Elimination Distance to Topological-minor-free Graphs is FPT

04/20/2021 ∙ by Akanksha Agrawal, et al. ∙ National University of Singapore Indian Institute of Technology Hyderabad The Regents of the University of California University of Warwick 0

In the literature on parameterized graph problems, there has been an increased effort in recent years aimed at exploring novel notions of graph edit-distance that are more powerful than the size of a modulator to a specific graph class. In this line of research, Bulian and Dawar [Algorithmica, 2016] introduced the notion of elimination distance and showed that deciding whether a given graph has elimination distance at most k to any minor-closed class of graphs is fixed-parameter tractable parameterized by k [Algorithmica, 2017]. There has been a subsequent series of results on the fixed-parameter tractability of elimination distance to various graph classes. However, one class of graph classes to which the computation of elimination distance has remained open is the class of graphs that are characterized by the exclusion of a family F of finite graphs as topological minors. In this paper, we settle this question by showing that the problem of determining elimination distance to such graphs is also fixed-parameter tractable.

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1 Introduction

A popular methodology for studying the parameterized complexity of problems is to consider parameterization by distance from triviality [GuoHN04]. In this methodology, the idea is to try and lift the tractability of special cases of generally hard computational problems, to tractability of instances that are “close” to these special cases (i.e., close to triviality) for appropriate notions of “distance from triviality”. This way of parameterizing graph problems has led to a rich collection of sophisticated algorithmic and lower bound machinery over the last two decades.

One direction in which this approach has been extended in recent years is by enhancing existing notions of distance from triviality by exploiting some form of structure underlying vertex modulators rather than just the size bound. This line of exploration has led to the development of several new notions of distance from triviality [EibenGS15, BulianD16, BulianD17, GanianRS17, GanianOR17, EibenGHK19]. Of primary interest to us in this line of research is the notion of elimination distance introduced in [BulianD16]. Bulian and Dawar [BulianD16] introduced the notion of elimination distance in an effort to define tractable parameterizations that are more general than the modulator size for graph problems. We refer the reader to Section 2 for a formal definition of this parameter. In their work, they focused on the Graph Isomorphism (GI) problem and showed that GI is fixed-parameter tractable (FPT) when parameterized by the elimination distance to graphs of bounded degree. In follow-up work, Bulian and Dawar [BulianD17] showed that deciding whether a given graph has elimination distance at most to any minor-closed class of graphs is fixed-parameter tractable parameterized by (i.e., can be solved in time ) and asked whether computing the elimination distance to graphs of bounded degree is fixed-parameter tractable.

Recently, Lindermayr et al. [LindermayrSV20] showed that computing elimination distance to bounded degree graphs is fixed-parameter tractable when the input is planar. However their approach is specifically adapted to planar graphs (in fact, more generally, -minor free graphs) and they note that their approach does not appear to extend to general graphs. Following this work, Agrawal et al. [AgrawalKP0021] showed that the problem is (non-uniformly) fixed-parameter tractable on general graphs. In fact, they proved a more general result and obtained the result for elimination distance to graphs of bounded degree as a consequence.

In this work, we further this line of research by showing that elimination distance to graphs that exclude a fixed family of graphs as topological minors is also fixed-parameter tractable.

Elimination to  admits an FPT algorithm running in time , where is a constant independent of and , is the maximum over the number of vertices in graphs in , and is a function on and .

2 Preliminaries

For a graph , we use and to denote its vertex set and edge set respectively. For a graph , whenever the context is clear, we use and to denote and , respectively. Consider a graph . For , denotes the graph with vertex set and the edge set . By , we denote the graph . Let . Then, and denote the set of open neighbors and closed neighbors of in . That is, and . For a set , and . In all the notations above, we drop the subscript whenever the context is clear. For a connected component in , by slightly abusing the notation, we use , , and to denote , , and , respectively.

A path in is a subgraph of where is a set of distinct vertices and , where for some . The above defined path is called as path. We say that the graph is connected if for every , there exists a path in . A connected component of is an inclusion-wise maximal connected induced subgraph of .

A rooted tree is a tree with a special vertex designated to be the root. Let be a rooted tree with root . We say that a vertex is a leaf of if has exactly one neighbor in . Moreover, if , then is the leaf (as well as the root) of .111A root is not a leaf in a tree, if the tree has at least two vertices. A vertex which is not a leaf, is called a non-leaf vertex. Let such that and is not contained in the path in , then we say that is the parent of and is a child of . A vertex is a descendant of ( can possibly be the same as ), if there is a path in , where is the parent of in . Note that when , then , as the parent of does not exist. That is, every vertex in is a descendant of .

A rooted forest is a forest where each of its connected component is a rooted tree. For a rooted forest , a vertex that is not a root of any of its rooted trees is a leaf if it is of degree exactly one in . The depth of a rooted tree is the maximum number of vertices in a root to leaf path in . The depth of a rooted forest is the maximum depth among its rooted trees.

[Forest embedding]A forest embedding of a graph is a pair , where is a rooted forest and is an injective function, such that for each , either is a descendant of , or is a descendant of . The depth of the forest embedding is the depth of the rooted forest .

We remark that in the above definition, it is sufficient (and also equivalent) to consider the function as bijective. However, for the sake of notational convenience, which will be clear in the technical sections of the paper, we give the above definition for injective .

Next, we recall the notion of elimination-distance introduced by Bulian and Dawar [BulianD16]. We rephrase their definition (ensuring equivalence), to facilitate its usage for us.222Observe here that the definition of -decomposition used in [BulianD16] and [DBLP:conf/iwpec/Agrawal020, MSRstacs2020] are equivalent to our definition.

[Elimination Distance and -decompositions]Consider a family of graphs and an integer . An -decomposition of a graph is a tuple , where and is a rooted forest of depth at most , such that the following conditions are satisfied:

  1. is a forest embedding of ,

  2. each connected component of belongs to , and

  3. for every connected component of , there exists a leaf in such that for every vertex , and every edge , either or is a vertex in the unique path in from to , where is the root of the connected component in containing the vertex .

In the above, we say that is an -ed modulator to .

The elimination distance of to (or the -elimination distance of ) is the smallest integer for which admits an -decomposition.

Unbreakable Graphs. To formally introduce the notion of unbreakability, we rely on the definition of a separation. Consider a graph . A pair where is a separation if there is no edge , such that and . The order of is . For integers , if admits no separation of order at most such that and , then is -unbreakable.

Topological minors: Roughly speaking, a graph is a topological minor of if contains a subgraph on vertices, where the edges in correspond to (vertex disjoint) paths in . We now formally define the above. Let be the set of all paths in . We say that a graph is a topological minor of if there are injective functions and such that for all , and are the endpoints of , for distinct , the paths and are internally vertex-disjoint, and there does not exist a vertex such that is in the image of and there is an edge where is an internal vertex in the path .

Next we state a result that will be useful in our algorithm.

[[fomin2020hitting]] Let be the maximum over the number of vertices in graphs in . There is a computable function and an algorithm, which given a graph and an integer , in time , either outputs of size at most such that , or correctly concludes that such a set does not exist.333The algorithm of Fomin et al. [fomin2020hitting] only solves the decision version of this problem, but we can use the well-known self-reducibility property of NP-complete problems to obtain such a set.

Counting Monadic Second Order Logic. The syntax of Monadic Second Order Logic (MSO) of graphs includes the logical connectives variables for vertices, edges, sets of vertices and sets of edges, the quantifiers and , which can be applied to these variables, and five binary relations:

  1. , where is a vertex variable and is a vertex set variable;

  2. , where is an edge variable and is an edge set variable;

  3. where is an edge variable, is a vertex variable, and the interpretation is that the edge is incident to ;

  4. where and are vertex variables, and the interpretation is that and are adjacent;

  5. equality of variables representing vertices, edges, vertex sets and edge sets.

Counting Monadic Second Order Logic (CMSO) extends MSO by including atomic sentences testing whether the cardinality of a set is equal to modulo where and are integers such that and . That is, CMSO is MSO with the following atomic sentence: if and only if , where is a set. We refer to [ArnborgLS91, Courcelle90, Courcelle97] for a detailed introduction to CMSO.

We will crucially use the following result of Lokshtanov et al. [LokshtanovR0Z18] that allows one to obtain a (non-uniform) FPT algorithm for CMSO-expressible graph problems by designing an FPT algorithm for the problem on unbreakable graphs.

[Theorem 1, [LokshtanovR0Z18]] Let be a CMSO sentence and let be a positive integer. There exists a function , such that for every there is an , if there exists an algorithm that solves CMSO on -unbreakable graphs in time , then there exists an algorithm that solves CMSO on general graphs in time .

3 Overview of the Algorithm

Consider any (fixed) finite family of connected graphs , and let be the maximum over the number of vertices in graphs in . By , we denote the family of graphs that do not contain any graph in as a topological minor. Recall that in the Elimination to problem, given a graph and an integer , the objective is to determine whether has elimination distance at most to . We will assume (without explicitly stating it) that no graph in is an isolated vertex, as otherwise, , and thus, the problem Elimination to  is exactly the same as testing if the input graph has tree-depth at most . Reidl et al. [DBLP:conf/icalp/ReidlRVS14] designed an algorithm, which given a graph and an integer , in time bounded by , decides whether has treedepth at most . Due to the above result, the interesting cases are those, when no graph in is an isolated vertex.

The first part of our algorithm is similar to the first part of the algorithm of Agrawal et al. [MSRstacs2020], where the problem under consideration is Elimination to , where is the family of graphs with maximum degree at most . In that paper we first show that the problem under consideration is CMSO expressible, and thus from Proposition 2, to obtain an FPT algorithm for the problem, it is enough to design an FPT algorithm for the problem when for the given instance , is an -unbreakable graph, where is the function from the proposition. Throughout the remainder of this paper, we let denote the function given by Proposition 2. Our previous paper further reduces the above task to designing an FPT algorithm for an extension version of the problem.

We will show that our problem is CMSO expressible (Lemma 5, Section 5), and thus from Proposition 2, to obtain an FPT algorithm for the problem, it is enough to obtain an FPT algorithm for the problem when for the input instance , is an -unbreakable graph. We call the problem Elimination to restricted to -unbreakable graphs, as -Unbreakable ED to . We will obtain an FPT algorithm for -Unbreakable ED to  with the help of an extension version of the problem, where (roughly speaking) we know the structure of forest in the desired -decomposition for . To explain the above in more details, we introduce some notations.

A tuple , where , is a rooted forest and is an injective function, is a pseudo solution for if satisfies item 1 and 3 of Definition 2. Note that for any -decomposition (if it exists), is also a pseudo solution for .

In the next (and the only lemma statement in this section), we give a useful property which will be crucial in defining the extension version of the problem for -unbreakable graphs. Intuitively speaking, it states that either has at most vertices, or any pseudo solution for it can contain at most vertices and there will be exactly one large connected component in the remainder of the graph.

Consider a graph , an integer , and any pseudo solution for , where is -unbreakable graph. Then, at least of the following holds:

  1. , or

  2. and there is exactly one connected component in that has at least vertices.

We will prove the above lemma in Section 5. If the given instance of -Unbreakable ED to , has at most vertices, then we can easily test whether it admits a -decomposition in time bounded by . Due to the above, the interesting case occurs when has more than vertices. For the above case, Lemma 3 guarantees that any -ed modulator to for , has at most vertices. This allows us to guess the structure of the forest. We remark here that although Definition 2

We will now intuitively explain the extension version of our problem, that will be called -Unbreakable Extension ED to  (-Ubr ExtED to , for short). An input to the above problem will be a graph , an integer , a tuple (possibly ) and a leaf , where is an -unbreakable graph, is a forest on at most nodes, and is an injective function. The objective will be to see if we can “extend” to a -decomposition for . Moreover, we additionally require that the unique large connected component in has all its neighbors mapped in -to-root path in . We will briefly discuss why we include in the problem definition, shortly. Notice that, due to Lemma 3, to design an FPT algorithm for -Unbreakable ED to , it is enough to design an FPT algorithm for -Ubr ExtED to .

We will now explain the working of our algorithm for -Ubr ExtED to . Roughly speaking, the key idea in our algorithm is: i) either we can find a connected set of bounded size and with bounded neighborhood size that contains a topological minor from , or ii) any arbitrary minimum sized deletion set of size at most , whose removal results in a graph with no topological minor from , can be “patched” with the input partial decomposition to give a desired -decomposition for . To attain the above guarantee, we also need to remember the leaf-to-root path in the forest, where the neighbors of the unique large connected component after removal of a -ed modulator must lie, and this is where the leaf plays a role. The crucial property that we mentioned is algorithmically exploitable. In particular, we can compute the set of all bounded sized connected sets with bounded neighborhood size, in the allowed amount of time. Thus, with the above at hand, we can test (in FPT time), if any of these bounded sized sets contain a topological minor from , and if it does, we can branch. On the other hand if an arbitrary solution of size at most can be patched to obtain the desired decomposition, we can can relegate the computation of such a set to the algorithm of Fomin et al. [fomin2020hitting], for hitting topological minors. As our algorithm is recursive, at every step we want to ensure that the neighbors of connected components in lie in one leaf-to-root path. Furthermore, we also need to ensure that the neighborhood of the unique large connected component in lies in -to-root path in . So we will have two more branching rules which will help us ensure the above two properties. With the above steps, we will be able to guarantee that the running time of our algorithm is bounded by , where is a constant independent of and , is the maximum over the number of vertices in graphs in , and is a function on and .

The rest of the paper is organised as follows. In Section 4, we formally define the problem -Unbreakable Extension ED to , and assuming that Lemma 3 is correct, we design an FPT algorithm for this problem. We present our proof for Lemma 3 in Section 6, which is obtained by combining the properties of -unbreakable graph and existence of balanced separators in an appropriately constructed weighted trees. The focus of Section 5 will be to prove Theorem 1. To this end, we will first establish that Elimination to  is CMSO expressible. After this, we will design an FPT algorithm for -Unbreakable ED to , using the FPT algorithm for -Ubr ExtED to . The above together with Proposition 2 will lead us to a proof of Theorem 1.

4 FPT Algorithm for -Unbreakable Extension ED to

Consider a fixed finite family of connected graphs , and let be the maximum over the number of vertices in graphs in . We begin by defining our extension problem for unbreakable graphs,444The pair of integer , from the unbreakability definition will be clear shortly. -Unbreakable Extension ED to  (-Ubr ExtED to , for short), which will lie at the heart of obtaining FPT algorithm for Elimination to .

(Problem Definition) -Unbreakable Extension ED to  (-Ubr ExtED to )
Input: A graph , an integer , a tuple , and a leaf in , such that is an -unbreakable graph, , , is a rooted forest of depth at most , , and is an injective function.555 may not necessarily be a pseudo solution for .
Question: Either return a -decomposition of ,666Note that the in tuple is exactly the same as the in tuple . such that the following conditions are satisfied:

  1. ,

  2. , and

  3. Let be the unique connected component in of size at least (see Lemma 3). For each (not that must be in ), is an ancestor of in .777We use the convention that for each , is both an ancestor and a descendant of itself in .

Or, correctly report that no satisfying the above conditions exists.

In the above definition, if exists, then we say that it is a solution for the instance .

The objective of this section is to prove the following lemma.

-Ubr ExtED to  admits an algorithm running in time , where is a constant independent of and , is the maximum over the number of vertices in graphs in , and is a computable function of and .

We prove the above lemma by exhibiting an FPT algorithm for -Ubr ExtED to . Let be an instance of -Ubr ExtED to .

We design a branching algorithm for the problem, and we will use as the measure to analyse the running time of our algorithm. We start with some simple cases, when we can directly resolve the given instance. (The time required for the execution of our base cases and branching rules will be provided in the runtime analysis of the algorithm.)

Base Case 1: If there is an edge , where , and and are not in ancestor-descendant relationship in , then return that the instance has no solution.

Base Case 2: If is a solution for the given instance, then return it as a solution. Moreover, if the above statement is not true and , then return that the instance has no solution.

The correctness of the above base cases immediately follows from the definition of the problem -Ubr ExtED to .

Intuitively speaking, our next base case deals with the case when an arbitrary solution of size at most can be mapped to the (unfilled) nodes in a root-to-leaf of , to obtain a solution. We will later prove that if our branching rules (to be described later) and Base Cases 1 and 2 are not applicable, then indeed we can obtain a solution to our instance using this base case. Before formally describing our next base case, we introduce a notation.

Consider some . Roughly speaking we define the set of -Ubr ExtED to -instances, , where each such instance is obtained by adding to and mapping them to nodes in .888This notation will be useful while creating recursive instances and in the description of next base case. Formally, we add each tuple to , where , is an injective function, and for each , and are in ancestor-descendant relationship in .999If such a tuple does not exist, we have .

We use Proposition 2 to find the minimum integer (if it exists), such that for the input , Proposition 2 returns a set of size at most such that . We are now ready to describe our next base case.

Base Case 3: Return a solution if possible, by adding vertices in to and map its vertices to nodes in . Formally, if for any , is a solution for , then return it as a solution.

Note that if the above base case returns a solution, then by the construction of and the definition of -Ubr ExtED to , it follows that it is indeed a correct solution.

Our first two branching rule attempt to fix “neighborhood” inconsistencies, below we begin by explaining what we mean by an inconsistency that our first branching rules attempts to fix. Suppose there are (not necessarily distinct) vertices in the same connected component of , such that and have neighbors and in , respectively, where and are not in ancestor-descendant relationship in . Furthermore, consider any path in (which exists, since and are in the same connected component of ). We remark that if , then is a path on one vertex. For the above, we say that is an inconsistent triplet. We will observe that for any solution for the given -Ubr ExtED to -instance, must hold. In fact, we can establish a stronger statement than the above, which guarantees that at least one among the first and the last many vertices in , belongs to . Intuitively speaking, the above property can be obtained because at least one of or cannot belongs to the unique connected component of size at least in , for any solution (see Lemma 3). The above leads us to the following branching rule.

Branching Rule 1: For an inconsistent triplet , let be the first and the last vertices in . For any and , if the instance of -Ubr ExtED to  has a solution , then return as a solution for the -Ubr ExtED to  instance . Otherwise return that the instance has no solution.

Branching Rule 1 is correct.

Proof.

Consider an inconsistent triplet , and let be the first and the last vertices in . We need to show that is a solution for the instance , if and only if for some and , is a solution for the -Ubr ExtED to  instance . Notice that to prove the above statement it is enough to argue that for any solution to the -Ubr ExtED to  instance , we have . We now focus on the proof of above statement.

Consider a solution for the instance . If , then from item 3 of Definition 2 we can obtain that (note that ). By the definition of inconsistent triplet, is a path in . Thus, if , then we can obtain that . Next we consider the case when . By item 3 of Definition 2, and must be in different connected components in . Moreover, by using Lemma 3 we can obtain that, at most one of or can belong to a connected component in that contains at least vertices. Assume that is in a connected component with less than vertices in (the other case is symmetric). From the above we can obtain that . Let be the first vertex in (starting from ) that does not belong to . As is a path in , must belong to . Moreover, as and , we obtain that is among the first vertices in the path . That is, . ∎

We will hereafter assume that Branching Rule 1 (and the base cases) are not applicable. Thus, for any connected component in , for each (note that ), and must be in ancestor-descendant relationship in . Thus from Lemma 3, there is exactly one connected component, denoted by , in that contains at least vertices.

Our next branching rule will focus on fixing neighborhood inconsistencies arising from , which we intuitively explain below. Consider the case when there is that has a neighbor , such that is not an ancestor of in . For a solution to the given instance, let be the unique connected component in that has at least vertices. Note that . Notice that one of the following must hold, either belongs to , or it belongs to a connected component in . From the above we will be able to establish that any (not necessarily spanning) tree of , rooted at with exactly vertices, it must hold that . In the above, we call the pair a problematic pair. The discussions above lead us to the following branching rule.

Branching Rule 2: Consider the case when there is a problematic pair . For any and , if the instance of -Ubr ExtED to  has a solution , then return as a solution for the -Ubr ExtED to  instance . Otherwise return that the instance has no solution.

We next prove the correctness of our branching rule.

Branching Rule 2 is correct.

Proof.

Consider a problematic pair . We need to show that is a solution for the instance , if and only if for some and , is a solution for the -Ubr ExtED to  instance . Notice that to prove the above statement it is enough to argue that for any solution to the -Ubr ExtED to  instance , we have . We now focus on the proof of above statement.

Consider a solution for the instance . Also, let be the unique connected component in that has at least vertices (see Lemma 3). Towards a contradiction suppose that . From the above we obtain that the connected component in containing has at least vertices. This implies that is connected component of size at least and it has a neighbor in such that is not an ancestor of . This violates the item that need to be satisfied for any solution to the -Ubr ExtED to  instance , a contradiction. ∎

Notice that when Branching Rule 1 is not applicable, for each connected component in , all neighbors of (if any) in must be mapped to nodes of (by ) in one root-to-leaf path. Furthermore, when Branching Rule 2 is not applicable, then all the neighbors of (if any) are mapped to nodes of that are ancestors of .

If none of the previous bases cases or branching rules are applicable, we want to guarantee that either there exists a (connected) set of vertices that contains some as a topological minor, or the instance has no solution. Notice that if we are somehow able to obtain such a , then for any solution , we can obtain that . Thus with such a at hand, we can devise a branching rule. Unfortunately, it is unclear how we can obtain such a , without any additional property. One additional property along with which such a becomes easily identifiable, is when its neighborhood size is bounded. We are actually able to guarantee that if none of the previous rules apply, either the instance has no solution, or a with an additional guarantee of bounded neighborhood size, always exists. The proof of the above, is one of the main (and very technical) steps in our algorithm. To formalise the above, we introduce some notations.

For a graph and integers , a set is a -connected set in , if is connected, and . Below we state a result regarding computation of -connected sets in a graph.

[Lemma 3.1 [DBLP:journals/combinatorica/FominV12], Lemma 2.5 [DBLP:journals/siamdm/AgrawalFLST20]] For a graph and integers , the number of -connected sets in is bounded by . Moreover, all such sets can be enumerated in time bounded by .

We say that a solution for is a minimum solution, if for any solution for the instance, we have .

Towards proving the existence of a connected with the desired properties (or concluding that there is no solution), we will crucially exploit the properties of a minimum solution. In the next two lemma we obtain some useful properties that any minimum solution satisfies, when the previous base cases and branching rules are not applicable. Recall that is the unique component of a size at least in .

Figure 1: Illustration of sets used in proof of Lemma 4

Suppose that Base Cases 1-3 and Branching Rules 1-2 are not applicable for . Furthermore, for each connected component in , where , we have . Then for any minimum solution for , does not contain any vertex from .

Proof.

Towards a contradiction, suppose that . Let . We will argue that is a solution for the instance, thus contradicting that is a minimum solution. By the construction of and our assumption that is a solution for the instance, to prove the desired claim, it is enough to argue that item 1 and 2 of Definition 2 is satisfied for .

Note that and . Refer to Figure 1 for an illustration of the above sets. Thus we can obtain that, a connected component in is: i) either a connected component in that is different from , or ii) it is a connected component in . Thus we can obtain that each connected component in is in . Hence item 2 of Definition 2 holds for . As Branching Rule 1 is not applicable and no vertex in is adjacent to a vertex in , we can obtain that item 3 of the definition is satisfied by . This concludes the proof. ∎

Suppose that Base Cases 1-3 and Branching Rules 1-2 is not applicable for . Furthermore, for each connected component in , where , we have . Then, there is no minimum solution for the instance such that for each , is an ancestor of .

Proof.

Towards a contradiction, suppose that there is a minimum solution for the instance such that for each , is an ancestor of . Let . Notice that . Moreover, we have that . Then we claim that Base Case is applicable, which is a contradiction. Since and , we know that by Proposition 2, we get a vertex subset such that and . Let . Since for each connected component (where ) in , , we have that . Let (note that ). Recall that we have . We construct a function as follows: i) for each , set , and ii) for each , we set . Notice that . We will now argue that is a solution for the given instance, thus contradicting the assumption that Base Case 3 is not applicable.

By the construction of to prove that is a solution, it is enough to show that items 1,2, and 3 in Definition 2 holds for . Since , and we mapped all the vertices in by to the path, is a forest embedding of . Thus, satisfies item . Consider a connected component in . Note that either , or is a connected component in . As each connected component in is in , if , then . By the premise of the lemma, each connected component in that is different from is in . Thus, for the case when is a connected component in that is different from , we have that . Thus, we proved that item 2 of Definition 2 is satisfied for . We will now prove item 3 of the definition for . Consider a connected component in . Note that when is a connected component in , then , and thus item 3 of the definition must hold for it (recall the assumption that Branching Rule 1 is not applicable).

Now consider the case when we have . Notice that . As Branching Rule 2 is not applicable and , for each vertex , () must be an ancestor of . Moreover, by the construction of , for any , is an ancestor of . Thus, we can conclude that item 3 of Definition 2 is satisfied by .∎

Figure 2: Illustration of various sets and connected components used in proof of Lemma 4. The connected component in contain vertices from all the purple connected components and all the (pre-images) of blue and green nodes in . Many edge connections are not shown to simplify the picture. The thick dotted black curves denote connections between small purple connected components to the (pre-images of) vertices in , whereas, the light dotted grey curves denote connections of small purple connected components to (pre-images of) vertices in .

We are now ready to prove that a desired type of exists.

Suppose that Base Cases 1-3 and Branching Rules 1-2 are not applicable for . Then, either there is no solution for the instance, or there is an -connected set in , such that has some as a topological minor.

Proof.

If has no solution, then the claim trivially follows. We now consider the case when has a solution, and let be a minimum solution for it. As is -unbreakable and is a -decomposition of , using Lemma 3 we can obtain that there is exactly one connected component, in that has at least vertices. Recall that is the (unique) connected component in that has at least vertices, thus, .

Firstly consider the case when there is some connected component in , such that and contains some graph in as a topological minor. Note that , as (also see Lemma 3). Note that . From the above discussions we can obtain that is an -connected set in , such that and contains some as a topological minor.

Now we consider the case when each connected component in that is different from , belongs to . From Lemma 4 we can obtain that . The above together with non-applicability of Base Case 2 implies that . Let . From Lemma 4 we can obtain that . Our next objective will be to obtain a subset (to be denoted by ) of and a subset (to be denoted by ) of , so that a connected component in satisfies the requirement of the lemma.

Recall that . Let