DeepAI

# Quasi-polynomial time approximation schemes for the Maximum Weight Independent Set Problem in H-free graphs

In the Maximum Independent Set problem we are asked to find a set of pairwise nonadjacent vertices in a given graph with the maximum possible cardinality. In general graphs, this classical problem is known to be NP-hard and hard to approximate within a factor of n^1-ε for any ε > 0. Due to this, investigating the complexity of Maximum Independent Set in various graph classes in hope of finding better tractability results is an active research direction. In H-free graphs, that is, graphs not containing a fixed graph H as an induced subgraph, the problem is known to remain NP-hard and APX-hard whenever H contains a cycle, a vertex of degree at least four, or two vertices of degree at least three in one connected component. For the remaining cases, where every component of H is a path or a subdivided claw, the complexity of Maximum Independent Set remains widely open, with only a handful of polynomial-time solvability results for small graphs H such as P_5, P_6, the claw, or the fork. We prove that for every such "possibly tractable" graph H there exists an algorithm that, given an H-free graph G and an accuracy parameter ε > 0, finds an independent set in G of cardinality within a factor of (1-ε) of the optimum in time exponential in a polynomial of log |V(G)| and ε^-1. That is, we show that for every graph H for which Maximum Independent Set is not known to be APX-hard in H-free graphs, the problem admits a quasi-polynomial time approximation scheme in this graph class. Our algorithm works also in the more general weighted setting, where the input graph is supplied with a weight function on vertices and we are maximizing the total weight of an independent set.

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## 1 Introduction

For an undirected graph , a vertex subset is independent if no two vertices of are adjacent. The size of the largest independent set in a graph, often denoted as , is one of the fundamental graph parameters studied in graph theory. Therefore, it is natural to study the computational task of computing , given , which we call the Maximum Independent Set problem (MIS). In the weighted generalization, Maximum Weight Independent Set (MWIS), the given graph is supplied with a weight function , and we ask for an independent set in with the maximum possible total weight . MIS is a classic problem that is known not only to be NP-hard, but also hard to approximate within a factor of for every , unless  [14, 20].

In light of these lower bounds, a lot of effort has been put into understanding the complexity of MIS and MWIS in restricted graph classes. While the celebrated Baker’s technique yields a polynomial-time approximation scheme (PTAS) for MWIS in planar graphs [2], MIS remains NP-hard in planar graphs of degree at most three and APX-hard in graphs of maximum degree at most three [9, 8, 12]. To extend these lower bounds to other graph classes, the following observation due to Poljak [18] is very useful: if is created from by subdividing one edge twice, then . Thus, if we fix any graph that contains either a cycle, a vertex of degree at least four, or two vertices of degree three in one connected component, then starting from a graph of maximum degree at most three (where MIS is known to be APX-hard and is linear in the size of the graph) and subdividing each edge a sufficient number of times, we obtain a graph where computing is equally hard, while does not contain an induced subgraph isomorphic to . This implies that MIS remains APX-hard in -free graphs for every finite family of graphs such that every is not a disjoint union of paths and subdivided claws.111A graph is -free if it does not contain an induced subgraph isomorphic to . A graph is -free if is -free for every . A subdivided claw is a tree with one vertex of degree three and all other vertices of degree at most two.

However, when is a disjoint union of paths and subdivided claws, no hardness result on the complexity of MIS nor MWIS on -free graphs is known. In fact, it would be consistent with our knowledge if MWIS turns out to be polynomial-time solvable in -free graphs for all such graphs . Currently we seem very far from claiming such a result. Let be the path on vertices and the claw be the four-vertex tree with one vertex of degree three and three leaves. The class of -free graphs (known also as cographs) have a very rigid structure (in particular, they have clique-width at most ), and hence they admit a simple polynomial-time algorithm for MWIS [7]. Claw-free graphs also possess very strong structural properties and inherit many properties of their main subclass: line graphs. In particular, the augmenting-path algorithm for maximum cardinality matching generalizes to a polynomial-time algorithm for MWIS in claw-free graphs [17, 19]. This, in turn, can be generalized to so-called fork-free graphs [16], where the fork is constructed from the claw by subdividing one edge once. The case of -free graphs, after being open for a long time, was resolved positively in 2014 by Lokshtanov, Vatshelle, and Villanger [15] using the framework of potential maximal cliques. With a substantially larger technical effort, their approach has been generalized to -free graphs by Grzesik et al. [11]. The polynomial-time solvability of MWIS on -free graphs, or -free graphs where is any subdivision of the claw other than the fork, remains open.

Recently, evidence in favor of the tractability of MIS and MWIS at least in -free graphs has been found: there is a subexponential-time algorithm for the problem running in time on an -vertex -free graph [1, 3, 10]. The main insight is that the classical Gyárfás’ path argument, originally used to show that -free graphs are -bounded [13], implies that a -free graph admits a balanced separator consisting of at most vertex neighborhoods. Here, a balanced separator is a set of vertices whose removal results in a graph where every connected component has at most vertices.

#### Our results.

We provide a new evidence in favor of the tractability of MWIS in all cases of -free graphs where it is not known to be APX-hard.

###### Theorem 1.1.

For every graph whose every connected component is a path or a subdivided claw, there exists an algorithm that, given an -free graph with a weight function and an accuracy parameter , computes a -approximation to Maximum Weight Independent Set on in time exponential in a polynomial of and .

That is, in all the cases when MWIS is not known to be APX-hard on -free graphs, we prove that MWIS admits a quasi-polynomial time approximation scheme (QPTAS).

For an insight into the techniques standing behind Theorem 1.1, let us first focus on the case . Let be an input to MWIS with being -free and let be an accuracy parameter. Let be an independent set in of maximum possible weight. Fix a threshold and say that a vertex is -heavy if it contains at least a fraction of the weight of in its closed neighborhood, that is, . A simple coupon-collecting argument shows that there is a set of size such that all -heavy vertices are contained in . We investigate all the subcases corresponding to the possible choices of . Having fixed in a subcase, we can delete from the graph and from now on assume that there are no more -heavy vertices (except for isolated vertices that are easy to deal with).

Now the Gyárfás path argument, like e.g. in [10], asserts that in there exists a balanced separator for some . We simply delete from the graph and restart the whole algorithm on every connected component of . Since there are no -heavy vertices, we lose only a fraction of of the weight of in this step. Since every connected component of is of size at most , the depth of the recursion is at most . Consequently, throughout the recursion the total loss in the weight of the optimum solution is at most . Furthermore, it can be easily seen that the whole recursion tree has size bounded by , giving a quasi-polynomial running time bound of the whole algorithm.

To generalize this argument to the case of being a subdivided claw, an additional ingredient is needed: the Three-in-a-Tree Theorem by Chudnovsky and Seymour [6]. Let be a graph and let be three distinguished vertices. The Three-in-a-Tree Theorem provides a dichotomy: either we can find an induced tree in that contains , , and , or we can find a suitable decomposition of that somehow “separates” , , and witnesses that no such tree exists; this decomposition has a similar flavor to the decomposition for claw-free graphs [5]. By carefully combining this result with the Gyárfás path argument, we show that in an -free graph one can either find a balanced separator containing a small fraction of the weight of the optimum solution (e.g., consisting of a constant number of vertex neighborhoods) or a decomposition coming from the Three-in-a-Tree Theorem where every part is of significantly smaller size. Such a decomposition allows us to recurse on every part independently and then assemble the final result from partial results using a reduction to the maximum weight matching problem.

Having obtained the statement of Theorem 1.1 for being a path or a subdivided claw, we can generalize it to being a disjoint union of such graphs in a relatively simple and standard way.

In light of Theorem 1.1, we conjecture the following generalization.

###### Conjecture 1.2.

For every forest of maximum degree at most three, MWIS admits a QPTAS in the class of graphs that do not contain any subdivision of as an induced subgraph.

Our techniques stop short of proving Conjecture 1.2: we are able to prove it for containing at most three vertices of degree three. Note that this strictly generalizes the conclusion of Theorem 1.1 for being a subdivided claw.

Furthermore, as a side result we obtain a QPTAS for graphs excluding a long hole.

###### Theorem 1.3.

For every there exists an algorithm that, given a graph that does not contain any cycle of length at least as an induced subgraph, a weight function , and an accuracy parameter , computes a -approximation to Maximum Weight Independent Set on in time exponential in a polynomial of and .

The techniques of Theorem 1.3 allow us also to state the following graph-theoretical corollary that generalizes an analogous result for -free graphs [1, 10] and for graphs excluding any induced cycle of length at most  [4].

###### Theorem 1.4.

For every there exists a constant such that every graph that does not contain any cycle of length at least as an induced subgraph has treewidth bounded by , where is the maximum degree of .

#### Organization.

After brief preliminaries in Section 2, we present our framework in Section 3, with a number of technical proofs with smaller conceptual weight postponed to Section 8. In Section 4 we treat heavy vertices with a technical proof of a suitable abstraction of the argument postponed to Section 9. As a warm-up, the argument for -free graphs is described in Section 5. Section 6, the main technical part of the paper, considers the case of -free graphs where is a subdivided claw, with Theorem 1.1 inferred in Section 6.2. In Section 7 we prove Conjecture 1.2 for being a forest with at most three vertices of degree three. Finally, the proofs of Theorems 1.3 and 1.4 are presented in Section 10.

## 2 Preliminaries

For an (undirected, simple) graph and a vertex , denotes the (open) neighborhood of , and is the closed neighborhood of . We extend it to sets of vertices by and . Whenever the graph is not clear from the context, we clarify it by putting it in the subscript. For brevity, we sometimes identify subgraphs with their vertex set when this does not create any confusion: if is a subgraph of , then , , and are shorthands for , , and , respectively. By we denote a path on vertices. For a graph , is the family of connected components of .

### 2.1 Maximum Weight Independent Set

Let be a graph and let be a weight function. For a set we denote . The Maximum Weight Independent Set (MWIS) problem asks for an independent set maximizing . We say that an independent set is an -approximation for MWIS in if for every independent set in we have . In this work, given , , and an accuracy parameter , we ask for an independent set that is a -approximation. For simplicity, we will develop an algorithm that gives only a -approximation for some universal constant , as we can then use it with rescaled value of . We denote .

### 2.2 Extended strip decomposition and the three-in-a-tree theorem

Let be a graph. An extended strip decomposition of consists of the following:

1. a simple graph ,

2. a vertex set for every and subsets ,

3. a vertex set for every , and

4. a vertex set for every triangle in ,

with the following properties:

1. the vertex sets of , , and form a partition of ;

2. for every and every two distinct edges incident with , the set is fully adjacent to in ;

3. every edge is either contained in one of the graphs , , , or is one of the following types:

• , for two distinct edges , of incident with a common vertex ;

• and for some edge incident with a vertex ;

• and for some triangle in and an edge of this triangle.

The main result of [6] is the following.

###### Theorem 2.1 ([6]).

Let be a connected graph and let be a set of size at least two such that for every induced tree of , . Then there exists an extended strip decomposition of such that for every there exists a distinct vertex of degree one in with where is the unique edge of incident with . Furthermore, given and , such a decomposition can be computed in polynomial time.

Given a graph and an extended strip decomposition of , a vertex satisfying the property expressed in Theorem 2.1 will be called peripheral in . Concretely, is peripheral in if there exists a vertex of , said to be occupied by , such that has degree in and satisfies , where is the unique edge incident to in .

We will also need the notion of a trivial extended strip decomposition. Given a graph , a trivial extended strip decomposition consists of an edgeless graph that has a vertex for every connected component of and .

## 3 Disperser yields a QPTAS

Let be a graph and let be an extended strip decomposition of . For an edge , let be the set of all triangles of that contain . We define a number of atoms as follows. For every edge , we define the following atoms:

 A⊥e =η(e)∖(η(e,u)∪η(e,v)), Aue =η(u)∪η(e)∖η(e,v), Ave =η(v)∪η(e)∖η(e,u), Auve =η(u)∪η(v)∪η(e)∪⋃T∈T(e)η(T).

Furthermore, we define an atom for every and an atom for every triangle in . A trivial atom is an atom for an isolated vertex of with being a singleton containing an isolated vertex of .

Let be a weight function and let be reals. Let and let be an extended strip decomposition of . We say that is

• -shrinking if for every nontrivial atom of we have ;

• -safe if and, furthermore, for every nontrivial atom of it holds that ;

• -good if it is both -shrinking and -safe.

For a set , a weight function is defined as for every and for every .

###### Definition 3.1.

For a graph and a weight function an -disperser is a family such that:

• every member of is a pair of the form , where is an extended strip decomposition of ; and

• for every independent set in with there exists that is -good for and .

The main result of this section is that an algorithm producing dispersers with good parameters yields a QPTAS. The following definition encompasses the idea that a graph class admits efficiently computable dispersers.

###### Definition 3.2.

Let be a real, be a nonincreasing function, and be nondecreasing functions. A hereditary class of graphs is called -dispersible if there exists an algorithm that, given an -vertex graph and a weight function , runs in time and computes a -disperser for and of size at most .

The main theorem of this section is the following.

###### Theorem 3.3.

Let be a hereditary graph class with the following property: For every there exist functions where

 (δ(n))−1∈poly(logn,γ−1)andS(n),T(n)∈2poly(logn,γ−1)

and is computable in polynomial time given and , such that is -dispersible. Then MWIS restricted to graphs from admits a QPTAS.

From now on, hereditary classes satisfying the assumptions of Theorem 3.3 will be called QP-dispersible. Thus, Theorem 3.3 states that MWIS admits a QPTAS on every QP-dispersible class, while in the next sections we will prove that several classes are indeed QP-dispersible.

The above definitions are suited for all our results, but in some simpler cases we will construct dispersers that have a simpler form. More precisely, a disperser is strong if for each , is the trivial extended strip decomposition of . Recall that this means that simply decomposes into connected components: is an edgeless graph with vertices mapped bijectively to connected components of ; then the atoms of are exactly the connected components of . As for strong dispersers the decomposition is uniquely determined by , we will somewhat abuse notation and regard strong dispersers as simply families of sets , instead of pairs of the form . Intuitively, a strong disperser for is simply a family of subsets of vertices such that for every possible weight function , some member of the family is a balanced separator for that has a small weight by itself. The notions of QP-dispersibility lifts to strong QP-dispersibility by considering strong dispersers instead of regular ones.

In the rest of this section we highlight the main insights in the proof of Theorem 3.3. The remainder of the proof is postponed to Section 8.

#### Independent sets in extended strip decompositions.

Let be a graph and let be an extended strip decomposition of . Let and be two atoms of . We say that and are conflicting if they are potentially not disjoint; that is, for every

1. [(i)]

2. , , , and are pairwise in conflict;

3. both and conflict with and both and conflict with ;

4. and conflicts with and for every edge incident with , and similarly for the endpoint; and

5. and are in conflict for every .

Observe that if and are not conflicting then not only but also . Informally, two atoms and are not conflicting if and only if the definition of the extended strip decomposition ensures that they are disjoint and there is no edge of between and . A family of atoms of is independent if every two distinct elements of are not conflicting.

For an independent set in , we define the following family of atoms of :

• for every with and ,

• for every with but ,

• for every with but ,

• for every with ,

• for every such that for every incident with we have ,

• for every triangle in such that for all edges of we have or .

Observe that for every , may intersect at most one set for incident with . From this, a direct check verifies the following crucial observation:

###### Claim 3.4.

For every independent set in , the family is independent and .

In the other direction, if we are given an independent set for every atom of an independent family of atoms, then is an independent set in .

Thus, one can reduce finding a good approximation of maximum-weight independent set in to finding such good approximation in subgraphs for atoms , where is the sought maximum-weight independent set. In the definition of a disperser, if one recurses in the above sense on and for every in the disperser, the notion of -shrinking ensures that such recursion is of small depth, while the notion of -safety ensures that by sacrificing the set we lose only a small fraction of the optimum at every recursion step. However, there is one major obstacle to the above outline: we do not know the family . Instead, we can recurse on every atom of .

Then, we need an observation that assembling results from the recursion in the best possible way reduces to a maximum-weight matching problem in an auxiliary graph, in a similar fashion that finding maximum-weight independent set in line graphs corresponds to finding maximum-weight matching in the preimage graph. The remainder of the proof of Theorem 3.3 appears in full detail in Section 8.

## 4 Heavy vertices and strong dispersers

Let be a graph, be a weight function, and be an independent set. For a real , a vertex is -heavy (with respect to ) if . A simple coupon-collector argument shows the following.

###### Lemma 4.1.

Let be an -vertex graph for , be a weight function, be an independent set, and be a real. Then there exists a set of size at most such that contains all -heavy vertices with respect to .

###### Proof.

Let be the set of

-heavy vertices. We consider a probability distribution on

where a vertex is chosen with probability . For every , a vertex chosen at random according to this distribution satisfies with probability at least . Consequently, if is the set of vertices of each chosen independently at random according to this distribution, then for every the probability that is less than (here we used that and ). By the union bound, the probability that is positive. ∎

Next we prove a general-usage lemma that reduces the task of finding small dispersers to connected graphs where the neighborhood of every vertex is not -heavy with regards to some fixed maximum-weight independent set we are looking for. This is done essentially as follows: we first guess the set of -heavy vertices of size using Lemma 4.1, focus on the heaviest connected component of , and construct a suitable disperser for this component. This idea can be used to prove the following statement (full proof can be found in Section 9).

###### Lemma 4.2.

Let be a hereditary graph class. Suppose there is a polynomial such that given any and -vertex connected graph one can in polynomial time compute a family with consisting of pairs of the form , where and is an extended strip decomposition of , such that the following holds: For every weight function satisfying for each there exists such that

 w(A)⩽(1−p(σ))⋅w(G)andw(X)⩽σ⋅w(G−A)for every atom A of (H,η).

Then the class is QP-dispersible. Moreover, if it is always the case that all the extended strip decompositions appearing in the family are trivial (i.e. corresponding to the partition into connected components), then is strongly QP-dispersible.

## 5 Dispersers in Pt-free graphs

As a warm-up for more general results, in this section we focus on the class of -free graphs and prove the following.

###### Theorem 5.1.

For every , the class of -free graphs is strongly QP-dispersible.

The proof of Theorem 5.1 relies on a classical construction used by Gyárfás [13] to prove that -free graphs are -bounded, which is usually called the Gyárfás path. We choose the encapsulate this concept in the following claim, as we will reuse it later on.

###### Lemma 5.2.

Let be a real. Let be a connected graph endowed with a weight function , and let be any vertex of . Then there is an induced path in (possibly with and being empty) such that, denoting and for , the following holds:

1. [label=(P0),ref=(P0)]

2. unless ;

3. for every , we have ; and

4. for every , there is a connected component of such that and contains a neighbor of .

Moreover, given and one can compute in polynomial time a family consisting of induced paths in , each starting at , so that for every and weight function there exists satisfying the above properties for and .

###### Proof.

We first prove the existential statement and then argue how the reasoning can be turned into a suitable algorithm.

Call an induced subgraph of heavy if and light otherwise. We construct inductively so that after constructing , these vertices induce a path in and property 3 is satisfied for all . If no component of is heavy, we may finish the construction immediately by setting and as the empty path. Otherwise, we start by setting . Since and is connected, the unique (due to ) heavy component of is adjacent to and 3 is satisfied for .

For , the construction of is implemented as follows. By 3 for , there is a connected component of that is heavy and adjacent to . As , no other connected component of can be heavy. Since is an induced subgraph of , either every connected component of is light, or there is exactly one heavy connected component of that is moreover an induced subgraph of . In the former case, we may finish the construction by setting , as then 2 is satisfied. Otherwise, observe that is obtained from by removing vertices of , hence is a connected component of . Here observe that is non-empty, because is adjacent to . Consequently, there exists a vertex that is simultaneously adjacent to and to . Since , is not adjacent to any of the vertices . We conclude that the induced path can be extended by so that 3 is satisfied for .

Since is finite, the construction eventually finishes yielding a path satisfying both 2 and 3. We are left with arguing the algorithmic statement.

Observe that in the above reasoning, we used the constant and the function only in order to verify whether the construction should be finished, or to identify the heavy connected component of . Having identified , can be chosen freely among the common neighbors of and . Fix beforehand a total order of and assume that is always chosen as the smallest eligible vertex. Consider any run of the algorithm for and for let be the unique heavy connected component of . Since , subgraphs pairwise intersect. Since is a descending chain in the induced subgraph order and each is a connected component of , we conclude that is also a descending chain in the induced subgraph order. Consequently, there exists a vertex that is contained in each of . Now comes the main observation: knowing and having constructed , we may identify as the unique connected component of that contains . Thus, a path suitable for can be constructed knowing only and (given the total order fixed beforehand). Constructing such a path for every choice of and , of which there are at most many, yields the desired family . ∎

Note that in the statement of Theorem 5.2, graph is equal to unless is empty, when it is equal to .

Now Theorem 5.1 follows from a straightforward combination of Lemmas 4.2 and Lemma 5.2.

###### Proof of Theorem 5.1.

Without loss of generality assume . We argue that the class of -free graphs satisfies the prerequisites of Lemma 4.2. Thus we assume we are given a connected -free graph and a parameter . Consider applying Lemma 5.2 to and any vertex . We infer that in polynomial time we can construct a polynomial-size family of induced paths in satisfying in particular the following: for each weight function there exists such that for every , where if is non-empty and otherwise. Since is -free, every path in has less than vertices. Consequently, supposing for every vertex , we have for every , and in particular for every .

From construct a family by including, for every , a pair where is as above and is the trivial extended strip decomposition of . The reasoning of the previous paragraph shows that then the assumptions of Lemma 4.2 are satisfied for . Therefore, from Lemma 4.2 we conclude that the class of -free graphs is strongly QP-dispersible. ∎

## 6 Rooted subdivided claw

In this section we will focus on the classes of graphs excluding a claw subdivided a fixed number of times. We try to construct such subdivided claws with the use of Theorem 2.1. This provides us with extended strip decompositions of considered graphs.

We introduce a useful lemma that encapsulates the way we will use Theorem 2.1. We first need a definition.

###### Definition 6.1.

Let be a graph and let be such that . An extended strip decomposition shatters if the following condition hold: whenever is a triple of induced paths in that are pairwise disjoint and non-adjacent, and each of them has one endpoint in , then there is no atom in that intersects or is adjacent to each of .

###### Lemma 6.2.

Let be a graph and let be such that . Then one can in polynomial time find either an induced tree in containing all vertices of , or an extended strip decomposition of that shatters .

The proof of Lemma 6.2 is postponed to Section 6.1. Note that contrary to Theorem 2.1, Lemma 6.2 does not assume that the graph is connected.

We move to the main point of this section, which concerns classes excluding subdivided claws.

###### Definition 6.3.

A subdivided claw is a graph obtained from the claw and subdividing each of its edges an arbitrary number of times. The degree- vertices are then called the tips of the claw, while the unique vertex of degree is the center. A subdivided claw is a -claw if all its tips are at distance at least from its center. A graph is -free if it does not contain any -claw as an induced subgraph.

###### Theorem 6.4.

For every , the class of -free graphs is QP-dispersible.

Theorem 6.4 is a direct consequence of Lemma 4.2 and Lemma 6.5 below.

###### Lemma 6.5.

Fix an integer and . Let be a connected graph supplied with a weight function such that

 w(N[v])⩽σ8⋅w(G) for every v∈V(G). (1)

Let be any vertex of . Then there is either

1. [label=(C0),ref=(C0)]

2. an induced -claw in with one of the tips being , or

3. a subset of vertices and an extended strip decomposition of such that

 w(A)⩽(1−σ7)⋅w(G)andw(X)⩽σ⋅w(G−A)for every atom A of (H,η).

Moreover, given and one can in polynomial time either find conclusion 1, or enumerate a family of pairs such that for every weight function there exists satisfying 2 for .

###### Proof.

We first focus on proving the existential statement. At the end we will argue how the enumeration statement can be derived using the enumeration statement of Lemma 5.2.

Apply Lemma 5.2 to , , , and , yielding a suitable path , where (unless and is empty). As in Lemma 5.2, denote and for . For , let be the heaviest (w.r.t. ) connected component of . Then by 3 and 2 we have

 w(Di)>(1−σ)⋅w(G) for i⩽kandw(Dk+1)⩽(1−σ)⋅w(G). (2)

Also, as argued in the proof of Lemma 5.2, is an induced subgraph of for each with .

If , then conclusion 2 can be obtained by taking and to be the trivial extended strip decomposition of . This is because due to (1), while for every connected component of . Note that if , then in particular , so the above analysis can be applied as well. Hence, from now on assume that and .

Define and as the largest indices satisfying the following:

 w(Dp)>(1−σ5)⋅w(G)andw(Dq)>(1−σ3)⋅w(G).

By (2) and the discussion of the previous paragraph we have that and are well-defined and satisfy .

We now observe that indices have to be well-separated from each other, or otherwise we are done. For this, consider the following paths in :

 R1=(v0,v1,…,vp−2),R2=(vp,vp+1,…,vq−2),R3=(vq,vq+1,…,vk−1).

Note that he above path formally may be empty in case the index of the second endpoint is smaller than that of the first endpoint; in a moment we will see that this is actually never the case. We now verify that the neighborhood of each of these paths has to have a significant weight, or otherwise we are done.

###### Claim 6.6.

If we have

 w(N[R1])⩽σ6/2⋅w(G)orw(N[R2])⩽σ4/2⋅w(G)% orw(N[R3])⩽σ2/2⋅w(G),

then conclusion 2 can be obtained.

###### Proof.

We first consider the case when , which is slightly simpler. By assumption we have where is the heaviest connected component of . On the other hand, we have

 w(N[v0,v1,…,vp]) ⩽ w(N[R1])+w(N[vp−1])+w(N[vp]) ⩽ (σ6/2+2σ8)⋅w(G) ⩽ σ6⋅w(G).

Hence, we can obtain conclusion 2 by taking and the trivial extended strip decomposition of . Indeed, for every connected component of we have , implying also that .

Now, consider the case when . Observe that we also have , because and are disjoint. By assumption we have where is the heaviest connected component of . On the other hand, we have

 w(N[v0,v1,…,vq]) ⩽ w(N[R1])+w(N[R2])+w(N[vp−1])+w(N[vq−1]) ⩽ (σ5+σ4/2+2σ8)⋅w(G) ⩽ σ4⋅w(G).

Hence, we can obtain conclusion 2 by taking and the trivial extended strip decomposition of . Indeed, for every connected component of we have , implying also that .

Finally, consider the case when . As in the previous case, we have and . By the construction of we have where is the heaviest connected component of . On the other hand, we have

 w(N[v0,v1,…,vk]) ⩽ w(N[R1])+w(N[R2])+w(N[R3]+w(N[vp−1])+w(N[vq−1])+w(N[vk]) ⩽ (σ5+σ3+σ2/2+3σ8)⋅w(G) ⩽ σ2⋅w(G).

Hence, we can obtain conclusion 2 by taking and the trivial extended strip decomposition of . Indeed, for every connected component of we have