Linear kernels for edge deletion problems to immersion-closed graph classes

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roughly has to look like a constant-treewidth graph plus k additional vertices that can have arbitrary connections. Having exposed this structure, Fomin et al. [14] apply protrusion-based techniques that originate in the work on meta-kernelization [3,15]. Roughly speaking, the idea is to identify large parts of the graphs that have constant treewidth and a small interface towards the rest of the graph (so-called protrusions), which can then be replaced by smaller gadgets with the same combinatorial behaviour. Such preprocessing based on protrusion replacement is the base of all three aforementioned results for F-Minor Deletion. In the absence of a constant bound on the treewidth of an F-minor-free graph, the technique breaks completely. In fact, the kernelization complexity of Planarization, that is, F-Minor Deletion for F = {K 5 , K 3,3 }, is a notorious open problem.
An interesting aspect of the work of Fomin et al. [14] is that the exponent of the polynomial bound on the size of the kernel for F-Minor Deletion grows quite rapidly with the family F. Recently, it has been shown by Giannopoulou et al. [17] that in general this growth is unavoidable: unless NP ⊆ coNP/poly, for every constant η, the Treewidth-η Deletion problem (delete k vertices to get a graph of treewidth at most η) has no kernel with O(k η/4− ) vertices for any > 0. Since graphs of treewidth η can be characterized by a finite set of forbidden minors F η , at least one of which is planar, this refutes the hypothesis that all F-Minor Deletion problems admit polynomial kernels with a uniform bound on the degree of the polynomial. However, as shown by Giannopoulou et al. [17], such bounds can be achieved for some specific problems, like vertex deletion to graphs of constant tree-depth.
Immersion problems. Recall that a graph H can be immersed into a graph G (or that H is an immersion of G) if there is a mapping from vertices of H to pairwise different vertices of G and from edges of H to pairwise edge-disjoint paths connecting the images of its endpoints 3 . Such a mapping is called an immersion model. Just like the minor relation, the immersion relation imposes a partial order on the class of graphs. Alongside with the minor order, Robertson and Seymour [33] proved that graphs are well-quasi-ordered under the immersion order as well. This implies that for every graph class G that is closed under taking immersions, the set obs im (G) containing immersion minimal graphs that do not belong in G, is finite (we call obs im (G) the immersion obstruction set of G). Therefore G can be characterized by a finite set of forbidden immersions. The general intuition is that immersion is a containment relation that corresponds to edge cuts, whereas the minor relation corresponds to vertex cuts. Also, the natural setting for immersions is the setting of multigraphs. Hence, all the graphs considered in this paper may have parallel edges connecting the same pair of endpoints.
Recently, there has been a growing interest in immersion-related problems [28,10,16,25,18,20,36,4,9,1,21,11] both from the combinatorial and the algorithmic point of view. Most importantly for us, Wollan proved in [36] an analogue of the Excluded Grid Minor Theorem, which relates the size of the largest wall graph that is contained in a graph as an immersion with a new graph parameter called tree-cut width. By a subcubic graph we mean a graph of maximum degree at most 3. The following theorem follows from the work of Wollan [36]. In other words, for any family F of graphs that contains some planar subcubic graph, the tree-cut width of F-immersion-free graphs is bounded by a universal constant depending 57:4 Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes on F only. In Section 2 we discuss the precise definition of tree-cut width and how exactly Theorem 1 follows from the work of Wollan [36]. Also, note that if a family of graphs F does not contain any planar subcubic graph, then there is no uniform bound on the tree-cut width of F-immersion-free graphs. Indeed, wall graphs are then F-immersion-free, because all their immersions are planar and subcubic, and they have unbounded tree-cut width.
After the introduction of tree-cut width by Wollan [36], the new parameter gathered substantial interest from the algorithmic and combinatorial community [16,28,18,25]. It seems that tree-cut width serves the same role for immersion-related problems as treewidth serves for minor-related problems and, in a sense, it can be seen as an "edge-analogue" of treewidth. In particular, given the tree-cut width bound of Theorem 1 and the general approach of Fomin et al. [14] to F-Minor Deletion, it is natural to ask whether the same kind of results can be obtained for immersions where the edge removals are considered instead of vertex removals. More precisely, fix a finite family of graphs F containing some planar subcubic graph and consider the following F-Immersion Deletion problem: given a graph G and an integer k, determine whether it is possible to delete at most k edges of G in order to obtain a graph that does not admit any graph from F as an immersion. Notice that when F = {K 2 } and F = {K 3 }, the problem of computing the minimum size of such a set of edges can be solved in polynomial time, while it is NP-hard when F = {K − 4 } (see e.g. [5]). Parallel to the case of F-Minor Deletion, for every fixed k, the graph class G im k,F consisting of the graphs in the YES-instances (G, k) of F-Immersion Deletion is closed under taking of immersions 4 , therefore O im k = obs im (G im k,F ) is a finite set, by the well-quasiordering of graphs under immersions [33]. Together with the immersion-testing algorithm of Grohe et al. [22], this implies that F-Immersion Deletion admits (non-constructively) an FPT-algorithm. This naturally induces the parallel project of optimizing the performance of such FPT-algorithms for various instantiations of F. More concretely, is it possible to extend the general framework of Fomin et al. [14] to obtain efficient approximation, kernelization, and FPT algorithms also for F-Immersion Deletion? Theorem 1 suggests that the suitable analogue of the assumption from the minor setting that F contains a planar graph should be the assumption that at least one graph from F is planar and subcubic.
Our results. In this work we give a definitive positive answer to this question. The following two theorems gather our main results; for a graph G, by |G| and G we denote the cardinalities of the vertex and edge sets of G, respectively. Theorem 2 (Constant factor approximation). Let F be a finite family of connected graphs with at least one member being planar and subcubic. Given a graph G, in O( G 3 log G ·|G| 3 ) time one can output a subset of edges F ⊆ E(G) such that G − F is F-immersion-free and the size of F is at most c apx times larger than the optimum size of a subset of edges with this property, for some constant c apx depending on F only.
The constant-factor approximation can be generalized to work when F contains disconnected graphs as well, using the approach of Fomin et al. [14,13].
Theorem 3 (Linear kernelization and obstructions). Let F be a finite family of connected graphs with at least one member being planar and subcubic. Given an instance (G, k) of F-Immersion Deletion, in time O( G 4 log G · |G| 3 ) one can output an equivalent instance (G , k) with G ≤ c ker · k, for some constant c ker depending on F only. Moreover, every graph in O im k has at most c F · k edges (for a constant c F non-constructively depending on F).
Thus, Theorems 2 and 3 mirror the approximation and kernelization results and the obstruction bounds of Fomin et al. [14]. However, this mirroring is not exact as we show that, in the immersion setting, a stronger kernelization procedure can be designed. Namely, the size of the kernel given by Theorem 3 is linear, with only the multiplicative constant depending on the family F, whereas in the minor setting, the exponent of the polynomial bound on the kernel size provably must depend on F (under plausible complexity assumptions). This shows that the immersion and minor settings behave quite differently and in fact stronger results can be obtained in the immersion setting. Observe that using Theorem 3 it is trivial to obtain a decision algorithm for F-Immersion Deletion working in time O(c k fpt + G 4 log G · |G| 3 ) for some constant c fpt depending on F only: one simply computes the kernel with a linear number of edges and checks all the subsets of edges of size k.
Our techniques. Our approach to proving Theorems 2 and 3 roughly follows the general framework of protrusion replacement of Fomin et al. [14] (see also [3]). We first define protrusions suited for the problem of our interest. In fact, our protrusions can be seen as the edge-analogue of those introduced in [14] (as in [5]). A protrusion for us is simply a vertex subset X that induces an F-immersion-free subgraph (which hence has constant tree-cut width, by Theorem 1), and has a constant number of edges to the rest of the graph. When a large protrusion is localized, it can be replaced by a smaller gadget similarly as in the work of Fomin et al. [14]. However, we need to design a new algorithm for searching for large protrusions, mostly in order to meet the condition that the exponent of the polynomial running time of the algorithm does not depend on F. For this, we employ the important cuts technique of Marx [27] and the randomized contractions technique of Chitnis et al. [6]. All of these yield an algorithm that exhaustively reduces all large protrusions.
Unfortunately, exhaustive protrusion replacement is still not sufficient for a linear kernel. However, we prove that in the absence of large reducible protrusions, the only remaining obstacles are large groups of parallel edges between the same two endpoints (called thetas), and, more generally, large "bouquets" of constant-size graphs attached to the same pair of vertices. Without these, the graph is already bounded linearly in terms of the optimum solution size. The approximation algorithm can thus delete all edges except for the copies included in bouquets and thetas, reducing the optimum solution size by a constant fraction of the deleted set. It then exhaustively reduces protrusions in the remaining edges, and repeats the process until the graph is F-immersion-free.
To obtain a linear kernel we need more work, as we do not know how to reduce bouquets and thetas directly. Instead, we apply the following strategy based on the idea of amortization. After reducing exhaustively all larger protrusions, we compute a constant-factor approximate solution F apx . Then we analyze the structure of the graph G − F apx , which has constant tree-cut width. It appears that every bouquet (and theta) in G can be reduced up to size bounded linearly in the number of solution edges F apx that "affect" it. After applying this reduction, we can still have large bouquets in the graph, but this happens only when they are affected by a large number of edges of F apx . However, every edge of F apx can affect only a constant number of bouquets and hence a simple amortization arguments shows that the total size of bouquets is linear in |F apx |, so also linear in terms of the optimum.

57:6 Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes
We remark that this part of the reasoning (the above amortization argument in particular) are fully new contributions of this work. These deviate significantly from arguments by Fomin et al. [14], which aimed at a obtaining a polynomial kernel only, instead of linear. Also, we remark that, contrary to the work of Fomin et al. [14], all our algorithms are deterministic.
For the second part of Theorem 3, we show that protrusion replacements can be realized as immersions of the original graph. This implies that, in the equivalent instance (G , k) produced by our kernelization algorithm, the graph G is an immersion of G. Therefore any immersion-obstruction of G im k−1,F must already have a linear, in k, number of edges.
Application: immersion-closed parameters. We would like to highlight one meta-algorithmic application of our results, which was our original motivation. Suppose p is a graph parameter, that is, a function that maps graphs to N. We shall say that p is closed under immersion For a parameter p and a constant r, define the p-at-most-r Edge Deletion problem as follows: given a graph G and an integer k, determine whether at most k edges can be deleted from G to obtain a graph with the value of p at most r. We also define the associated para- Then the following meta-result can be derived from Theorems 2 and 3 and the fact that immersion is a well-quasi-order; a proof can be found in the full version of the paper. 5 . Then, for every constant r, the p-atmost-r Edge Deletion problem admits a constant-factor approximation and a linear kernel. Moreover, there is a constant c r , depending (non-constructively) on r, such that for every k, every graph H in obs im (G k,pr ) has at most c r · k edges.

Theorem 4. Let p be a graph parameter that is closed under immersion and under disjoint union and moreover is large on the class of walls
Natural parameters that satisfy the prerequisites of Theorem 4 include cutwidth, carving width, tree-cut width, and edge ranking; see e.g. [34,35,36,26,23] for more details. Theorem 4 mirrors a corollary by Fomin et al. [14] for the Treewidth-η Deletion problem asserting a constant-factor approximation, a polynomial kernel, a polynomial bound for the corresponding minor-obstruction set, and a single-exponential FPT algorithm, for every constant η.
Organization. This extended abstract focuses on sketching the proofs of Theorems 2 and 3. The proofs of statements marked with are omitted and can be found in the full version [19].

Preliminaries
For a positive integer p, we denote is a multiset of edges. Each edge connects two different vertices, called the endpoints of the edge (we do not allow loops). Note that there might be several edges (called parallel edges) between two vertices. An edge is incident to a vertex if it is one of its two endpoints. We write |G| for |V (G)| and G for |E(G)| (counting edges with multiplicities). For a subset of vertices X ⊆ V (G), G[X] is the subgraph induced by X. For a subset of edges Tree-cut width. A near-partition of a set X is a family of (possibly empty) subsets By rooting T at some vertex, we can talk about a rooted tree-cut decomposition. When G is disconnected, a tree-cut decomposition is a forest consisting of one tree for each connected component.
For each edge e = uv of T , T − uv has exactly two components which we call T uv and T vu , that contain u and v respectively. Since X is a near-partition, sets X V (Tuv) and X V (Tvu) form a near-partition of V (G) (provided G is connected). We define the adhesion of an edge e = uv of T , denoted adh T (e), as the set E G (X V (Tuv) , X V (Tvu) ). We omit the subscript if T is clear from the context. An adhesion is thin if it has at most 2 edges, bold otherwise.
We now move to the definition of tree-cut width. In fact, we do not give the original definition (it can be found in the full version of the paper), but we instead give an alternative definition that is easier to handle. Let G be a graph and (T, X = {X t : t ∈ V (T )}) be a tree-cut decomposition of G. For a node t of T , let w(t) be the number of edges incident to t that have bold adhesions. The width of the decomposition, denoted width (T, X ), is equal to max{max e∈E(T ) |adh(e)|, max t∈V (T ) |X t | + w(t)}. The tree-cut width of G, denoted tctw (G), is the minimum width of a tree-cut decomposition of G. The standard tree-cut width of G, denoted tctw(G), is similarly defined as the minimum width of a tree-cut decomposition of G, where width is defined slightly differently. We prove that both notions are equivalent. Ganian et al. [16] showed that bounded tree-cut width implies bounded treewidth. Besides, Kim et al. [25] showed that the dependency cannot be improved to subquadratic.
In our algorithms we need to adjust tree-cut decompositions to our needs, and hence we define the notion of a neat tree-cut decomposition. A neat tree-cut decomposition is a rooted tree-cut decomposition (T, X ) of a connected graph G that has the following properties: For every e = uv ∈ E(T ), the graphs G[X V (Tuv) ] and G[X V (Tvu) ] are connected. Suppose t is a node of T with parent s, such that the adhesion of st is thin. Then E G (X V (Tts) , X V (T t s ) ) = ∅ for every sibling t of t. The second property was used by Ganian et al. [16] under the name niceness. It appears that any tree-cut decomposition can be made neat without increasing the width by much; the proof follows closely the lines of [16, Lemma 1]. Lemma 7 ( ). Given a tree-cut decomposition of width ≤ k, a neat tree-cut decomposition of the same graph with width ≤ k 2 + 1 can be constructed in time O( G · |G| 2 · k 2 ).
The following result shows why neat tree-cut decompositions are useful: if a node of a neat decomposition has many neighboring nodes, then all but a constant number of them are connected to it via very simple adhesions.

57:8 Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes Lemma 8 ( ). Let G be a graph with a neat tree-cut decomposition T = (T, X ) satisfying width (T ) ≤ r. Let t ∈ V (T ). Then for all but at most 2r + 1 of the edges e of T incident to t, adh(e) is thin and all of its edges have an endpoint in the bag at t.
Finally, we need that the tree-cut width of a graph can be computed efficiently. For this, we can use the following 2-approximation algorithm of Kim et al. [25]. We remark that the width of the decomposition yielded is measured in terms of width(·) and not width (·), but the decomposition can be adjusted to have also width bounded by 2r using Lemma 5.
Theorem 9 (see [25]). There is an algorithm that, given a graph G and an integer r, runs in time 2 O(r 2 log r) · |G| 2 and either concludes that tctw(w) > r, or returns a tree-cut decomposition of G of width at most 2r.
For the whole paper we fix a finite family of graphs F with the following properties: all graphs of F are connected, and at least one is planar and subcubic. A graph G will be called F-immersion-free, or F-free for short, if G contains no graph from F as an immersion. By Theorem 1, the tree-cut width of an F-free graph is bounded by a constant depending on F only, so by Lemma 6 also its treewidth is bounded by a constant. By combining this with Bodlaender's algorithm [2] and Courcelle's Theorem [7], we obtain the following:

Lemma 10 ( ). It can be checked in linear-time whether a given graph is F-free.
Moreover, by combining the bound on tree-cut width of an F-free graph with Lemmas 5, 7 and Theorem 9, we obtain the following.

Lemma 11 ( ). There exists an algorithm that, given an F-free graph G, runs in time O( G · |G| 2 ) and computes a neat tree-cut decomposition
For a graph G, by OPT(G) we denote the minimum number of edges that need to be deleted from G in order to obtain an F-free graph.

Protrusions
Replacing protrusions. We now introduce the notion of a protrusion suited to the considered problem. In the sequel, we will only deal with 2b F -and 2-protrusions, where b F is the constructive bound on tctw guaranteed by Lemma 11.

Definition 12.
An r-protrusion of a graph G is a set X ⊆ V (G) such that |δ(X)| ≤ r and G[X] is F-free.
As in [14], the base for our kernelization algorithm is protrusion replacement. That is, we iteratively find a protrusion X that is large but has small δ(X), and replace it with a smaller gadget X that has the same behaviour. The following lemma formalizes this intuition.

Lemma 13 ( ).
There is a constant c F and algorithm that, given a graph G and a 2b Fprotrusion X in it with G[X] > c F , outputs in linear time a graph G with OPT(G) = OPT(G ) and G < G . Moreover, there is a linear-time algorithm working as follows: given a subset F of edges of G such that G − F is F-free, the algorithm computes a subset F of edges of G such that G − F is F-free and |F | ≤ |F |.
The proof of Lemma 13 follows closely the strategy used by Fomin et al. [14]: Every 2b F -protrusion can be assigned a type, where the number of types is bounded by a function depending on F only. The type of a protrusion can be computed efficiently due to protrusions having constant treewidth. Protrusions with the same type behave in the same way with respect to the problem of our interest, and hence can be replaced by one another. Therefore, we store a replacement table consisting of the smallest protrusion of each type, so that every larger protrusion can be replaced by a smaller representative stored in the table. The lifting algorithm finds, using dynamic programming, a partial solution in the large protrusion that has the same behaviour as the given partial solution in the replacement protrusion, while being not larger. Note that while a replacement table exists and can be hard-coded into the algorithm (as it depends on the fixed family F only), giving an explicit bound on the size of the graphs in it (and thus on c F in Lemma 13) would require additional arguments. The details can be found in the full version of the paper.
We henceforth define a replaceable protrusion in G as a 2b F -protrusion X with G[X] > c F , where c F is the constant given by Lemma 13.

Finding excessive protrusions.
Recall that a replaceable protrusion in a graph G is a 2b F -protrusion X with G[X] > c F . To find replaceable protrusions in the input graph, we need to assume some additional connectivity constraint (which will be implied from a connected tree-cut decomposition) -this is captured by the following definition. The larger protrusion size is needed to make any connected component of the protrusion replaceable.
Replaceable protrusions could be found easily if we allowed a (far worse) running time of the form G O(b F ) , but this would affect the running times in both our main results. With the above definition in hand, we use the techniques of important cuts, introduced by Marx [27] (see also the exposition in [8,Chapter 8.2]) and of randomized contractions by Chitnis et al. [6] instead. These two techniques allow us to reduce excessive protrusions: we use the randomized contractions technique to find a large enough subset of a presumed excessive protrusion, after which important cuts allow us to find a boundary that makes this subset a replaceable protrusion.

Lemma 15 ( ).
There is an algorithm that, given a connected graph G, runs in time O( G log G · |G| 2 ) and either correctly concludes that G does not contain any excessive protrusion, or it outputs some replaceable protrusion in G.
We remark that we only defined excessive protrusions in connected graphs. Note that if B is an excessive protrusion in a connected component H of G, it would not necessarily be an excessive protrusion in G, since G − B may have more components than H − B (they are however not adjacent to B). We will say that no component of G has an excessive protrusion if for each connected component H of G, there is no excessive protrusion in H.
By exhaustively (at most G times) executing the algorithm of Lemma 15 and replacing any obtained protrusion using Lemma 13, we can get rid of all excessive protrusions. We formalize this in the following lemma, which will serve as the abstraction of protrusion replacement in the sequel.

Lemma 16 (Exhaustive Protrusion Replacement).
There is an algorithm that, given a graph G, runs in time O( G 2 log G · |G| 2 ) and computes a graph G such that OPT(G) = OPT(G ), G ≤ G , and no connected component of G has an excessive protrusion.
Moreover, there exists a solution-lifting algorithm that works as follows: given a subset F of edges of G for which G − F is F-free, the algorithm runs in time O( G 2 ) and outputs a subset F of edges of G such that |F | ≤ |F | and G − F is F-free.

57:10
Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes 4

Constant-factor approximation
It would be ideal if just applying the Exhaustive Protrusion Replacement (Lemma 16) reduced the size of the graph to linear in OPT. Then, we would already have a linear kernel, and taking all its edges would yield a constant-factor approximation. Unfortunately, there are graphs with no excessive protrusions, where the size is not bounded linearly in OPT. To see this, observe that a large group of parallel edges is not a protrusion, so our current reduction rules will not reduce their multiplicity, even if they amount to 99% of the graph. Hence, we need to find a way to discover and account for such groups (we remark that reducing each to O(OPT) would be relatively easy, giving a quadratic kernel). More generally, the structures that turn out to be problematic are large groups of constant-size 2-protrusions attached to the same pair of vertices; a group of parallel edges is a degenerated case of this structure. To describe the problematic structures formally, we introduce the notion of a bouquet.
Bouquets. Let us define the following constant d F := max{2b F · c F + 2b F , 3MAX F } + 1 (where MAX F = max H∈F H ). A bouquet is a family of at least d F isomorphic 2-protrusions, while a theta is a set of at least d F parallel edges.
Definition 17. Consider a graph G, a set U ⊆ V (G) and a family of 2-protrusions {S i } i∈I such that for each i ∈ I: for all i, j ∈ I, with an isomorphism that maps each vertex of U to itself. We call such a family a bouquet attached to U if it is maximal under inclusion (i.e. there is no proper superfamily which is also a bouquet) and has at least d F elements. The edge set of the bouquet is the set of all edges incident to some S i .

Definition 18.
For two vertices u, v ∈ V (G), a theta attached to {u, v} is a set of edges between u and v that is maximal under inclusion and has at least d F elements.
The constant d F is chosen so that a protrusion containing a set to which a bouquet (or theta) is attached is large enough to be excluded as an excessive protrusion, and so that any immersion of a graph of F cannot simultaneously intersect all elements of a bouquet. Indeed, in any immersion of some H ∈ F in a graph G, the image of an edge of H is a path in G, which visits every vertex of the bouquet's attachment at most once, and hence intersects at most three elements of the bouquet. Thus in total, the immersion model intersects at most 3 · max H∈F H elements of the bouquet or theta, which is less than d F .
We now show that the number of edges of a graph with no excessive protrusions, no bouquets, and no thetas is linearly bounded in the optimum solution size, which formalizes the intuition that only those structures prevent the graph from being a linear kernel.

Lemma 19 ( ).
Let G be a connected graph without excessive protrusions, bouquets, or thetas. Then G is F-free, or G ≤ c · OPT(G), for some constant c depending on F only.
The proof of Lemma 19 goes roughly as follows. Take some optimum solution F . Then G − F is F-free, so, by Theorem 11, it has a neat tree-cut decomposition (T, X ) of width at most b F . For simplicity suppose that G − F is connected, so that T is a tree (the proof in the general case is essentially the same). Let M 0 be the set of all vertices of T whose bags contain a vertex incident to an edge of F ; then |M 0 | ≤ 2|F |. Compute the lowest common ancestor closure M of M 0 : start with M := M 0 , and iteratively add to M any lowest common ancestor of two nodes of M that is not yet included. As in Fomin et al. [14], we have that |M | ≤ 2|M 0 | ≤ 4|F |, and each component of T − M is adjacent to at most two nodes of M .
Let us consider some connected component T of T − M , and let X T be the union of the bags at the nodes of T . Suppose first that T is adjacent to exactly two nodes of M ; note that there are at most |M | − 1 such components T . Then one can easily see that G[X T ] ≤ 2b F c F , because otherwise X T would be an excessive protrusion. Here, we crucially use the first condition of the neatness of T to argue that G − X T has at most two connected components. Hence, the total number of edges in graphs G[X T ] for such components T is bounded by (|M | − 1) · 2b F c F , which is linear in |F | = OPT(G).
We are left with considering components T that are adjacent to exactly one node of M . We again have that G[X T ] ≤ 2b F c F for each such component T , because otherwise X T would be an excessive protrusion. However, a priori we do not have any bound on the number of such components. Suppose for a moment that a large number of such components T is adjacent to the same node t ∈ M . By Lemma 8, all but a constant number of them are connected to t via edges of the decomposition with thin adhesions. Moreover, for each of these adhesions, all the (at most 2) edges of the adhesion have both endpoints in the bag X t . Since the size of X t is bounded by a constant, and each X T induces a graph of constant size, we can infer that there is a constant number of isomorphism types for graphs G[X T ], together with the choice of the attachment points in X t . So if the number of the considered components T was very large, then some of them would form a bouquet, a contradiction.
This shows that, in fact, the number of components T adjacent to only one vertex of M is also bounded linearly in |M |, so also in OPT(G). The fact that G has no thetas is used to bound the number of edges contained in graphs induced by the bags of M . By combining all these bounds, we conclude the proof of Lemma 19.
Finding a constant-factor approximation piece by piece. To handle bouquets and thetas, we first show that they are disjoint, as otherwise they would constitute a large protrusion.
Lemma 20 ( ). Let G be a connected graph with no excessive protrusions. Then every two bouquets and/or thetas in G have disjoint edge sets. Furthermore, if a bouquet or theta is attached to U ⊆ V (G), then U is disjoint with all elements of any bouquet.
The following lemma is the crucial step for our approximation: we find a subset of edges ∆ with a guarantee that a constant fraction of ∆ is used in some optimum solution.
Lemma 21 ( ). Given a connected graph G with no excessive protrusion that is not F-free, one can find in in time O(|G| 3 ) a set ∆ ⊆ E(G) such that (for some c depending on F only): The set ∆ given by Lemma 21 is constructed as follows. First, we locate all thetas and bouquets in G; Lemma 20 ensures that they do not overlap. ∆ is defined as the set of all edges of G, with the exception that in each bouquet and in each theta we exclude from ∆ the edges of all but d F − 1 elements of the bouquet/theta. We show that the subgraph given by edges of ∆ satisfies the assumptions of Lemma 19, thus bounding |∆|. The bound is linear in the size of the intersection of ∆ with any optimal solution, allowing to conclude the claim.
To get a constant-factor approximation algorithm, we iteratively invoke the algorithm of Lemma 21 (extended to general graphs by considering each connected component separately). More precisely, we perform iteratively the following procedure, starting with G 1 = G. At step i, given a graph G i , we first run the algorithm of Lemma 16 to remove excessive protrusions from G i and obtain a new graph G i . Thus, in a sense, we reduce those parts of the graph where no more edges need to be deleted. Then, we apply the algorithm of Lemma 21 to G i , thus finding a set ∆ i with the following property: the deletion of ∆ i reduces OPT by some number p, while increasing the total number of edges deleted so far by at most c · p. We proceed to the next iteration with G i+1 := G i − ∆ i . Eventually, we arrive at the situation when the current graph G i is already F-free, in which case we stop. A constant-factor approximate solution can be then obtained by reverting the iterations: Proceeding from the last iteration to the first, we always add the extracted set ∆ i to the constructed solution, and roll-back the protrusion reductions performed by the algorithm of Lemma 16 while lifting the current solution using the solution-lifting algorithm. This concludes the proof of Theorem 2 (the formal description is in the full version of the paper).

Linear kernel
In the previous section we observed (Lemma 19) that the only structures in the graph that prevent it from being a linear kernel are excessive protrusions, bouquets, and thetas. Using the Exhaustive Protrusion Replacement (Lemma 16) we can get rid of excessive protrusions, but bouquets and thetas can still be present in the graph. It would be ideal if we could reduce the size of every bouquet or theta to a constant, but unfortunately we are unable to do this. Instead, bouquets and thetas will be reduced to constant size in the amortized sense. The next lemma is the crux of our approach: provided we know a "local" solution ∆ that isolates a bouquet into an F-free part, this bouquet can be pruned proportionally to the size of ∆ without changing OPT(G). The proof is by a simple replacement argument, and the same reasoning can also be applied to limit the sizes of thetas.

Lemma 22 ( ).
Let {X i } i∈I be a bouquet attached to U in G. Suppose ∆ ⊆ E(G) is such that all the connected components of G − ∆ that intersect U ∪ i∈I X i are F-free. Then OPT(G) = OPT(G ), where G is obtained from G by removing vertices of all except d F + |∆| elements of the bouquet.
We are now ready to show the main part of our reasoning: given some solution F , for example the one returned by the approximation algorithm of Theorem 2, we are able to reduce the graph, provided it is not already bounded linearly in |F |. Lemma 23 ( ). Let G be a connected graph with no excessive protrusions and let F ⊆ E(G) be such that G − F is F-free. Then either G ≤ c · |F | for some constant c depending on F only, or given G and F , one can compute in time O( G · |G| 2 ) a subgraph G of G such that OPT(G) = OPT(G ) and G < G .
The proof uses the following strategy based on the idea of amortization. Given a solution F , we use parts of F as local solutions to locally bound bouquets and thetas. More precisely, we first perform a similar structural analysis of G with F removed as in the proof of Lemma 19; see the sketch following its statement. There, we considered a neat tree-cut decomposition T = (T, X ) of G − F of width b F . In this decomposition, we highlighted a subset M ⊆ V (T ), the lca-closure of those nodes of T whose bags are incident to F . We concluded that the only parts not yet bounded linearly in terms of |F | were large bouquets and thetas with both attachment points contained in a bag X t of some node t ∈ M . For such a node t, define ∆(t) as the set containing: (i) all the edges of F incident to X t , and (ii) all the adhesions of edges of T incident to t that lead to components of T − M containing other vertices of M . Then ∆(t) easily satisfies the prerequisites of Lemma 22. Hence, d F + |∆(t)| can be used as a bound for reducing these bouquets and thetas. Summing these bounds through all nodes t ∈ M , we achieve an O(|F | + |M |) bound, thus O(|F |) due to |M | ≤ 4|F |.
With Lemma 23, we can now easily conclude the proof of Theorem 3 (the formal description is in the full version of the paper). First, we get rid of all excessive protrusions in the graph (Lemma 16) and compute an approximate solution F apx (Theorem 2). If |F apx | > c apx · k, the input is a NO instance. Otherwise, we give F apx to the algorithm of Lemma 23 which, provided the graph is still too large to be a kernel, computes a strictly smaller, but equivalent instance. We then recurse on this smaller instance, eventually returning a linear kernel.