also write . For any we will abuse notation slightly and write to denote the unique vertex with .
Treewidth, pathwidth, and grid minors.
A tree decomposition of a graph is a pair , where is a tree and is a family of vertex sets , such that the following two properties hold:
, and every edge of has both ends in some .
If and lies on the path in between and , then .
The width of a tree decomposition is , and the treewidth of is defined as the minimum width taken over all tree decompositions of . If is a path, then we can define the pathwidth of as the minimum width taken over all path decompositions of . We use and to denote the treewidth and pathwidth of respectively.
We recall the following result on the treewidth of planar graphs.
Lemma 3.2 (Robertson, Seymour and Thomas [RST94a]).
Any planar graph of treewidth contains some -grid minor for some .
From Lemma 3.2 we obtain the following.
Lemma 3.3 (Existence of a large grid minor in a -apex graphs).
Let be a -apex graph of treewidht . Then contains some -grid minor for some .
Let with such that is planar. We have . By Lemma 3.2 we have that contains the -grid as a minor for some . ∎
The following is a related result needed in this paper.
Lemma 3.4 (Eppstein [Epp14]).
Let . Let be the -grid, and , with . Then, contains the -grid as a minor, where .
Our proof needs to handle a graph of large tree-width. It is well-known that such a graph must have a “highly-connected” subset, which is often referred to as “well-linked”.
Definition 3.5 (Cut-linked set).
Let be a graph, let and . We say that is -cut-linked (in ), iff for any partition of into , we have
Graph on surfaces.
A drawing of a graph into a surface is a mapping that sends every vertex into a point and every edge into a simple curve connecting its endpoints, so that the images of different edges are allowed to intersect only at their endpoints. The Euler genus of a surface , denoted by , is defined to be , where is the Euler characteristic of . This parameter coincides with the usual notion of genus, except that it is twice as large if the surface is orientable. For a graph , the Euler genus of , denoted by , is defined to be the minimum Euler genus of a surface , such that can be embedded into .
Let be a graph and let be an embedding of into some surface . A simple non-contractible loop in is called a -noose if it intersects the image of only on vertices. We define the length of to be
Sparsest-cut and the multi-commodity flow-cut gap.
Definition 3.6 (Sparsest Cut).
Consider a graph . The sparsity of a cut equals
where and is the number of cut edges, that is, the number of edges from to . The Sparsest Cut problem asks to find a cut with smallest possible sparsity .
Let . For a graph , we say that some is a -balanced vertex separator if every connected component of contains at most vertices. We also say that some is a -balanced edge separator if every connected component of contains at most vertices. The following is the well-known for approximating the sparsest cut. The uniform sparsest-cut means the sparsest-cut problem with uniform weight on every edge.
Theorem 3.7 (Arora, Rao and Vazirani [Arv09]).
There exists a polynomial-time -approximation for uniform sparsest-cut. There is also a polynomial-time algorithm which given some graph that contains a -balanced edge separator of size , computes a -balanced edge separator of size .
Theorem 3.8 (Feige, Hajiaghayi and Lee [Fhl08]).
There exists a polynomial-time which given a graph outputs a -balanced vertex separator of of size at most , where is the size of the minimum -balanced vertex separator, for some universal constant .
We recall that for any graph , we have . Combined with Theorem 3.8 one obtains the following.
Corollary 3.9 (Feige, Hajiaghayi and Lee [Fhl08]).
There exists a polynomial-time algorithm which given a graph of treewidth outputs a tree decomposition of of width and a path decomposition of of width .
Finally, we need the following well-known theorem about the flow-cut gap for multicommodity flows.
Theorem 3.10 (Leighton and Rao [Lr99]).
The multi-commodity flow cut gap for product demands in -vertex graphs is .
Some of the algorithms presented in this paper are randomized. In order to simplify notation, we say that an algorithm succeeds with high probability when the failure probability is at most . When the target running time is polynomial, we need to be some sufficiently large constant. Similarly, when the target running time is , we need . Both guarantees can be achieved by repeating some algorithm that succeeds with constant probability, either at most or times respectively.
4 The main algorithm
In this Section we present the main algorithm for approximating minimum planarization. We first introduce some definitions. Intuitively, a patch is a small irrelevant subgraph, that is contained inside some disk in any optimal solution. The framing of a patch is a new graph, that does not contain the interior of the patch, and instead contains grid of constant width attached to the boundary of the patch. Computing a framing in this manner will allow us to extend an approximate solution to the whole graph via an embedding into a higher genus surface.
Definition 4.1 (Patch).
Let be a graph, let , and let be a cycle. Suppose that there exists a planar drawing of having as the outer face. Then we say that the ordered pair is a patch (of ).
Definition 4.2 (Framing).
Let be a graph and let be a patch of . Suppose that , and . Let be the graph with
where are new vertices, and
We refer to the graph as the -framing of (see Figure 1 for an example).
Using the above definitions, we can now define the concept of a pruning sequence. Intuitively, this consists of a sequence of operations that inductively simplify the graph until it becomes planar.
Definition 4.3 (Pruning sequence).
Let be a graph. Let be a sequence satisfying the following properties:
(1) For all , is a graph. Moreover and is planar.
(2) For all , exactly one the following holds:
(2.1) and . We say that is a deletion step (of ).
(2.2) , where is a patch in , and is the -framing of . We say that is a framing step (of ).
We also let .
We say that is a pruning sequence (for ). We also define the cost of to be
The next Lemma shows how to compute a pruning sequence of low cost. We remark that the algorithm for computing a pruning sequence calls recursively the whole approximation algorithm on graphs of smaller size. By controlling the size of these subgraphs, we obtain a trade-off between the running time and the approximation guarantee. We remark that this is the main technical contribution of this paper. The proof of the next Lemma uses several other results, and spans the majority of the rest of the paper.
Lemma 4.4 (Computing a pruning sequence).
Let be an -vertex graph, and let . Suppose that there exists an algorithm which for all , with , given an -vertex graph , outputs some , such that is planar, with , for some , in time , where is increasing and convex. Then the algorithm returns some pruning sequence for of cost at most . Moreover
where and denote the worst-case running time of and respectively on a graph of vertices.
The next Lemma shows that given a pruning sequence of low cost, we can efficiently compute an embedding into a surface of low Euler genus, after deleting a small number of vertices.
Lemma 4.5 (Embedding into a higher genus surface).
Let be a graph and let be a pruning sequence for . Then there exists a polynomial-time algorithm which given and outputs some , with , and an embedding of into some surface of Euler genus .
The next Lemma gives an efficient algorithm for planarizing a graph embedded into some surface of low Euler genus.
Lemma 4.6 (Planarizing a surface-embedded graph).
Let be a graph, and let be an embedding of into some surface of Euler genus . Then there exists a polynomial-time algorithm which given and , outputs some planarizing set for , with .
We are now ready to present our main results in this paper, which combines the above results.
Let , and . Then there exists a -approximation algorithm for the minimum vertex planarization problem with running time
Let be the parameter from Lemma 4.4, with . Using the algorithm from Lemma 4.4 we compute a pruning sequence of with . Using the algorithm from Lemma 4.5, in polynomial time, we compute some , with , and an embedding of into some surface of genus . Using the algorithm from Lemma 4.6 we compute some , such that is planar, with . It follows that is a planarizing set of , with . Thus the resulting approximation factor is at most .
The total running time is dominated by the running time of the algorithm from Lemma 4.4. Thus we have , which concludes the proof. ∎
By Theorem 4.7, we easily obtain the following results.
5 Computing a doubly cut-linked set
In this Section we show that when the treewidth is sufficiently large, we can efficiently compute a separation , and some large , such that is well-linked on both sides of the separation. This result will form the basis for our algorithm for computing a grid minor in the subsequent Sections.
Let be a -vertex -apex graph of treewidth . In this section, we shall, in polynomial time, compute some separation of of order at most and some with that is -cut-linked in both and . Moreover let be the graph obtained from by removing all edges in , that is . Then is also -cut-linked in .
To this end, we begin with the following lemma, which merges two disjoint cut-linked sets.
Lemma 5.1 (Merging two cut-linked sets).
Let and . Let be a graph and let that are all -cut-linked in , with . Suppose further that for any there exist a collection of pairwise edge-disjoint paths in between and , with , such that all paths in have distinct endpoints. Then is -cut-linked in .
Let and . For any let , , and let , . Let also . Let and . We distinguish between the following cases:
Case 1: Suppose that or . Then
Case 2: Suppose that or . This is similar to Case 1.
Case 3: Suppose that .
Case 3.1: Suppose that . Then
Case 3.2: Suppose that . We may assume w.l.o.g. that and thus ; the case is identical by swapping and . We may also assume w.l.o.g. that because the case is identical by swapping and . Since , it follows that and .
Case 3.2.1: Suppose that . Since , we get . Thus
Case 3.2.2: Suppose that . Then . There exist at most paths in with both endpoints in . Thus there exist at least paths in with at least one endpoint in . Moreover there exist at most paths in with both endpoints in . Therefore there exist at least paths in with one endpoint in and one endpoint in . Each such path must contain an edge in . Therefore we have
Case 3.3: Suppose that . This is similar to Case 3.2.
We conclude that in either case we have , as required. ∎
The next two lemmas are concerning results for the -grid. These two lemmas are needed in the rest of this section.
Let and let be the -grid. Let be the set of vertices in the first row of . Then is -cut-linked in .
Let and let . Let and . Assume w.l.o.g. that . Suppose that for every the column of containing intersects both and . Then we get . Similarly, if for every the column of containing intersects both and then . It remains to consider the case where there exist and such that the column (resp. ) of containing (resp. ) does not intersect both and . Since and we get and . Since every row in intersects both and (because there exist and such that the column (resp. ) of containing (resp. ) does not intersect both and ), it follows that every row in contains an edge in , and thus . In either case we have obtained that , concluding the proof. ∎
Lemma 5.3 (Linking two grid minors in a grid).
Let be a grid and let . For each let be some -grid and suppose that is a minor of with minor mapping . Assume that . For each let be the set of vertices in the first row of , and for each pick some . For each let . Then is -cut-linked in .
Let . Let be a cut in . Define a cut in as follows: Let be an edge crossing , and let with . We cut all the edges in that are incident to . Since has maximum degree 4, it follows that . Since is 1-cut-linked in , we have . By construction, if a pair of vertices is separated by the cut in , then the pair of vertices , is also separated by the cut in . Thus . Putting everything together we get