DeepAI

Turing Kernelization for Finding Long Paths in Graph Classes Excluding a Topological Minor

The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether k-Path admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length k, using an oracle that answers queries of size poly(k)? We show this can be done when the input graph avoids a fixed graph H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and K_3,t-minor-free graphs. Moreover, we show that k-Path even admits a polynomial Turing kernel when the input graph is not H-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H-topological-minor-free graph that does not contain a k-path, has a separation that can safely be reduced after communication with the oracle.

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1 Introduction

Suppose that Alice is a polynomial-time agent faced with an input to an NP-hard problem that she wishes to solve exactly. To facilitate her in this process, she can ask questions to an all-knowing oracle. These will be answered truthfully and instantly, but the oracle is memory-less and will not take previous questions into account when answering the next one. How large do these questions have to be, to allow Alice to find the answer to her problem? Clearly, the answer can be established by sending the entire input to the oracle, who determines the answer and sends it to Alice. Could there be a more clever strategy? Alice can attempt to isolate a small but meaningful question about the behavior of her input, such that after learning its answer, she can reduce to a smaller input without changing the outcome. Iterating this process solves her problem: when it has become sufficiently small, it can be posed to the oracle in its entirety.

Such problem-solving strategies can be rigorously analyzed using the notion of Turing kernelization that originated in parameterized algorithmics. The parameter makes it possible to express how the size of the questions that Alice asks, depends on properties of the input that she is given. (See Section 3 for a formal definition.)

Understanding the power of Turing kernelization is one of the main open research horizons in parameterized algorithmics. There is a handful of problems for which a nontrivial Turing kernelization is known [1, 3, 4, 6, 14, 17, 19, 20, 26, 28]. On the other hand, there is a hierarchy of parameterized complexity classes which are conjectured not to admit polynomial Turing kernels [16]. Arguably, the main open problem (cf. [5, 4, 16]) in this direction is to determine whether the -Path problem (determine whether an undirected graph has a simple path of length ) has a polynomial Turing kernel. In earlier work [18], the first author showed that -Path indeed admits polynomial Turing kernels on several graph classes. In this work, we develop Turing kernels for -Path in a much more general setting.

Our results

Our algorithmic contributions are twofold. First of all, we extend the Turing kernelization for -Path to much broader families of sparse graphs. Whereas the earlier work could only deal with -minor-free graphs, claw-free graphs, and bounded-degree graphs, we show that a Turing kernelization exists on -minor-free graphs for all fixed graphs . We even lift the kernelization to -topological-minor-free graphs, thereby capturing a common generalization of the bounded-degree and -minor-free cases.

Theorem 1.1.

For every fixed graph , the -Path problem, restricted to graphs excluding as a topological minor, admits a polynomial Turing kernel. Furthermore, the kernel runs in time and invokes calls to the oracle.

Our second contribution is the following theorem. By a novel algorithmic approach, we obtain a Turing kernelization even when the input graph does not belong to the desired restricted graph class itself, but contains a small known vertex modulator whose deletion places the graph in such a graph class.

Theorem 1.2.

For every fixed graph , the -Path problem, on instances consisting of a graph , integer , and a modulator such that is -topological-minor-free, admits a polynomial Turing kernel, when parameterized by and .

Techniques

To explain our approach, we briefly recall the idea behind the Turing kernelization for -Path on planar graphs. At the core lies a win/win: there is a polynomial-time algorithm that either (i) establishes that a planar graph  has a -path (a simple path on vertices), or (ii) finds a separation  in  with the following property: the size of  is polynomially bounded in , but large enough that after marking a witness structure for each reasonable way in which a -path might intersect , some vertex remains unmarked. Using bounded-size oracle queries to mark the witness structures, this allows the problem to be simplified by removing an unmarked vertex from  without changing the answer.

Theorem 1.1 is established by lifting this win/win approach to -(topological)-minor-free graphs. This requires an adaptation of the decomposition theorems of Robertson and Seymour [25] (for minors) and of Grohe and Marx [15] (for topological minors), to obtain the following. Every -free graph that does not have a -path, has a tree decomposition of constant adhesion and width . A reducible separation can be found by inspecting this tree decomposition. To establish this result, we exploit known theorems stating that triconnected -vertex graphs that exclude  as a minor for some  [8], contain paths of length  for some . Roughly speaking, this allows us to infer the existence of a -path if there is a large embedded part in the nearly-embeddable graph corresponding to a bag of the graph minors decomposition, since graphs embeddable in a fixed surface are -minor-free for some . We use lower bounds on the circumference of graphs of bounded degree [7, 27] to achieve a similar conclusion from the existence of a large bounded-degree bag in the topological-minor-free decomposition. Several technical steps are needed to translate this into the desired win/win, due to the existence of vortices, virtual edges, and the lack of a direct polynomial-time algorithm to compute the decomposition.

To prove Theorem 1.2, we introduce a new algorithmic tool for finding irrelevant vertices for the -Path problem in the presence of a modulator  in the input graph . Since Theorem 1.1 can be applied to find a -path in  if one exists, the challenge is to detect a -path in  that jumps between  and  several times. The absence of a -path in  implies it has a tree decomposition of width  and constant adhesion. Using Theorem 1.1 as a subroutine, along with a packing argument, we can compute a vertex set  of size polynomial in  with the following guarantee. If there is a -path, then there is a guarded -path  in which each successive pair of vertices in  are connected by a subpath through  that intersects . Using the tree decomposition of , the standard ancestor-marking technique allows us to identify a vertex subset  of  that is adjacent to constantly many vertices from . Unless  is already small, we can find such a set  that is sufficiently large to be reducible but small enough that we may invoke the oracle for questions about it. We can then reduce the graph without losing the existence of a guarded -path, by marking a witness for each sensible way in which a constant-size subset from  can connect to prescribed vertices in  through . The fact that  only has constantly many neighbors in  implies that there are only polynomially many relevant choices. We may then safely remove the unmarked vertices.

Organization

After preliminaries in Section 2, we give a generic Turing-style reduction rule for -Path in Section 3. In Section 4 we show that an -minor-free graph either has a -path or a separation that is suitable for reduction. In Section 5 we extend this to topological minors. Finally, in Section 6 we present a Turing kernel applicable when the input graph has a small modulator to a suitable graph class.

2 Preliminaries

Notation.

All graphs we consider are finite, simple, and undirected. A separation of a graph is a pair , such that and there are no edges between and . The order of the separation is . A graph is triconnected if it is connected and cannot be disconnected by deleting fewer than three vertices. When referring to the size of a graph in our statements, we mean the number of vertices.

A tree decomposition of a graph  is a pair  where  is a rooted tree and  is a function that assigns to every node  a subset  of called a bag such that:

• ;

• for each edge , there is a node with ;

• for each , the nodes induce a (connected) subtree of .

The width of is . Its adhesion is . We also call the set the adhesion of , for every edge of . For a decomposition of and a node , the torso, denoted , is the graph obtained from by adding an edge between each pair of vertices in , for every neighbor of in (so each adhesion induces a clique in the torso). Added edges not present in are called virtual edges. For a subtree we write for the union of bags in .

For an edge , let be the connected component of that contains . Let . Observe that the properties of a tree decomposition imply that is a separation with .

A decomposition is connected if for every and its child , if is the subtree of rooted at , we have (i) that is connected, and (ii) that  has edges to every vertex of the adhesion . It is straightforward to turn any decomposition into a connected one without increasing its width nor adhesion. For (i), as long as there exists a pair violating the requirement, make a distinct copy of for each connected component of , restrict the bags of to the vertices of only, and attach as a subtree with the root being a child of . For (ii), while there is a vertex  that has no neighbors in , simply remove  from all bags in .

We will also need the following non-standard complexity measure of a tree decomposition . For every , the number of distinct adhesions for is called the adhesion degree of . The maximum adhesion degree over all nodes is the adhesion degree of the decomposition . Observe that if a tree decomposition has width less than and adhesions of size at most , then its adhesion degree is at most

 h∑i=0(ℓi)≤(1+ℓ)h.

However, in sparse graph classes we can prove a much better bound on the adhesion degree due to linear bounds on the number of cliques in such graphs; cf. Lemma 4.3.

A path decomposition is a tree decomposition where is a path; we will denote it simply as a sequence of bags .

For an integer , a tangle of order in a graph is a family of separations of order at most such that, for every separation of order at most in , exactly one of the separations and belongs to . If we call the small side of the separation and the large side. Furthermore, we require that for every three separations , , in , we have .

3 Turing kernels

In this section we introduce a general toolbox and notation for proving our Turing kernel bounds.

3.1 Definitions and the auxiliary problem

For a parameterized problem and a computable function , a Turing kernel of size is an algorithm that solves an input instance of in polynomial time, given access to an oracle that solves instances of with . A Turing kernel is a polynomial one if is a polynomial.

If we are only interested in distinguishing between NP-complete problems admitting a polynomial Turing kernel from the ones that do not admit such a kernel, we can assume that the oracle solves an arbitrary problem in NP, not necessarily the -Path problem. Indeed, note that by the definition of NP-completeness, an oracle to a problem in NP can be implemented with an oracle to -Path with only polynomial blow-up in the size of the passed instances.

In our work, it will be convenient to reduce to the Auxiliary Linkage problem, defined as follows. The input consists of an undirected graph , an integer , a set of terminals , and a number of requests ; a request is a set of at most two terminals. A path in is said to satisfy a request if and every vertex of is an endpoint of . With such an input, the Auxiliary Linkage problem asks for a sequence of paths such that satisfies for every , , and every vertex of is contained in at most one path (i.e., the paths are vertex-disjoint, except that they may share an endpoint, but only if the requests ask them to do so).

We remark that Auxiliary Linkage is a more general problem than -Path: an instance with , , , , and asks precisely for a -path in .

Clearly, the decision version of the Auxiliary Linkage problem belongs to the class NP. By using its self-reducibility (cf. [18, Lemma 2]), we assume that the oracle returns a sequence of paths in case of a positive answer. That is, in all subsequent bounds on the number of Auxiliary Linkage oracle calls, the bound adheres to the number of calls to an oracle that returns the actual paths ; if one wants to use a decision oracle, one should increase the bound by the blow-up implied by the self-reducibility application (i.e., at most for calls on a graph ).

3.2 Generic reduction rule

We now show a generic reduction rule for the -Path problem. We start with a few definitions.

Definition 3.1.

For a graph , a subset , and a simple path in , an -traverse of is a maximal subpath of that contains at least one vertex of and has all its internal vertices in .

Note that if is an -traverse of , then every endpoint of is either an endpoint of or lies in . See Figure 1.

Definition 3.2.

Let be a graph, , and let be an integer. A set is called a -guard of if the following implication holds: if admits a -path, then there exists a -path in that is either contained in or such that every -traverse of has at least one endpoint in .

Given a graph , a set , and a -guard of , a -path satisfying properties as in the above definition is called guarded (w.r.t. , , and ). If the integer and the set are clear from the context, we call such a set simply a guard.

Observe that is always a guard, but sometimes we will be able to find smaller ones. Of particular interest will be guards of constant size, as our kernel sizes will depend exponentially on the guard size. To describe our single reduction rule, we show how solutions to Auxiliary Linkage can be used to preserve the existence of guarded -paths.

Assume we are given a graph , a set , an integer , and a -guard of . Let and . Furthermore, assume that admits a -path, and let be a guarded one w.r.t.  and . Let be the -traverses of , let for , let , , and let . Observe that is a feasible solution to the Auxiliary Linkage instance ; the instance is henceforth called induced by and . Furthermore, it is easy to see that if is a different feasible solution to , then a path obtained from by replacing every subpath with is also a guarded -path in .

The crucial observation is that a small guard limits the number of -traverses.

Lemma 3.3.

The number  of traverses of the guarded -path  is bounded by .

Proof.

Every vertex of can be an endpoint of at most two traverses. If , then none of the traverses are contained in , and thus every traverse has at least one endpoint in the guard . ∎

Lemma 3.3 in turn limits the number of possible instances that can be induced by a guarded -path, for a fixed set and guard . Note that we have and . Furthermore, unless and , we have , , and every set needs to have at least one element of ; there are at most choices for such a set . Consequently, the number of possibilities for the instance is at most

 (k+1)⋅(1+2h∑r=0hr(ℓ+1)r)≤(k+1)⋅(h(ℓ+1))2h+1=:p(k,ℓ,h). (1)

Reduction rule.

If , then we can apply the following reduction rule. For each Auxiliary Linkage instance out of at most reasonable instances for -traverses of a guarded -path in , we invoke an oracle on the instance , and mark the vertices of the solution if the oracle finds one. The whole process will mark at most vertices, thus at least one vertex of will remain unmarked. We delete any such vertices.

The observation that on a guarded -path one can replace a solution to the instance induced by and by a different solution provides safeness of this reduction. Finally, note that the reduction invokes at most calls to the oracle; each call operates on a subgraph of the graph with and .

We shall apply the Reduction Rule for a medium-sized set and a guard set of constant size formed from adhesions of a tree decomposition. For most of the paper we will use with a constant (depending on the excluded (topological) minor, in the results of Sections 4 and 5). Only in Section 6, when dealing with a modulator such that has an appropriate structure, it will be important to consider potentially containing all of , with a guard set of constant size disjoint from .

3.3 Separation oracles

The natural way of using our reduction rule is to find in a graph a large (but not too large) part of the graph with a small (preferably, constant) boundary. Let us first make an abstract definition of an algorithm finding such a separation.

Definition 3.4.

For a graph class , a constant , and a computable coordinate-wise nondecreasing function , an algorithm is called a -separation oracle if, given a graph and integers and , in time it finds a separation in of order at most with , or correctly concludes that contains a -path.

For all considered graph classes, we will be able to provide a separation oracle with being a polynomial. This, in turn, allows the following generic Turing kernel.

Lemma 3.5.

Let be a -separation oracle for a hereditary graph class . Take . Then, the -Path problem restricted to graphs from can be solved:

• [-,itemsep=4pt]

• in time ,

• using at most calls to Auxiliary Linkage

• each call on an induced subgraph of the input graph of size at most .

Proof.

Let

 p=k⋅p(k,h,h)≤k(k+1)(h(h+1))2h+1≤2k2(2h)4h+2≤k2^h.

As long as , we proceed as follows. Invoke algorithm on . If claims that admits a -Path, we simply output the answer yes. Otherwise, let be the separation output by . Apply the Reduction Rule for , , and . Note that as , the Reduction Rule deletes at least one vertex of . Furthermore, the Reduction Rule invokes at most

 p(k,h,h)≤k^h

calls to the oracle, each call on an induced subgraph of of size at most

 |A′|≤q(k,p)=q(k,k2^h).

Once we obtain , we solve the instance using a single call to Auxiliary Linkage with , , and . The bounds follow, as there are at most applications of the Reduction Rule, and each call to the oracle takes time to prepare the instance and parse the output. ∎

Note that for any graph class where separations as in Definition 3.4 exist, there exists a trivial separation oracle which finds them, running in time : one iterates over every candidate for and, for fixed set , a straightforward knapsack-type dynamic programming algorithm checks if one can assemble of the desired size from the connected components of .

However, this running time bound is unsatisfactory, as it greatly exceeds the number of used oracle calls. For all considered graph classes we prove a much stronger property than just merely the prerequisites of Lemma 3.5, in particular providing a more efficient separation oracle. We provide necessary definitions in the next section.

3.4 Decomposable graph classes

The following definition captures the key concept of this section.

Definition 3.6.

For a constant and a computable nondecreasing function , a graph class is called -decomposable if for every positive integer and every that does not admit a -path, the graph admits a tree decomposition of width less than and adhesions of size at most .

A standard argument shows that in a decomposable graph class, given the decomposition with appropriate parameters, it is easy to provide a separation oracle.

Lemma 3.7.

Assume we are given a graph and a tree decomposition of of width less than , adhesion at most , and adhesion degree at most . Then, given an integer such that , one can in time find a separation of order at most such that

 p<|A|≤w+p⋅a.
Proof.

Root the tree in an arbitrary node, and for let be the subtree of rooted in . Let be the lowest node of such that ; such a node can be computed in linear time in the size of and .

Group the children of according to their adhesions . Due to the bound on the adhesion degree, there are at most groups. For every adhesion , let be the set of the children of with . Define

 VS=⋃t′∈Xt,SX(Tt′).

We consider now two cases. First, assume that for every adhesion . Then, by the adhesion degree bound, we have

 |X(Tt)|≤|X(t)|+ap≤w+ap.

Consequently, we can return the separation with and .

In the other case, there exists an adhesion with . We greedily take a minimal subset such that

 V′S:=⋃t′∈Yt,SX(Tt′)

is of size greater than . By the minimality of , for every we have and, consequently . Thus, we can return the separation for and , as then . ∎

A critical insight is that the decomposition used by Cygan et al. [9] to solve the Minimum Bisection problem in fact provides an approximate decomposition in a decomposable graph class. Let us first recall the main technical result of [9].

Definition 3.8.

A vertex set of a graph is called -unbreakable if every separation of order at most satisfies or .

Theorem 3.9 ([9]).

There is an algorithm that given a graph and integer runs in time and outputs a connected tree decomposition of such that: (i) for each , the bag is -unbreakable in , and (ii) for each the adhesion has at most vertices and is -unbreakable in .

Lemma 3.10.

Let be a graph and suppose there exists a decomposition of of width less than , adhesion , and adhesion degree . Let be a tree decomposition of such that for each , the bag is -unbreakable in . Then .

Proof.

Consider a bag of the second decomposition, or any -unbreakable set . For every edge of the -decomposition, removing it partitions into subtrees and containing and , respectively. The induced separation on has order at most , so one of the sides, say , contains at most vertices of , by definition of unbreakability. Let us orient the edge away from (the ‘smaller’ side). By orienting each edge of this way, we find a single node of such that all incident edges point to it. That is, for every neighbor of , we have .

While can have many neighbors, we can group those neighbors according the adhesion to which they correspond. By the adhesion degree bound, there are at most such groups. If for any such group , the union contained more than vertices of , then the group can be partitioned into two parts with more than vertices of each. This would give a separation of order of with too many vertices of on both sides, contradicting its unbreakability. Therefore, there are at most groups, each containing at most vertices of , thus the size of is bounded by

Thus, the decomposition computed by the algorithm of Theorem 3.9 approximates the desired decomposition of a decomposable graph class.

Corollary 3.11.

Let be a -decomposable graph class. Then, for every and every integer , one can in time either correctly conclude that admits a -path, or find a tree decomposition of of width at most and adhesion at most .

Let us now combine all the above. That is, given an integer and a graph from a hereditary -decomposable graph class , we start by computing the tree decomposition of Corollary 3.11 (or conclude there is a -path). In general this approximated decomposition has adhesion degree . We use this decomposition to find separations of any induced subgraphs of using the algorithm of Lemma 3.7 in time times linear in the size of and the computed decomposition. This gives a -separation oracle with and , for any hereditary -decomposable graph class. By plugging it into Lemma 3.5, we obtain the following.

Corollary 3.12.

Let be a hereditary -decomposable graph class. Then, the -Path problem, restricted to graphs from , can be solved in time using calls to Auxiliary Linkage on induced subgraphs of the input graph of size .

In the next section, we prove that -minor-free graphs are -decomposable by analyzing the Global Structure Theorem of -minor-free graphs due to Robertson and Seymour. A subsequent section provides an analogous result for graphs excluding a fixed topological minor. In both cases we also get better bounds on the adhesion degree of the approximate decomposition outputted by Theorem 3.9, improving the bounds in the final kernel.

We would like to remark that we do not want to claim in this paper the idea that, in the context of -(topological)-minor-free graphs, the decomposition of Theorem 3.9 should be related to the decomposition of the Global Structure Theorem via an argument as in the proof of Lemma 3.10. In particular, this observation appeared previously in a work of the second author with Daniel Lokshtanov, Michał Pilipczuk, and Saket Saurabh [21].

4 Excluding a minor

In this section we tackle proper minor-closed graph classes, that is, we prove Theorem 1.1 for graph classes excluding a fixed minor, by proving the following.

Theorem 4.1.

For every fixed graph , the -Path problem restricted to -minor-free graphs can be solved in time using calls to Auxiliary Linkage on instances being induced subgraphs of the input graph of size .

Our main technical result is the following:

Theorem 4.2.

For every graph , the class of -minor-free graphs is -decomposable for and .

By plugging the above into Corollary 3.12, we obtain the desired polynomial Turing kernel, but with worse bounds than promised by Theorem 4.1. To obtain better bounds, we need to recall the folklore bound on the adhesion degree in sparse graph classes; for completeness, we provide a full proof in Appendix A.1.

Lemma 4.3.

Let be a graph not containing as a topological minor, and let be a connected tree decomposition of of width less than and adhesion . Then the adhesion degree of is bounded by for some integer depending only on and .

This way, we conclude that -minor-free graphs without -paths have tree decompositions of width , adhesion and adhesion degree . We can use the algorithm of Theorem 3.9 for to find (by Lemma 3.10) an approximate decomposition of width , adhesion , and thus, again using Lemma 4.3, of adhesion degree . Theorem 4.1 follows from Lemma 3.5 if we find separations using the algorithm of Lemma 3.7 applied to this decomposition.

Thus, it remains to prove Theorem 4.2. For the proof, we use the graph minors structure theorem, decomposing an -minor-free graph  into parts ‘nearly embeddable’ in surfaces (precise definitions are given in the next subsection). By carefully analyzing details of the structure, we either find a large triconnected embedded part, which must contain a long path by the following theorem of Chen et al. [8], or we tighten the graph structure to give a tree decomposition where all parts are small (polynomial in ) and adhesions (‘boundaries’) between them are of constant size.

Theorem 4.4 ([8]).

There is a constant such that for every integer , every triconnected graph on vertices embeddable in a surface of (Euler) genus contains a cycle of length at least .

We note that Chen et al. phrase the theorem (more generally) for -minor-free graphs, but a folklore edge-counting argument shows that graphs embeddable on a surface of genus are -minor-free for (see e.g. [2]).

Two intertwined problems that arise with this approach is that torsos of decompositions are not necessarily triconnected, and long paths in them do not necessarily imply long paths in the original graph, because of virtual edges added in torsos. Torsos can be made triconnected if their near-embeddings include cycles or paths around each vortex, but these may use virtual edges in essential ways. On the other hand, the decomposition can be modified so that virtual edges can be replaced with paths in the original graph, but this requires changes that remove virtual edges, hence potentially removing paths around vortices and destroying triconnectedness.

Because of that, we need to go a little deeper and use a local, strong version of the structure theorem from Graph Minors XVII [25]. For the same reason we cannot use existing algorithms for finding the graph minors decompositions. Instead, we only prove the existence of a tree decomposition of bounded adhesion, small width, and with nearly embeddable bags.

Global and local graph minor structure theorems

We now define near-embeddability and the graph minors -decomposition.

Definition 4.5.

For an integer , an -near embedding of consists of:

1. [(i)]

2. a set of at most vertices (called the apex set);

3. a family of edge-disjoint subgraphs of , where:

• is called the embedded part,

• for some are called (large) vortices,

• the intersection is called the society of vortex ,

• vortices are pairwise vertex-disjoint,

4. an embedding of in a surface of genus at most such that for , the society is embedded on the boundary of a disk whose interior is empty (i.e., does not intersect the embedding or other disks), called the disk accomodating ;

5. a linear ordering of each vortex society , corresponding to its natural ordering around its disk (for some choice of direction and starting point);

6. for each large vortex , a path decomposition of of width at most such that , for .

We denote such an -near embedding as , with the embedding and path decompositions only implicit in the notation.

Definition 4.6.

A (graph minors) -decomposition of a graph consists of:

• a rooted tree decomposition of of adhesion at most ;

• for each , an -near embedding ( of .

In this subsection we prove the following variant of the Global Structure Theorem which implies Theorem 4.2.

Theorem 4.7.

For every graph , there is a constant such that the following holds, for any integer : any graph excluding as a minor and without a -path has an -decomposition of width at most .

We deduce Theorem 4.7 from a similarly modified variant of the Local Structure Theorem. An -near local embedding of a graph is defined similarly to an -near embedding, but we allow an arbitrary number of ‘small vortices’ and we allow the path decompositions of all vortices to have only bounded adhesion instead of bounded width; thus arbitrarily complicated graphs can hide behind vortices. We additionally require each large vortex to be surrounded by a certain path. Formally:

Definition 4.8.

A comb is a union of a path with some mutually vertex-disjoint (possibly trivial) paths that have their first vertex and no other vertex on . The last vertices of those paths are called the teeth of the comb, they are naturally ordered by .

Definition 4.9.

For an integer , an -near local embedding of consists of:

1. [(i),series=localStructure]

2. a set of at most vertices (called the apex set);

3. a family of edge-disjoint subgraphs of , where:

• is called the embedded part,

• for some are called large vortices,

• are called small vortices,

• vortices intersect only in : for ,

• the intersection is called the society of vortex ,

• large vortices are pairwise vertex-disjoint,

• small vortices have societies of size ;

4. an embedding of in a surface of genus at most such that for , the society is embedded on the boundary of a disk whose interior is empty (i.e., does not intersect the embedding or other disks), called the disk accomodating ;

5. a linear ordering of each society , corresponding to its natural ordering around its disk (for some choice of direction and starting point);

6. for each large vortex , a path decomposition of of adhesion at most such that for and .

7. for each large vortex , a comb in whose teeth are vertices of , in the same order.

We denote such an -near local embedding as , with the embedding and path decompositions only implicit in the notation. The near embedding is said to respect a tangle if the large side of a separation in is never contained in a vortex or in a bag of the decomposition of a vortex .

Definition 4.10.

In a graph , a tangle controls an -minor if there is a minor model in (defined by branch sets inducing vertex-disjoint connected subgraphs in and with an edge between and whenever ) such that no branch set is fully contained in a small side of a separation in .

The following Local Structure Theorem follows from [25], as explained in [12] (we note that while [12] assumes that is -minor-free, the original statement in [25] only requires that controls no -minor – we will need this stronger version when dealing with topological minors). Since this is crucial for our approach, we stress that the comb contains a path of length at least (a scrupulous reading of [25] implies even stronger statements, but this will suffice).

Theorem 4.11 ([12]).

For every graph there exist integers such that: for every graph and every tangle in of order that controls no -minor, there is an -near local embedding of which respects .

We now improve the statement of the Local Structure Theorem 4.11 in steps: first requiring small vortices to be ‘well attached’, then making the torsos triconnected, and finally deducing bounds on width in terms of longest path length.

Lemma 4.12.

In Theorem 4.11, we can additionally assume that:

1. [(i),resume=localStructure]

2. for each small vortex and every , there is a path in between and , with no internal vertices in .

Proof.

Property 1 is guaranteed by (9.1) in [25]. (As explained in [12], the border cells of [25] are translated into bags of large vortex decompositions, so small vortices arise only from internal cells; for an internal cell , is translated into a small vortex , while is translated into , giving exactly the statement we want. We also note that the proof of (9.1) simply partitions a vortex with no - path into two vortices with at most two vertices in their societies). ∎

Definition 4.13.

For an -near local embedding of , define to be the graph obtained from by adding an edge between every two consecutive vertices in each society , and a new vertex for each society of a large vortex, with edges to every vertex of the society. The embedding of is naturally extended to an embedding of (the new vertices and edges embedded in place of the accommodating disks).

Lemma 4.14.

In Lemma 4.12, we can additionally assume that:

1. [(i),resume=localStructure]

2. is triconnected.

Proof.

Suppose has a separation of order . Since societies of small vortices induce cliques in , they are contained in or . Similarly for each large vortex , its society , together with the new vertex in with as neighborhood, induces a wheel in , which is triconnected, hence contained in or . In the language of [25], this translates back into a partition , of cells, whose intersection corresponds to at most two embedded vertices (that is, ). But this contradicts (11.1) in [25], which states that such an intersection has size at least 3. ∎

Lemma 4.15.

In Lemma 4.14, we can additionally assume that there is a constant depending on only such that:

1. [(i),resume=localStructure]

2. for any integer , if does not contain a path of length , then .

Proof.

For a given graph , let be the constant given by Theorem 4.11. By Theorem 4.4, there is a universal constant such that any triconnected graph embeddable in a surface of genus at most contains a cycle of length at least . Let and . For an -minor-free graph , suppose the -near local embedding given by Lemma 4.14 has . Then is a triconnected graph (by property 1) embedded in a surface of genus at most (by Definition 4.13) with at least as many vertices, so must contain a cycle of length at least .

If the society of any vortex of the -near local embedding of has at least vertices, then contains a path of at least that length (in the comb from property 6), in which case the lemma follows. Otherwise, there are at most large vortices, hence at most vertices in their societies, and at most  virtual vertices that were inserted as the centers of the wheels. Let be the longest subpath of between any two such vortex-related vertices (or any subpath not visiting any such vertices, if there are less than two). Then is a path of length at least in whose edges in came only from small vortices, by the definition of (Definition 4.13). By property 1, these edges can be replaced with paths (each of length at least 2) in the corresponding small vortices, giving a path in . Only one or two consecutive edges can come from the same small vortex (since their societies have at most 3 vertices), so the resulting path has length at least .

The constant from Theorem 4.4 given by Chen et al. [8] is , hence the universal constant here is . ∎

Finally, we deduce Theorem 4.7 (the Global Structure Theorem) from Theorem 4.11 (the Local Structure Theorem) by a standard induction, exactly as done by Diestel et al. [10, Theorem 4]. (Note that in case the excluded graph is planar, we can already conclude the theorem trivially from the fact that there is a tree-decomposition of width  [23]). The only difference is that in the Local Structure Theorem we add the bound on the size of the embedded part from Lemma 4.15. In the proof from [10], every bag of the created decomposition is either constructed as a set of size bounded as , or is constructed from an -near local embedding by taking the vertices of: the embedded part , the set of apices , and for each large vortex, the intersection of every two consecutive bags of its path decomposition, each of size at most . The number of bags in a path decomposition of a large vortex is equal to the size of its society, and large vortex societies are disjoint subsets of . Therefore every tree decomposition bag constructed in the proof has size at most , by Lemma 4.15. This proves the additional condition we require in Theorem 4.7.

5 Excluding a topological minor

In this section we tackle graph classes excluding a topological minor, that is, we prove Theorem 1.1 by proving the following.

Theorem 5.1.

For every fixed graph , the -Path problem restricted to -topological-minor-free graphs can be solved in time using calls to Auxiliary Linkage on instances being induced subgraphs of the input graph of size .

This follows as before from the following decomposability theorem. Note the exponent in the polynomial bound on width (bag size) now depends on .

Theorem 5.2.

For every graph , the class of -topological-minor-free graphs is