A deterministic polynomial kernel for Odd Cycle Transversal and Vertex Multiway Cut in planar graphs

10/02/2018
by   Bart M. P. Jansen, et al.
Utrecht University
TU Eindhoven
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We show that Odd Cycle Transversal and Vertex Multiway Cut admit deterministic polynomial kernels when restricted to planar graphs and parameterized by the solution size. This answers a question of Saurabh. On the way to these results, we provide an efficient sparsification routine in the flavor of the sparsification routine used for the Steiner Tree problem in planar graphs (FOCS 2014). It differs from the previous work because it preserves the existence of low-cost subgraphs that are not necessarily Steiner trees in the original plane graph, but structures that turn into (supergraphs of) Steiner trees after adding all edges between pairs of vertices that lie on a common face. We also show connections between Vertex Multiway Cut and the Vertex Planarization problem, where the existence of a polynomial kernel remains an important open problem.

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1 Introduction

Kernelization provides a rigorous framework within the paradigm of parameterized complexity to analyze preprocessing routines for various combinatorial problems. A kernel of size for a parameterized problem and a computable function is a polynomial-time algorithm that reduces an input instance with parameter of problem to an equivalent one with size and parameter value bounded by . Of particular importance are polynomial kernels, where the function is required to be a polynomial, that are interpreted as theoretical tractability of preprocessing for the considered problem . Since a kernel (of any size) for a decidable problem implies fixed-parameter tractability (FPT) of the problem at hand, the question whether a polynomial kernel exists became a “standard” tractability question one asks about a problem already known to be FPT, and serves as a further finer-grained distinction criterion between FPT problems.

In the recent years, a number of kernelization techniques emerged, including the bidimensionality framework for sparse graph classes [12] and the use of representative sets for graph separation problems [22]. On the hardness side, a lower bound framework against polynomial kernels has been developed and successfully applied to a multitude of problems [1, 5, 7, 13]. For more on kernelization, we refer to the survey [24] for background and to the appropriate chapters of the textbook [3] for basic definitions and examples.

For this work, of particular importance are polynomial kernels for graph separation problems. The framework for such kernels developed by Kratsch and Wahlström in [22, 23], relies on the notion of representative sets in linear matroids, especially in gammoids. Among other results, the framework provided a polynomial kernel for Odd Cycle Transversal and for Multiway Cut

with a constant number of terminals. However, all kernels for graph separation problems based on representative sets are randomized, due to the randomized nature of all known polynomial-time algorithms that obtain a linear representation of a gammoid. As a corollary, all such kernels have exponentially small probability of turning an input yes-instance into a no-instance.

The question of deterministic polynomial kernels for the cut problems that have randomized kernels due to the representative sets framework remains widely open. Saket Saurabh, at the open problem session during the Recent Advances in Parameterized Complexity school (Dec 2017, Tel Aviv) [29], asked whether a deterministic polynomial kernel for Odd Cycle Transversal exists when the input graph is planar. In this paper, we answer this question affirmatively, and prove an analogous result for the Multiway Cut problem.

Theorem 1.1.

Odd Cycle Transversal and Vertex Multiway Cut, when restricted to planar graphs and parameterized by the solution size, admit deterministic polynomial kernels.

Recall that the Odd Cycle Transversal problem, given a graph and an integer , asks for a set of size at most such that is bipartite. For the Multiway Cut problem, we consider the Vertex Multiway Cut variant where, given a graph , a set of terminals , and an integer , we ask for a set of size at most such that every connected component of contains at most one terminal. In other words, we focus on the vertex-deletion variant of Multiway Cut with undeletable terminals. In both cases, the allowed deletion budget, , is our parameter. (A deterministic polynomial kernel for Edge Multiway Cut in planar graphs is known [27, Theorem 1.4].)

Note that in general graphs, Vertex Multiway Cut admits a randomized polynomial kernel with terminals [22], and whether one can remove the dependency on from the exponent is a major open question in the area. Theorem 1.1 answers this question affirmatively in the special case of planar graphs.

Our motivation stems not only from the aforementioned question of Saurabh [29], but also from a second, more challenging question of a polynomial kernel for the Vertex Planarization problem. Here, given a graph and an integer , one asks for a set of size at most such that is planar. For this problem, an involved -time fixed-parameter algorithm is known [20], culminating a longer line of research [20, 21, 26]. The question of a polynomial kernel for the problem has not only been posed by Saurabh during the same open problem session [29], but also comes out naturally in another line of research concerning vertex-deletion problems to minor-closed graph classes.

Consider a minor-closed graph class . By the celebrated Robertson-Seymour theorem, the list of minimal forbidden minors of is finite, i.e., there is a finite set of graphs such that a graph belongs to if and only if does not contain any graph from as a minor. The -Deletion problem, given a graph and an integer , asks to find a set of size at most such that has no minor belonging to , i.e., . If contains a planar graph or, equivalently, has bounded treewidth, then the parameterized and kernelization complexity of the -Deletion problem is well understood [11]. However, our knowledge is very partial in the other case, when contains all planar graphs. The understanding of this general problem has been laid out as one of the future research directions in a monograph of Downey and Fellows [6]. The simplest not fully understood case is when is exactly the set of planar graphs, that is, , and the -Deletion becomes the Vertex Planarization problem. The question of a polynomial kernel or a -time FPT algorithm for Vertex Planarization remains open [15, 29].

In Section 6, we observe that there is a simple polynomial-time reduction from Planar Vertex Multiway Cut to Vertex Planarization that keeps the parameter unchanged. If Vertex Planarization would admit a polynomial kernel, then our reduction would transfer the polynomial kernel back to Planar Vertex Multiway Cut. In the presence of Theorem 1.1, such an implication is trivial, but the reduction itself serves as a motivation: a polynomial kernel for Planar Vertex Multiway Cut should be easier than for Vertex Planarization, and one should begin with the first before proceeding to the latter. Furthermore, we believe the techniques developed in this work can be of use for the more general Vertex Planarization case.

Figure 1: When all terminals (blue squares) lie on the infinite face, a solution to Vertex Multiway Cut (black circles) becomes a Steiner forest (red dashed connections) in the overlay graph.

Techniques

On the technical side, our starting point is the toolbox of [27] that provides a polynomial kernel for Steiner Tree in planar graphs, parameterized by the number of edges of the solution. The main technical result of [27] is a sparsification routine that, given a connected plane graph with infinite face surrounded by a simple cycle , provides a subgraph of of size polynomial in the length of that, for every , preserves an optimal Steiner tree connecting .

Both Odd Cycle Transversal and Vertex Multiway Cut in a plane graph translate into Steiner forest-like questions in the overlay graph of : a supergraph of that has a vertex for every face of , adjacent to every vertex of incident with . To see this, consider a special case of Planar Vertex Multiway Cut where all terminals lie on the infinite face of the input embedded graph. Then, an optimal solution is a Steiner forest between some tuples of vertices on the outer face lying between the terminals, cf. Figure 1. Following [27], this suggest the following approach to kernelization of vertex-deletion cut problems in planar graphs:

  1. By problem-specific reductions, reduce to the case of a graph of bounded radial diameter.

  2. Using the diameter assumption, find a tree in the overlay graph that has size bounded polynomially in the solution size, and that spans all “important” objects in the graph (e.g., neighbors of the terminals in the case of Multiway Cut or odd faces in the case of Odd Cycle Transversal).

  3. Cut the graph open along the tree. Using the Steiner forest-like structure of the problem at hand, argue that an optimal solution becomes an optimal Steiner forest for some choice of tuples of terminals on the outer face of the cut-open graph.

  4. Sparsify the cut-open graph with a generic sparsification routine that preserves optimal Steiner forests, glue the resulting graph back, and return it as a kernel.

However, contrary to the Steiner tree problem [27], these Steiner forest-like questions optimize a different cost function than merely number of edges, namely the number of vertices of , with the “face” vertices being for free. This cost function is closely related to (half of) the number of edges in case of paths and trees with constant number of leaves, but may diverge significantly in case of trees with high-degree vertices.

For this reason, we need an analog of the main technical sparsification routine of [27] suited for our cost function. We provide one in Section 3. To this end, we re-use most of the intermediate results of [27], changing significantly only the final divide&conquer argument.

The application of the obtained sparsification routine to the case of Odd Cycle Transversal, presented in Section 4, follows the phrasing of the problem as a -join-like problem in the overlay graph due to Fiorini et al [10]. For the sake of reducing the number of odd faces, we adapt the arguments of Suchý [30] for Steiner tree.

The arguments for Vertex Multiway Cut are somewhat more involved and presented in Section 5. Here, we first use known LP-based rules [4, 14, 16, 28] to reduce the number of terminals and neighbors of terminals to and then use an argument based on outerplanarity layers to reduce the diameter.

2 Preliminaries

A finite undirected graph  consists of a vertex set  and edge set . We denote the open neighborhood of a vertex  in  by . For a vertex set  we define its open neighborhood as .

For vertex subsets  of a graph , we define an -cut as a vertex set  such that no connected component of  contains both a vertex of  and a vertex of . An  cut  is minimal if no proper subset of  is an -cut, and minimum if it has minimum possible size.

2.1 Planar graphs

In a connected embedded planar (i.e. plane) graph , the boundary walk of a face is the unique closed walk in obtained by going along the face in counter-clockwise direction. Note that a single vertex can appear multiple times on the boundary walk of and an edge can appear twice if it is a bridge. We denote the number of edges of this walk by ; note that bridges are counted twice in this definition. The parity of a face is the parity of . Then a face is odd (even) if its parity is odd (even). The boundary walk of the outer face of is called the outer face walk and denoted .

We define the radial distance in plane graphs, based on a measure that allows to hop between vertices incident on a common face in a single step. Formally speaking, a radial path between vertices  and  in a plane graph  is a sequence of vertices  such that for each , the vertices  and  are incident on a common face. The length of the radial path equals , so that a trivial radial path from  to itself has length . The radial distance in plane graph  between  and , denoted , is defined as the minimum length of a radial -path.

For a plane graph , let  denote the set of faces of . For a plane (multi)graph , an overlay graph  of  is a graph with vertex set  obtained from  as follows. For each face , draw a vertex with identity  in the interior of . For each connected component  of edges incident on the face , traverse the boundary walk of  starting at an arbitrary vertex. Every time a vertex  is visited by the boundary walk, draw a new edge between  and the vertex representing , without crossing previously drawn edges. Doing this independently for all faces of  yields an overlay graph . Observe that an overlay graph may have multiple edges between some  and , which occurs for example when  is incident on a bridge that lies on . The resulting plane multigraph  is in general not unique, due to different homotopies for how edge bundles may be routed around different connected components inside a face. For our purposes, these distinctions are never important. We therefore write  to denote an arbitrary fixed overlay graph of . Observe that  forms an independent set in .

Apart from the overlay graph, we will also use the related notion of radial graph (also known as face-vertex incidence graph). A radial graph of a connected plane graph  is a plane multigraph  obtained from  by removing all edges with both endpoints in . Hence a radial graph of  is bipartite with vertex set , where vertices are connected to the representations of their incident faces. From these definitions it follows that  is the union of  and , which explains the terminology.

We need also the following simple but useful lemma.

Lemma 2.1.

Let be a connected graph, let and assume that for each vertex , there is a terminal that can reach by a path of at most edges. Then contains a Steiner tree of at most  edges on terminal set , which can be computed in linear time.

Proof.

Observe that there exists a spanning forest in  where each tree is rooted at a vertex of , and each tree has depth at most . Such a spanning forest can be computed in linear time by a breadth-first search in , initializing the BFS-queue to contain all vertices of  with a distance label of . Consider the graph  obtained from  by contracting each tree into the terminal forming its root. Since  is connected, is connected as well. An edge  between two terminals in  implies that in  there is a vertex in the tree of  adjacent to a vertex of the tree of . So for each edge in , there is a path between the corresponding terminals in  consisting of at most  edges.

Compute an arbitrary spanning tree of the graph , which has  edges since  has  vertices. As each edge of the tree expands to a path in  between the corresponding terminals of length at most , it follows that  has a connected subgraph  of at most  edges that spans all terminals . To eliminate potential cycles in , take a spanning subtree of  as the desired Steiner tree. ∎

Lemma 2.2 ([19, Lemma 1]).

Let  be a planar bipartite graph with bipartition  for . If all distinct  satisfy , then .

3 Sparsification

3.1 Overview

A plane partitioned graph is an undirected multigraph , together with a fixed embedding in the plane and a fixed partition where is an independent set. Consider a subgraph of a plane partitioned graph . The cost of is defined as , that is, we pay for each vertex of in the part . We say that connects a subset if and is contained in a single connected component of .

Our main sparsification routine is the following.

Theorem 3.1.

Given a connected plane partitioned graph , one can in time find a subgraph in , with the following properties:

  1. contains all edges and vertices of ,

  2. contains edges,

  3. for every set there exists a subgraph of that connects and has minimum possible cost among all subgraphs of that connect .

In the subsequent sections, given a connected plane graph , we will apply Theorem 3.1 to a graph that is either the overlay graph of without the vertex corresponding to the outer face, or the radial graph of . In either case, is the set of face vertices and , i.e., we pay for each “real” vertex, not a face one. If the studied vertex-deletion graph separation problem in turns into some Steiner problem in , then we may hope to apply the sparsification routine of Theorem 3.1.

After this brief explanation of the motivation of the statement of Theorem 3.1, we proceed with an overview of its proof. We closely follow the divide&conquer approach of the polynomial kernel for Steiner Tree in planar graphs [27].

We adopt the notation of (strictly) enclosing from [27]. For a closed curve on a plane, a point  is strictly enclosed by  if  is not continuously retractable to a single point in the plane punctured at . A point is enclosed by if it is strictly enclosed or lies on . The notion of (strict) enclosure naturally extends to vertices, edges, and faces of a plane graph being (strictly) enclosed by ; here a face (an edge) is strictly enclosed by if every interior point of a face (every point on an edge except for the endpoints, respectively) is strictly enclosed. We also extend this notion to (strict) enclosure by a closed walk in a plane graph in a natural manner. Note that this corresponds to the natural notion of (strict) enclosure if is a cycle or, more generally, a closed walk without self-intersections.

We start with restricting the setting to being bipartite and being a simple cycle. Theorem 3.1 follows from Lemma 3.2 by simple manipulations.

Lemma 3.2.

The statement of Theorem 3.1 is true in the restricted setting with being a connected bipartite simple graph with being a simple cycle and being one of the bipartite color classes (so that is an independent set as well).

We now sketch the proof of Lemma 3.2.

First observe that the statement of Lemma 3.2 is well suited for a recursive divide&conquer algorithm. As long as is large enough, we can identify a subgraph  of  such that:

  1. The number of edges of is ;

  2. For every set there exists a subgraph of that connects , has minimum possible cost among all subgraphs of that connect , and for every finite face of , if is the subgraph of consisting of the edges and vertices embedded within the closure of , then one of the following holds:

    1. for some universal constant ;

    2. does not contain any vertex of degree more than that is strictly inside .

Similarly as in the case of [27], we show that such a subgraph is good for recursion. First, we insert into the constructed sparsifier . Second, we recurse on for every finite face of that satisfies Point 2a. Third, for every other finite face (i.e., one satisfying Point 2b), we insert into a naive shortest-paths sparsifier: for every two vertices , we insert into a minimum-cost path between and in . Property 1 together with the multiplicative progress on in Point 2a ensure that the final size of is polynomial in , with the exponent of the polynomial bound depending on and the constant hidden in the big- notation in Property 1.

The main steps of constructing are the same as in [27]. First, we try minimum-size (i.e., with minimum number of edges, as opposed to minimum-cost) Steiner trees for a constant number of terminals on . If no such trees are found, the main technical result of [27] shows that one can identify a cycle in of length with the guarantee that for any choice of , there exists a minimum-size Steiner tree connecting that does not contain any Steiner point strictly inside . In [27] such a cycle is used to construct a desired subgraph with the inside of being a face satisfying the Steiner tree analog of Point 2b. In the case of Lemma 3.2, we need to perform some extra work here to show that — by some shortcutting tricks and adding some slack to the constants — one can construct such a cycle with the guarantee that the face inside satisfies exactly the statement of Point 2b: that is, no “Steiner points” with regards to minimum-cost trees, not minimum-size ones.

In other words, the extra work is needed to at some point switch from “minimum-size” subgraphs (treated by [27]) to “minimum-cost” ones (being the main focus of Lemma 3.2). In our proof, we do it as late as possible, trying to re-use as much of the technical details of [27] as possible. Observe that for a path in , the cost of equals up to an additive error. Similarly, for a tree with a constant number of leaves, the cost of is up to an additive error bounded by a constant. Hence, as long as we focus on paths and trees with bounded number of leaves, the “size” and “cost” measures are roughly equivalent. However, if a tree in contains a high-degree vertex , the cost of may be much smaller than half of the number of edges of : a star with a center in has cost one and arbitrary number of edges. For this reason, the final argument of the proof of Lemma 3.2 that constructs the aforementioned cycle using the toolbox of [27] needs to be performed with extra care (and some sacrifice on the constants, as compared to [27]).

3.2 From Lemma 3.2 to Theorem 3.1

We start with the formal proof of Theorem 3.1 from Lemma 3.2.

Proof of Theorem 3.1..

Let be a connected plane partitioned graph . First, for every edge with , we subdivide it with a new vertex . In this manner, becomes bipartite with and being its bipartition classes, the minimum possible cost of connecting subgraphs does not change (since the cost does not count vertices in ), and the length of at most doubled (so the bound remains the same).

Second, if is not a simple cycle, then the graph induced by is a collection of simple cycles of length at least three, a collection of cycles of length two, and a set of bridges. For every , let be the subgraph of enclosed by . Note that .

For every , we turn into a simple graph by dropping multiple edges, but never dropping an edge of . That is, for every multiple edge of , we delete all but one of its multiple copies, but we always keep the one contained in if it exists. Since is a simple cycle of length at least three, no two edges of connect the same pair of vertices. Consequently, and satisfies the conditions of Lemma 3.2. We apply the algorithm of Lemma 3.2, obtaining a graph of size .

For , observe that trivially satisfies the conditions of Theorem 3.1 for : the only nontrivial subset  consists of the two vertices on , for which an edge of  forms a minimum-cost connecting subgraph. Consequently,

satisfies the required properties: the size bound follows from the fact that while the covering property follows easily from the fact that every minimum-cost connecting subgraph splits into minimum-cost connecting subgraphs in each . ∎

We continue with a formal proof of Lemma 3.2. In Section 3.3 we recall the main technical results of [27] we re-use here. In Section 3.4 we show how to find the aforementioned cycle . Finally, we wrap up the argument in Section 3.5.

In this section we implicitly identify every graph with its set of edges. In particular, is a shorthand for while means that is a subgraph of . For a tree and two vertices , by we denote the unique path from to in . For a cycle embedded on a plane and two vertices , by we denote the counter-clockwise path along from to (which is a trivial path if ).

3.3 Toolbox from the Steiner Tree kernel

We start with briefly recalling the content of Section 4 of [27]. A brick is a connected plane graph whose outer face is surrounded by a simple cycle . A subbrick of a plane graph is a subgraph that is a brick and consists of all edges of enclosed by . A brick covering of a plane graph is a collection of subbricks of such that every finite face of is a finite face of some brick in as well. A brick covering is a brick partition if every finite face of is a face of exactly one brick in . The total perimeter of a brick covering is , and a brick covering is -short if . Furthermore, for a constant , a brick covering is -nice if for every we have .

A connected subgraph of a plane graph is called a connector, and the vertices of that are incident to at least one edge of are anchors of . A connector is brickable if the boundary of every finite face of is a simple cycle, that is, these boundaries form subbricks of . Thus, a brickable connector induces a brick partition of . Note that if is a tree with every leaf lying on , then is a brickable connector in . Another important observation is that , so if , then is -short. For brevity, we say that a brickable connector is -short or -nice if is -short or -nice, respectively.

A technical modification of the algorithm of Erickson et al. [9] gives the following.111Technically speaking, Theorem 4.4 of [27] is stated with only , but a quick inspection of its proof shows that its two ingredients, Lemmata 9.4 and 9.6, are already stated for arbitrary .

Theorem 3.3 (Theorem 4.4 of [27]).

Let be fixed constants. Given a brick , in time one can either correctly conclude that no -short -nice tree exist, or find a -short -nice brick covering of .

A direct adaptation of the proof of Lemma 4.5 of [27] gives the main recursive step of the algorithm of Lemma 3.2.

Lemma 3.4 (essentially Lemma 4.5 of [27]).

Let be constants. Let be a brick and let be a -short -nice brick covering of . Assume that the algorithm of Theorem 3.1 was applied recursively to bricks in , yielding a graph for every . Furthermore, assume that for every we have for some constants and such that . Then satisfies the conditions of Theorem 3.1 for and .

Proof.

The condition that and the size bound follows exactly the same as in the proof of Lemma 4.5 of [27]. For the last condition of Theorem 3.1, the proof is essentially the same as in [27], but we repeat it for completeness. Consider a set and a subgraph that connects it in with minimum possible cost. Without loss of generality, assume that, among subgraphs of connecting of minimum possible cost, contains minimum possible number of edges that are not in . We claim that there are no such edges; by contrary, let be such an edge. Since is a brick covering, let be a brick containing . Let be the connected component of that contains , where is the subgraph . Clearly, is a connector in , and let be its anchors. By the properties of , there exists a subgraph that connects and is of cost not larger than the cost of . Consequently, connects , has cost not larger than the cost of , and has strictly less edges outside than . This is the desired contradiction. ∎

We now move to carves and mountains (Sections 5 and 6 of [27]). For a constant , a -carve in a brick is a pair such that is a path in of length at most with both endpoints on , and is a path on between the endpoints of with length at most . The interior of a -carve is the subgraph of enclosed by the closed walk . The main result of Section 5 of [27] is the following.

Theorem 3.5 (Theorem 5.7 of [27]).

For any and , if a brick has no -short -nice tree, then there exists a finite face of that is never in the interior of a -carve in . Furthermore, such a face can be found in time.

Section 6 of [27] treats mountains, a special case of -carves. The definitions of [27] are general to accommodate the edge-weighted setting as well; here we are content with only the unweighted setting. Let be a brick and let . A -carve with endpoints and of (so that is the counter-clockwise traverse along from to ) is a -mountain with summit if for and we have that is a shortest path in the subgraph of enclosed by and is a shortest path in the subgraph enclosed by . The main structural result of Section 6 of [27] is the following.

Theorem 3.6 (Theorem 6.3 of [27]).

Let and . Let be a brick that does not admit a -short -nice tree and let such that the counter-clockwise walk along from to has length strictly less than . Then, there exists a closed walk in of length at most that contains and such that, for each finite face of , is enclosed by if and only if is enclosed by some -mountain with endpoints and . Furthermore, the set of faces enclosed by can be found in time.

3.4 Finding the middle cycle

Armed with the toolbox from the previous section, we now re-engineer the argument of Section 7 of [27] to our setting. This is the place where the arguments of this work and [27] mostly diverge, as here we build the interface between the cost and size (number of edges) measures.

Let be a brick. Consider a set and a subgraph of connecting of minimum possible cost. Without loss of generality, we can assume that is a tree and, furthermore, all its leaves are in . Furthermore, assume that contains a vertex that is not a leaf. Then can be partitioned into two trees and , , . Observe that if is a tree that is a subgraph of connecting of minimum possible cost, then is a subgraph of connecting of minimum possible cost for every . Furthermore, if for , is a subgraph of connecting of minimum possible cost, then is a subgraph of connecting of minimum possible cost. Since , we infer that and is also a minimum-cost subgraph of connecting .

The paragraph above motivates the following definitions. A tree such that is exactly the set of leaves of is called boundary-anchored. A boundary-anchored tree that is a minimum-cost subgraph of connecting is called a minimum-cost tree (connecting ).

From the discussion above, the following is immediate.

Lemma 3.7.

Let be a bipartite brick with bipartition classes and . Let and let be a minimum-cost subgraph of that connects that, among all minimum-cost subgraphs of connecting , has minimum possible number of edges. Then, is a tree with all its leaves in , and can be decomposed as , where and for every , is a minimum-cost tree connecting . Furthermore, for every , has minimum possible number of edges among all minimum-cost subgraphs of connecting .

We also have the following bound.

Lemma 3.8.

Let be a bipartite brick with bipartition classes and . Let and let be a minimum-cost tree connecting . Then the cost of is at most and . In particular, is a -short brickable connector.

Proof.

For the first claim, it suffices to observe that is a subgraph connecting any of cost . For the second claim, root in an arbitrary vertex, and compute as follows. First, there are at most edges of that are incident to a leaf of belonging to . For every edge that is not as above, if connects with its parent , then charge to if and otherwise charge to any of the children of (which exist and are in ). In this manner, every edge of that is not incident with a leaf of in is charged to some vertex in and every vertex is charged at most twice: once by the edge from to its parent, and one possibly from the parent of to the grandparent of . Since the number of vertices of is the cost of , which is at most , the bound on follows. Finally, following the observations preceding Theorem 3.3, is a brickable connector that is -short. ∎

We are now ready to find the cycle whose existence was promised in Section 3.

Theorem 3.9 (analog of Theorem 7.1 of [27]).

Let be a fixed constant. Let be a brick that does not admit a -short -nice tree, is bipartite with bipartition classes and , and has perimeter at least . Then one can in time compute a simple cycle in with the following properties:

  1. the length of is at most ;

  2. for each vertex , the distance from to in is at most and, furthermore, there exists a shortest path from to that does not contain any edge strictly enclosed by ;

  3. encloses , where is any arbitrarily chosen face of promised by Theorem 3.5 that is not enclosed by any -carve;

  4. for any , there exists a minimum-cost subgraph of connecting such that no vertex of degree at least is strictly enclosed by .

Proof.

For two vertices , by we denote the path that is the counter-clockwise traverse of from to .

We start by computing a set of pegs in the following greedy manner. We start with arbitrary and traverse starting from twice, once clockwise and once counter-clockwise. In each pass, we take as a next peg the first vertex in that is at distance (along ) larger than from the previously placed peg. With two passes, we have , which is a constant. Furthermore, we have that for every there exist pegs with and

In the above, the first inequality stems from the way we place pegs (including the requirement than pegs are in ) and the second inequality is implied by . Note that is the first peg in clockwise direction from and is the first peg in the counter-clockwise direction from .

We set . For any , , , we apply Theorem 3.6 to , , and , obtaining a set of faces . We define . Clearly, . We define to be the connected component of that contains , where is the dual of without the infinite face. Furthermore, let be the closed walk in around .

Consider a face . By definition, there exists with and by Theorem 3.6 there exists a -mountain enclosing . Since is a simple path, there exists a path in from to a face incident to an edge of , with all faces on enclosed by . This implies that is a simple cycle.

Furthermore, since every edge on is an edge of or some edge of the walk obtained from Theorem 3.6, we have that

The above estimation is also the sole need for introducing the pegs. If one picks

to be the union of for every with then the above estimate will be cubic, not linear in . The linear dependency on is essential for the final polynomial bound on the size of the kernel.

So far, all computations can be done in time due to being of constant size and the time bounds of Theorems 3.5 and 3.6. We now compute the subgraph induced by all vertices of that are within distance at most from . This subgraph can be easily computed by breadth-first search in linear time.

Since for the sake of defining , we used -mountains for , all vertices of any such -mountain are contained in and, consequently, is contained in . Let be the subgraph of enclosed by ; note that is a brick. Furthermore, let be the face of that contains . We define to be some shortest cycle in separating from the infinite face of . Since corresponds to a minimum cut in a dual of , can be computed in linear time [8]. We claim that satisfies the desired conditions.

Clearly, the length of is at most the length of (as is a good candidate for ), and hence satisfies the desired length bound. Also, encloses by definition. For the second property, the fact that is a cycle in ensures that every is within distance from . Let be a shortest path from to that minimizes the number of edges strictly enclosed by . We claim that there are no such edges; by contradiction, assume that there exists a subpath of with endpoints and all edges and internal vertices strictly enclosed by . If or , then we reach a contradiction with the choice of by replacing with or respectively on . Otherwise, if , then, as and thus , either or is a strictly better candidate for , a contradiction.

We are left with the last desired property. Consider a set and a subgraph of connecting of minimum possible cost. Furthermore, we choose that satisfies the following minimality property: has minimum number of edges among all minimum-cost subgraphs connecting and, subject to that, has minimum number of edges strictly enclosed by . We claim that this choice of satisfies the desired properties.

Lemma 3.7 implies that it suffices to consider the case when is actually a minimum-cost tree.

Since every tree in with all leaves on is a brickable connector, Lemma 3.8 implies that every minimum-cost tree in is -short. Since does not admit a -short -nice tree, every minimum-cost tree in is not -nice.

Consequently, we infer that is not -nice, that is, there exists a brick with . Since is a minimum-cost tree, there exist such that .

Observe that connects ; hence

Consequently, as , we have that

(1)

Let be the union of and the set of vertices of that are of degree at least in . Let be the vertices on with

such that and are as long as possible. Note that it may happen that , but lie on in this order and , .

Let be the vertex of on that is closest to and similarly define on . Let be the subtree of being the connected component of containing , rooted at , where is the edge of incident with but not lying on . Traverse in clockwise direction from , and let be the last vertex of encountered (before returning back to ); note that it may happen that if .

Note that from (1) we infer that and, consequently,

(2)

where the last inequality follows from the fact that .

Claim 3.10.

Either or and .

Proof.

Assume otherwise. Following (2), is a -carve. We have , as otherwise would be a -short -nice tree in . By (2), this implies , so . Let be a closed walk and let be constructed from by first deleting all edges enclosed by , and then adding instead. Clearly, connects as well. Let be the set of vertices of that are not enclosed by . Since is enclosed by , and , we have that

Here, we again used the fact that . The above inequality contradicts the choice of . ∎

In the second case of Claim 3.10 (i.e., and ), we have that is a -carve using (2). Similarly as before, this implies that , as otherwise is a -short -nice tree. We now use the pegs and . Let and . Note that the placement of pegs ensure that . Consequently, , and is a -carve. Note that the path contains no vertex of (as is a minimum-cost tree) and thus is a simple path.

Figure 2: Situation in the proof of Claim 3.11.
Claim 3.11.

is a -mountain with as a summit.

Proof.

Assume the contrary; see Figure 2 for an illustration. Let and . Assume that there exists a path from to , enclosed by , that is strictly shorter than ; the proof for the symmetric case of a short path from to is analogous and thus omitted. Since and overlap on , this implies that there exists a subpath of from to that is strictly shorter than and is enclosed by . Let be the other endpoint of . Construct a graph from as follows: replace in all edges enclosed by by