Edge-Disjoint Paths in Eulerian Digraphs

Disjoint paths problems are among the most prominent problems in combinatorial optimization. The edge- as well as vertex-disjoint paths problem, are NP-complete on directed and undirected graphs. But on undirected graphs, Robertson and Seymour (Graph Minors XIII) developed an algorithm for the vertex- and the edge-disjoint paths problem that runs in cubic time for every fixed number $p$ of terminal pairs, i.e. they proved that the problem is fixed-parameter tractable on undirected graphs. On directed graphs, Fortune, Hopcroft, and Wyllie proved that both problems are NP-complete already for $p=2$ terminal pairs. In this paper, we study the edge-disjoint paths problem (EDPP) on Eulerian digraphs, a problem that has received significant attention in the literature. Marx (Marx 2004) proved that the Eulerian EDPP is NP-complete even on structurally very simple Eulerian digraphs. On the positive side, polynomial time algorithms are known only for very restricted cases, such as $p\leq 3$ or where the demand graph is a union of two stars (see e.g. Ibaraki, Poljak 1991; Frank 1988; Frank, Ibaraki, Nagamochi 1995). The question of which values of $p$ the edge-disjoint paths problem can be solved in polynomial time on Eulerian digraphs has already been raised by Frank, Ibaraki, and Nagamochi (1995) almost 30 years ago. But despite considerable effort, the complexity of the problem is still wide open and is considered to be the main open problem in this area (see Chapter 4 of Bang-Jensen, Gutin 2018 for a recent survey). In this paper, we solve this long-open problem by showing that the Edge-Disjoint Paths Problem is fixed-parameter tractable on Eulerian digraphs in general (parameterized by the number of terminal pairs). The algorithm itself is reasonably simple but the proof of its correctness requires a deep structural analysis of Eulerian digraphs.


INTRODUCTION
The -disjoint paths problem, that is, the problem of deciding for a given (directed) graph and a set ≔ { 1 , . . ., } of sources and ≔ { 1 , . . ., } of targets, whether there is a set of mutually edge-or vertex-disjoint paths-depending on whether we are talking about the vertex-or Edge-Disjoint Paths problem-connecting the sources to the targets, is one of the most fundamental problems in the area of graph algorithms.By Menger's theorem, or network ow algorithms, this problem can be solved in polynomial time on undirected and directed graphs if we are only interested in a set of disjoint paths each having one end in and the other end in .But the situation changes completely if we require that the paths connect each source to its corresponding target : this problem is NP-complete for edge-disjoint and vertex-disjoint paths, on directed and undirected graphs.
On undirected graphs , Robertson and Seymour developed an algorithm for the -Vertex-Disjoint-Paths and the -Edge-Disjoint-Paths problem which runs in time O ( 3 ) for any xed number of terminal pairs [23] where = | ( )|.Rephrased in the terminology of parameterized complexity, they showed that the problem is xed-parameter tractable parameterized by the number of terminal pairs, i.e., it runs in fpt-time witnessed by a running time of the form ( ) for some constant ∈ N and some computable function : N → N. The complexity has subsequently been improved to quadratic time in [18].Robertson and Seymour developed the algorithm as part of their celebrated series of papers on graph minors.While the correctness proof is still long and di cult, relying on large parts of the graph minors series, the algorithm itself is beautifully concise and essentially facilitates a reduction rule that reduces any input instance to an equivalent instance of bounded treewidth which in turn can be solved using standard dynamic programming techniques.
On directed graphs-henceforth called digraphs-the -Vertexand -Edge-Disjoint-Paths problems are considerably more dicult.As shown by Fortune, Hopcroft, and Wyllie [8], both problems are NP-complete already for = 2 terminal pairs.This implies that they are not xed-parameter tractable and not even in the class (under the usual complexity theoretical assumptions that we tacitly assume throughout the introduction), i.e., they are widely believed to be unsolvable in polynomial time for any xed ≥ 2.
Furthermore, Slivkins [27] showed that the problems remain [1]hard-and therefore (presumably) not xed-parameter tractablealready on acyclic digraphs, which is directed graphs not admitting any cycles.On the positive side, Cygan, Marx, Pilipczuk, and Pilipczuk [6] proved that the -Vertex-Disjoint Paths problem is xed-parameter tractable on planar digraphs.Interestingly, Chitnis proved that the edge-disjoint version remains [1]-hard [4], refuting its xed-parameter tractability under the aforementioned standard assumptions.Eulerian Digraphs.A well-studied class of digraphs whose complexity often turns out to be somewhere between undirected and general directed graphs is the class of Eulerian digraphs.A digraph = ( , ) is Eulerian if the in-degree of each vertex equals its out-degree or, equivalently, if it is the union of a set of edge disjoint cycles.See [2,Chapter 4] for a recent survey on Eulerian digraphs.It has been observed in [17] that the correctness proof of the algorithm for the -Edge-Disjoint Paths problem can be simplied for undirected Eulerian graphs.This already suggests that the 'Eulerian' property could make a di erence for the -Edge-Disjoint-Paths problem also on digraphs.Indeed, Johnson [15] pioneered the structural analysis of Eulerian digraphs with emphasis on solving the -Edge Disjoint Paths problem in his dissertation.He proved a structure theorem for internally 6-connected Eulerian digraphs (a notion that will not be of further relevant to this exposition) in the same avour as the undirected structure theorem proved by Robertson and Seymour, following the same line of argumentation as their proof.Unfortunately, his results have never been published.
In the literature on Eulerian digraphs the Edge-Disjoint-Paths problem is often studied in the following formulation.

De nition 1.1 (Edge-Disjoint Paths problem). The Edge-Disjoint
Paths problem is the problem to decide, given two digraphs and with ( ) ⊆ ( ) and ( ) ∩ ( ) = ∅ as input, whether contains a set L of pairwise edge-disjoint paths which contains for each edge ( , ) ∈ ( ) an − -path ∈ L that is edge-disjoint from .
When xing the number of terminal pairs we are interested in, i.e., for xed ≔ | ( )|, we refer to the problem as the -Edge-Disjoint Paths problem.An equivalent formulation is to decide, given and as above, whether + contains a set of pairwise edge-disjoint cycles each containing exactly one edge of , where + ≔ ( ( ), ( ) ∪ ( )).
We call the supply and the demand digraph.The vertices incident to an edge in are called terminals.It is easily seen that this formulation is (qualitatively) equivalent to the specication of the disjoint paths problem by a single digraph and pairs ( 1 , 1 ), . . ., ( , ) of terminals.One advantage of the presentation with separate demand and supply graphs is that this makes it possible to classify the complexity of the problem relative to the structure of the demand graph.
Unfortunately, the Edge-Disjoint-Paths problem remains NPcomplete on Eulerian digraphs.In fact, Marx [22] proved that the problem is already NP-complete if is an acyclic directed grid graph and + is Eulerian.
On the positive side, Slivkins [27] proved that the problem is xed parameter tractable in the case that is acyclic and + is Eulerian.Further, Frank [9] showed that the problem can be solved in polynomial time if + is Eulerian and consists of two sets of parallel edges or is the union of two stars.Moreover, he showed that in these cases the directed cut criterion is su cient for the existence of a solution; that is, the problem can be solved by deciding whether there exists a set of vertices ⊂ ( ) such that the number of edges in with a head in and tail in ¯ ≔ ( ) \ is less than the number of edges in having a tail in and head in ¯ .
Polynomial time algorithms for a few other special cases (for ≤ 3) if + is Eulerian have been developed in [10,14,28], but in the last nearly 30 years no signi cant progress on determining the complexity of the general -Edge-Disjoint-Paths problem on Eulerian digraphs has been made.However, Johnson [15] proved in his dissertation that given an Eulerian digraph (Euler-)embedded in some surface Σ-we will make this precise shortly-such that there exists a disc Δ ⊂ Σ containing many concentric edge-disjoint cycles of alternating orientation (with respect to the orientation of the disc), then the most deeply nested cycles are irrelevant to the instance.That is, one may delete any such cycle from the graph without altering the outcome of the instance.Unfortunately, since Johnson's work has never been published said result has not been peer-reviewed thus far.
As That is, there is a computable function and an algorithm with running time ( ) • O (1) , which, given an -vertex digraph and a 2 -vertex digraph with ( ) ⊂ ( ) with parameter ≔ | ( )| such that + is Eulerian, decides correctly whether or not contains a set of pairwise edge-disjoint paths which contains an −path for each edge ( , ) ∈ ( ).
We start with recalling some concepts, notation and results relevant for the exposition and continue with a high level overview of the algorithm and its correctness proof in section 3.

PRELIMINARIES AND NOTATION
Throughout this exposition we use standard graph theoretic notation as in [2,7] and assume the reader to be familiar with common graph theoretic concepts and notation.For example, given a subset ⊂ ( ) we write [ ] to denote the subgraph induced by .Given a directed graph = ( , ), we call the graph resulting from when forgetting the edge directions its underlying undirected graph.Further we call a graph pseudo-Eulerian (of order ∈ N) if there exists a graph satisfying | ( )| = such that + is Eulerian.Although our main theorem talks about Eulerian digraphs, most of the results solely rely on the fact that is pseudo-Eulerian, a notion turning out to be central to our results (although not apparent from this exposition).Eulerian graphs are commonly known for the existence of Eulerian cycles, where we de ne cycles slightly di erent than the standard literature.
Note that, by de nition, vertices may be visited several times in a cycle; what we call a cycle is usually referred to as a closed walk in the standard literature.A cycle is Eulerian if it visits all of the edges in ( ).
Further notions of great importance to the paper are induced cuts.
Beautifully, the order of induced cuts is always even for Eulerian digraphs, and given any such the number of edges in ( , ¯ ) with a head in equals the number of edges with a tail in , revealing a nice symmetrical property for induced cuts in Eulerian digraphs.
We further assume the readers to be familiar with general graph embedding concepts such as planarity.We say that is a plane graph if we assume to be given together with a planar embedding.Throughout this exposition we will frequently talk about digraphs embedded on a xed surface Σ, where surfaces in our setting are compact 2-dimensional manifolds possibly with boundary.In these cases we will often work with a speci c type of graph-embedding which we call Euler-embedding.Let + be Eulerian of degree at most four.Then is called Euler-embedded (in some surface Σ) if the embedding contains no strongly planar vertex.A vertex ∈ ( ) of the embedded digraph is called strongly planar if it is of degree four and we can draw a simple closed curve in the surface Σ around the vertex such that intersects exactly all edges adjacent toexactly once and only these-such that visits rst both in-edges and then both out-edges (up to a cyclic rotation) at .
Given , such that + is Eulerian and is a demand graph with | ( )| = for some ∈ N we say that + encodes an instance of the Eulerian Edge-Disjoint Paths problem.That is, given and we are to decide whether there exist edgedisjoint paths 1 , . . ., -we call the collection L = { 1 , . . ., } a -linkage-such that connects to for some ( , ) ∈ ( ) for every 1 ≤ ≤ .(Note that a priori = , = and = are possible for any 1 ≤ , ≤ .)Given such an instance, if the respective edge-disjoint paths exist we call it a YES-instance, otherwise we call it a NO-instance.
Moreover, we assume the readers to be familiar with the notion of undirected treewidth and will at times talk about directed treewidth (see [16,19] for de nitions and results) although the exact de nitions of either notions will not be of importance.The reason why we will not need the directed treewidth is that Eulerian digraphs of bounded degree admit 'high' undirected treewidth if and only if they admit 'high' directed treewidth and thus both notions are qualitatively the same (this is not true in general directed graphs).
Theorem 2.3 (Theorem 2.2 in [16]).Let + be Eulerian and let be its directed treewidth.Let be the undirected treewidth of Figure 1: A cylindrical wall of order 4. The perimeter of the wall is depicted using thick edges.
An important result proved in [19] that is central to our arguments is that high directed treewidth guarantees the existence of a large cylindrical wall W. See g. 1 for a de nition by picture.We refer to the paths marked 1 , . . ., as wall-cycles and to the paths 1 , . . ., 2 as the horizontal paths.We say that the wall-we omit 'cylindrical' whenever we talk about walls in digraphs-is of order or simply a × -wall with a natural extension to ×walls (note that a -wall contains 2 horizontal paths).The vertices in the intersection of horizontal paths and wall-cycles are called coordinate vertices.We may write W = ( 1 , . . ., ; 1 , . . ., 2 ) to mean a × -wall with a speci ed ordering of the wall-cycles and horizontal paths in a xed plane drawing as in g. 1.Thus, tying to the above, whenever we will say 'high treewidth' this implies a 'large cylindrical wall as a subgraph'.
Further, it was shown in [3] that computing such a wall is xedparameter tractable parameterized by the treewidth.In particular, this means that for Eulerian directed graphs we can nd a large cylindrical wall in fpt-time parameterized by the undirected treewidth.Summarising we get the following.Theorem 2.4.There is a computable function : N → N such that every Eulerian digraph + of directed treewidth at least ( ) for ≔ | ( )| contains a cylindrical wall W of order with (W)∩ ( ) = ∅ as a subgraph that can be found in fpt-time on .
In order to use theorem 2.3 and facilitate many of the arguments made in the paper we rst reduce the Edge-Disjoint Paths problem to the class of Eulerian digraphs of maximum degree four via an easy reduction.Lemma 2.5.Let + be Eulerian where is the demand graph with | ( )| = ∈ N. Then + can be reduced in polynomial time to a new instance ′ , ′ such that ′ + ′ is Eulerian and in which each non-terminal vertex has degree 4, all terminals have degree 2, no terminal vertex is part of two edges in ( ) and + is a YES-instance if, and only if, ′ + ′ is a YES-instance.
Throughout the rest of the exposition we will tacitly assume our graphs + to be Eulerian and such that every vertex ∈ ( ) \ ( ) is of degree four and every vertex in ( ) is of degree two unless stated otherwise.

STRUCTURE OF THE PROOF
We continue with the algorithm proving the main theorem 1.2 before giving a high level description of the proof of its correctness.Let + be an Eulerian directed graph of maximum degree four, where is the demand graph encoding an instance of the Edge-Disjoint Paths problem with | ( )| = ∈ N.That is, we want to decide whether there exists a -linkage L = { 1 , . . ., } where connects to for ( , ) ∈ ( ) for every 1 ≤ ≤ .
The Main Idea of the algorithm is to keep reducing the instance using the irrelevant vertex technique-in our case irrelevant cycle technique-until the graph has bounded undirected treewidth whence the instance can be solved in fpt-time using standard techniques (e.g. using Courcelle's Theorem [5]; more precisely an adaptation of it for directed graphs [1]).In a nutshell the irrelevant cycle technique works as follows: as long as the treewidth of is not bounded by a function in , we are able to locate a cycle in the graph whose deletion does not change the existence of a solutionby deleting a cycle we mean the graph − ≔ ( ( ), ( ) \ ( )) after removing possibly isolated vertices.We can therefore repeat this process until, eventually, the graph is of bounded treewidth and proceed as discussed.
Our goal in this paper is to show that the Edge-Disjoint Paths problem on Eulerian digraphs is xed-parameter tractable and not to optimise the running-time of our algorithm.In our correctness proof we establish several results that can be used more explicitly in the algorithm to get a better running-time performance and sometimes we deliberately chose to bloat up the numbers at the cost of a worse overall running time when granting slicker and conciser proofs, circumventing case distinctions or ddling with structural details.
Readers familiar with the work of Robertson and Seymour [23,25,26] on Graph Minors will see the similarities between our algorithm and our approach to prove its correctness and the line of argumentation presented in their papers.To bring the readers not familiar with their work onboard, we will brie y summarise the broad line of argumentation behind the arguments that have relevance for this exposition.
The Undirected Case.There are two classes of graphs central to the work of Robertson and Seymour: • Cliques on ∈ N vertices denoted by : a graph on vertices where every pair of vertices is connected by an edge, • Grids of order : a graph consisting of disjoint paths = ( 1 , . . ., ) of length ( − 1) such that is connected to +1 for every 1 ≤ ≤ − 1, 1 ≤ ≤ .
Robertson and Seymour have shown that given any graph of high treewidth, the graph contains a large grid as a minor-taking a subgraph and contracting edges in that subgraph results in a large grid-and in particular a large wall as a subgraph (the underlying undirected graph of g. 1 after deleting the bent edges is an example of an undirected wall).They then continue to analyse how the rest of the graph is attached to that wall, using the wall as a kind of skeleton for the rest of the graph; like drawing a graph on squared paper, using the corners of the squares as vertices (and adding any missing vertices to the drawing of course).A central result being that, either the remainder of the graph has many attachments to the wall connecting many di erent parts 'reasonably far apart' from each other, in which case they were able to nd a large clique minor in the graph, or, if we cannot nd such a clique minor, then a large part of the wall is 'almost' planar embeddable; it is what they call at.
To understand the intuition behind the de nition of atness, it is important to rst think about how one can construct a clique-minor starting from a wall.The crucial concept to the existence of large clique-minors is the existence of many disjoint 'non-planarities', i.e., crosses.Of course, when working with drawings, the exact edges that cross in the drawing depend on the drawing at hand.This leads to the following de nition of crosses.
Intuitively the idea here is, that when trying to draw in a disc with the vertices Ω drawn on the boundary of the disc in its prescribed order, the paths 1 , 2 cross in the drawing in the common sense.Thus, when trying to nd a clique-minor, one tries to locate many distinct crosses spread all over a plane wall, using the wall to connect them.However, it is not hard to see that given a pair ( , Ω) as above it may well be the case that there is no Ω-cross in despite being highly non-planar.To see this, think of a plane wall, Ω being its four corners in clockwise order, where we attach a large to a single vertex somewhere inside the wall.This example can be strengthened as we see next.
Flat Walls.The duality between nding a large clique-minor or a at wall is one of the many cornerstone results in the graph minor series, a theorem that was later on named the Flat Wall Theorem [20].In a nutshell, a wall W ⊆ can be thought of as at if its perimeter (the outer cycle when drawing it in the plane) is separating in -marking the outside and inside of the wall in a sense-such that the inside is 'cross-free'.More precisely, deleting ( ) from let ⊂ \ ( ) be the unique component with (W) ∩ ( ) ≠ ∅.Then we say that W is at in if there exists no Ω-cross in the 'inside of the wall', that is for ( [ ∪ ], Ω) where Ω contains the four corners of the wall in clockwise order.The graph [ ∪ ] is formally referred to as the compass of W; note that W ⊆ [ ∪ ].It turns out that being at has rather strong impacts on the topological structure of the compass as we brie y discuss next.
(Non)-Planarity and the Two-Paths Theorem.Planar graphs behave very rigidly when it comes to routing problems: if a path traverses an undirected wall completely (it has both its ends on the perimeter), it intuitively cuts the wall into two disjoint parts since no other path is allowed to cross the rst one when we require the paths to be vertex-disjoint.Non-planarities allow for edges that can 'hop over' the rst path making it possible for pairs and sets of paths to cross: take a wall and add two crossing edges connecting diagonally opposite corners for each face, then the resulting graph is highly non-planar, it contains large clique-minors, and allows to cross paths easily using said non-planarities.In fact, the relation between planarity and rigidness in routing can be measured by what is called the Two-Paths Theorem which sees a proof in the graph-minors series due to Robertson and Seymour.Essentially, the Two-Paths Theorem proves that given a pair ( , Ω), either we can 'nicely embed' into a disc drawing Ω on its boundary respecting its order, or admits an Ω-cross.By a 'nice embedding' we mean that can be planar embedded in a disc up to ≤ 3-separations, which are replaced by a respective clique on ≤ 3 vertices in the embedding.
A very important implication of the Two-Paths Theorem is that at walls can be nicely embedded xing Ω to be the corners of the wall as above.That is, given a wall with corners 1 , 2 , 1 , 2 in clock-wise order and perimeter , the Two-Paths Theorem implies that either we nd two 'crossing paths' connecting 1 to 1 and 2 to 2 in the compass of the wall refuting its atness, or we can embed the compass (after replacing ≤ 3-separations by respective cliques on the ≤ 3-vertices) into a disc with the corners of the wall embedded on the boundary of the disc.The fact that it is 3separations crucially helps with solving the Disjoint Paths problem: no two vertex-disjoint paths starting and ending outside of the at wall can enter the wall and both use (enter and leave) a part attached via a ≤ 3-separation to the wall, for there are not enough vertices to disjointly enter and leave said part.
The Structure Theorem.With the Flat-Wall Theorem in hand, Robertson and Seymour proved that given an integer ∈ N and a graph that does not admit a -minor, then we can decompose the graph into chunks that can be glued together at vertices in a tree-like fashion such that two parts overlap in only a few vertices with respect to -a tree-decomposition of low adhesion-such that every chunk can be 'almost' embedded-up to a bounded number of apices and vortices which we will introduce shortly-in a surface Σ of genus bounded in , where every chunk 'uses up' the surface.By 'using up the surface' we mean that the graph cannot be embedded in a lower surface unless we delete an large (not boundable in ) number of vertices-the embedding has high representativity.That is, given such a chunk ′ , after deleting a few special vertices called apices (think of them as vertices connected to all other vertices of the graph, hence introducing a lot of non-planarities but they can be used by at most a single path), we can embed the graph in a surface of genus bounded in up to ≤ 3-separations and a few highly local non-planar regions called vortices that may be harder to disconnect.One may think of a vortex as an 'untamed' subgraph of that given the above embedding is drawn in a disc Δ ⊂ Σ (although not planar) where one usually by cuts a hole into Σ along the boundary of Δ and pushes , and thus the local non-planarities, into the the hole such that is attached to the boundary of the hole (drawing its attachment vertices on the boundary of the hole).Further the above embedding guarantees the vortices to be of bounded depth, i.e., given any such vortex there is no large linkage between any two halves of the boundary of the hole (being Δ) no matter how we choose the halves; so the boundary is rather loosely connected through the hole.In order to get to the above structure theorem one can start by embedding a at wall using the Flat-Wall Theorem and then extend the embedding from there by introducing handles, cross-caps, apices and vortices (see also [21]).
Routing Paths Disjointly.Given the above structure theorem, the fpt-algorithm for nding vertex-disjoint paths works in three major steps: Either the graph has a -minor, in that case Robertson and Seymour locate a vertex of the -minor that is irrelevant to the problem (think of a large clique and trying to route two paths.Then it seems intuitive that the paths do not need all of the vertices of the clique, for every vertex is connected to every vertex and thus no path needs to enter the clique twice).Thus we may assume that the graph has no large clique-minor left after deleting said vertices, in which case we can nd the above described embedding for the graph in fpt-time.
In a next step Robertson and Seymour proved that, given the 'quasi-embedding' from the structure theorem above, one can locate a vertex deeply nested inside the at wall that is irrelevant to the instance and simply delete it (this proof is very technical and far from easy; in the planar case it seems intuitive that it should be true, for no path should need to use much of a wall as one cannot cross any paths in it).The proof works via several inductions again using three major steps.First it is shown that the theorem holds true for planar graphs containing a large wall (which is clearly at).This is then leveraged to graphs embeddable on xed surfaces by induction on the genus of the surface.In a last step the proof is extended to the 'quasi-embeddings' by proving that solutions do not enter and leave vortices too often, reducing it to the embedded case, in a sense 'killing vortices'.(We omit a discussion about apices as we will not encounter that problem ourselves).
Finally one repeats both steps above, deleting vertices until the graph has no large wall left and is thus of low treewidth; use Courcelle's theorem from here.
Back to Eulerian Digraphs While many of the arguments and techniques we use are highly inspired and follow the same line of reasoning as Robertson and Seymour's, we note that our proofs, the respective constructions, and ideas behind them are in no way easily derivable from the results presented in [23,25,26].For example the standard graph-minor structure theorem due to Robertson and Seymour [24] is of no direct use to us.One reason being that there is no straightforward argument how undirected clique-minors help in routing directed edge-disjoint paths; in particular edge-contractions do change the instance.Also, knowing that there are no undirected clique-minors left is no real help either, as given a drawing of + in the plane in no way yields enough structure to forbid certain edge-disjoint linkages in the graph.To see this note that strongly planar vertices may still help in crossing paths in the same spirit as discussed above-we will make this more precise shortly-a phenomenon that in the (undirected) vertex-disjoint case only appears if the graph itself is non-planar (as given by the Two-Paths Theorem).But taking any drawing of a non-planar graph and adding vertices at points of crossing edges results in a planar graph which has at least the possible edge-disjoint paths as the non-planar graph and at most an undirected 4 -minor.However, vertices of degree four that are not strongly planar lose said intuition and do not allow to cross paths in the same spirit; both strongly planar vertices and non-strongly planar vertices cannot be distinguished in the underlying undirected graph.Hence it is not obvious how to leverage the undirected graph-minor structure theorem-and in fact we do not in this paper-neither is it obvious how to use clique-minors (not even directed clique minors) for routing, which is why we do not.Note further that many results in the graph-minor structure theory rely on inductive reasoning, separating the graphs into smaller graphs, deleting parts of the vertices and edges, splitting vertices, contracting subgraphs; arguments that cannot easily be transferred to the Eulerian setting, for they may destroy the Eulerianness of the graph or, in the latter case, augment the set of solutions by creating new ways to route the paths edge-disjointly.This problem required us to develop new techniques that are tailored towards Eulerian digraphs and the edge-disjoint case, bringing to light a deeper structural understanding of both.
More generally, the fact that the graphs in question are directed makes algorithmic problems often harder, and especially in the eld of structural graph theory the direction of edges has turned out to be a major nuisance in the past; for example there is no directed analogue of the aforementioned Two-Paths Theorem which in the undirected setting, as elaborated above, is used in [23] to prove nice embedding properties for the at wall given the absence of large clique-minors.Also, perhaps somewhat counter-intuitively at rst, and certainly in contrast to the undirected case, given a cylindrical wall with many disjoint non-planarities (that is, crosses) that are pairwise 'far apart' on the wall does in general not yield a directed clique-minor [12]; whereas on undirected graphs it does, an observation that lies at the core of the undirected Flat-Wall Theorem [20,23].In particular, topological obstructions do not as easily infer the existence of large clique minors.Fortunately, most nuisances seem to disappear when focusing on Eulerian digraphs and there are other routing devices that help with routing paths (edge-)disjointly.

Revisiting Notation
We start with introducing the most relevant notions needed throughout the remainder of this exposition.
In our setting the irrelevant cycles found by the algorithm are either cycles of some large Router-a collection of edge-disjoint cycles that pairwise intersect-or cycles deeply nested inside an Euler-embedded at swirl-a collection of edge-disjoint concentric cycles that alternatingly change their orientation (with respect to the orientation of the plane they are embedded in).

De nition 3.2 (Routers and Swirls
is a graph consisting of edgedisjoint cycles with 1 ≤ ≤ such that they pairwise intersect.
and two consecutive cycles and +1 have di erent orientation with respect to a given orientation of the plane for 1 ≤ < ≤ .
Let be an Eulerian digraph and W a large cylindrical wall in .A -tile ⊂ W is a connected subgraph whose underlying undirected graph is an undirected × 2 -wall (see g. 3).We say that an -swirl S = 1 ∪ . . ., ∪ is induced by W if there is antile ⊂ W such that ⊂ S; see g. 2 for an example.Intuitively, routers and induced swirls will be to Eulerian digraphs what cliques and walls are to undirected graphs.
Next we de ne the notions of crosses relevant to the exposition, the rst of which are wall-usable crosses, i.e., crosses that are only readily usable when going with the ow of the cylindrical wall.De nition 3.3 (Wall-usable crosses).Let ∈ N and let W = ( 1 , . . ., ; 1 , . . ., 2 ) be a plane cylindrical -wall.Let ⊂ W be some tile of the wall with corners 1 , 2 , 1 , 2 visited in clockwise order such that 1 , 2 lie on a common wall-cycle ℓ and 2 , 1 lie on a common wall-cycle for some 1 ≤ ℓ < ≤ .Then we say that there is a wall-usable -cross if there exist edge-disjoint paths 1 , 2 ⊆ − (W − ) such that connects to for = 1, 2.
When investigating swirls induced by walls, they do not have any apparent ow direction.This leads to the following notion of swirl-usable crosses which, in turn, comes with a natural de nition of at swirls.De nition 3.4 (Swirl-usable crosses and at swirls.).Let be Eulerian and S = ( 1 ∪ . . .∪ ) ⊆ be an -swirl induced by an -tile ⊂ with denoting the outer-cycle of S. We dene S [ ] ≔ ∪ where is the unique component of − containing 1 .Let 1 , 2 , 1 , 2 be the four corners of appearing in clockwise order given a plane embedding of S. Then S admits a swirl-usable -cross if there exist two disjoint paths 1 , 2 ⊆ S [ ] such that connects to or vice-versa for = 1, 2. We call S at if it does not admit a swirl-usable -cross.
In particular the above reveals that a plane graph may contains swirl-usable crosses if it contains strongly planar vertices.It turns out that if it does not contain (well-connected) strongly planar vertices, then there is no swirl-usable cross.This is a directed version of the Two-Paths Theorem for Eulerian digraphs as we will clarify later.

The Algorithm
Our algorithm exploits that, given a large router in our graph , that router contains some cycle whose deletion does not change the instance.In particular we present an algorithmic approach that starts with a cylindrical wall W and either nds said router grasped by W-think of this as the router being well connected to the wallor a at swirl induced by W; this is given as the Flat-Swirl Theorem (see section 3.3 for more details).Then, after at most | ( )| steps there is no Router left and either the treewidth of the graph is low (bounded in ) or the treewidth of the graph remains high.If the treewidth of the graph is low (with respect to ) the problem can be solved using a variant of Courcelle's theorem adapted to directed graphs as pursued in [1].Thus, assume the treewidth is still high.The Flat-Swirl theorem 3.5 then implies the existence of (as well as an algorithmic way to nd) a large at swirl S in .Given S we show that there exists an irrelevant cycle deeply nested inside the at swirl.This is presented as theorem 3.12, the proof of which needs a lot of preparation and machinery.In either case, router or at swirl, we are able to inductively reduce the input instance to an equivalent instance of low treewidth in fpt-time on , which in turn is an instance that we can solve in fpt-time as discussed.We proceed by giving the algorithm, proving the main theorem 1.2 of this paper, referring to theorems we will only introduce and discuss subsequently.Proof of theorem 1.2.Let + be an instance of the directed Eulerian Edge-Disjoint Paths problem such that + is of maximum degree four and every terminal vertex is of degree two.Let ≔ | ( )|.Let 1 ( ) ≔ 3.9 ( ) and 2 ( ) ≔ 2ℎ 3.12 ( ).Given 1 , 2 de ne 1 ( ) ≔ 3.5 ( ; 1 , 2 ).And nally let ( ) ≔ 2.4 ( 1 ( )).
The following algorithm decides the instance in fpt-time on .
1. Determine whether tw( + ) ≤ 6 • ( ), which can be done in fpt-time on .If this is the case, then we can solve the instance using a version of Courcelle's theorem for directed graphs [1] in fpt-time on .Otherwise continue with Step 2. 2. Since tw( + ) ≥ 6 • ( ), theorem 2.3 implies that dtw( + ) ≥ ( ).Using theorem 2.4 we deduce that there is an 1 ( ) × 1 ( )-wall W in (away from ) which can be found in fpt-time on .Using the Flat-Swirl theorem 3.5 we deduce that either we can nd a at 2 ( )-swirl S induced by a same-sized tile ⊂ W or a 1 ( )-router R grasped by W in away from ( ) in fpt-time on .If we nd a router go to Step 3; else proceed with Step 4.
3. If we have found a 1 ( )-router R, use the irrelevant cycle theorem 3.9 for routers to nd and delete a cycle ⊂ R that is irrelevant to the instance in fpt-time on .That is, the graph ( − ) + is Eulerian and an equivalent instance to + .After having successfully reduced the instance, go back to Step 1 and start over.4. If we have found a at 2 ( )-swirl S induced by some samesized tile use theorem 3.8 to construct an equivalent instance ′ + with | ( ′ ) ∪ ( ′ )| ≤ | ( ) ∪ ( )| together with a separation ( , ) with ∪ = ′ in polynomial-time such that ′ [ ] contains a at 2 ( )-swirl S ′ with ( ) ∩ ′ [ ] = ∅ and ′ [ ] can be Euler-embedded in a disc.Finally, using the embedding of ′ [ ], the irrelevant cycle theorem 3.12 for at swirls yields an irrelevant cycle ⊂ ′ to the instance nested deeply inside S ′ in fpt-time on .Thus ( ′ − ) + is a reduced and equivalent instance; go back to Step 1 and start over.We continue with a dissection of the above proof, providing further (high-level) details concerning each of the steps; the rst of which is self-explanatory and rather trivial.It is noteworthy that for general directed graphs the directed treewidth may not be bounded by the undirected treewidth in any meaningful way, e.g.acyclic grids which have directed treewidth 1, but contain a large underlying undirected grid and hence have high undirected treewidth.Thus, the Eulerianness (as well as the assumption on the bounded degree) is crucial for the rst step of the algorithm.Note further that there is no direct analogue to Courcelle's Theorem [5] for directed treewidth, again highlighting that the Eulerianness is crucial for our algorithm.

The Flat-Swirl Theorem
The Flat-Swirl Theorem is in the same spirit as the undirected Flat-Wall Theorem as seen in [23] or the directed Flat-Wall Theorem as pursued in [12] and the many more grid-like theorems lying at the heart of graph structure Theorems [11,13].
In a nutshell the Flat-Swirl Theorem states that given high (un)directed treewidth in an Eulerian digraph of maximum degree four, we either nd a large router or a large at swirl.The proof of the theorem has three major steps.In a rst instance, given high directed treewidth we nd a large cylindrical wall W in fpt-time on using theorem 2.4.We continue with analysing how the rest of attaches to W.
Finding and Untangling a Swirl.Given an Eulerian graph containing a -wall W = ( 1 , . . ., ; 1 , . . ., 2 ), we rst prove that either we nd a large swirl induced by the wall (not necessarily at), or we nd a large router grasped by the wall.The topological gadgets of interest in this step are the wall-usable crosses.The swirl we construct in a rst instance will be what we call tangled-not to be confounded with the notion of tangles introduced by Robertson and Seymour, but to be taken in the gurative sense-that is, the swirl when taken as a subgraph may still contain (wall-or swirl-) usable crosses.
To nd said swirl, note that the coordinates of the wall W ⊂ have degree 3 in W, while said vertices have degree 4 in ; there is an in-edge or out-edge missing for each coordinate vertex highlighted in red in g. 3. Using this insight we analyse how the remaining paths in − W that start (or end) in wall-coordinates attach to the wall.Given an embedding of we may assume that the cycles of the wall run collectively in the same direction, say clock-wise, lies 'left of' +1 and, similarly, the horizontal path lies 'above' +1 .Finally we denote by , ∈ ( ) ∩ ( ) the coordinate-vertices, i.e., vertices of degree 3; see g. 1 and g. 3. Suppose a path , starts in , , then either the path , ends in a vertex 'above' , and close to it, given the embedding, say , −1 ∈ ( ) ∩ ( −1 ), in which case we call , an up-path, or it ends somewhere further away or below, say in +2, +1 ∈ ( +2 ) ∩ ( +1 ), then , is what we will call The green area is a subwall, the blue area forms a band, and the intersection of both marked in dark blue is a tile.
a jump; see g. 4 for a schematic representation of both.Carefully analysing the possible types of jumps (there are Type 0, and jumps) we then deduce that having 'a lot' of edge-disjoint jumps (whichever type) witnesses the existence of routers, while the absence of jumps implies the existence of many up-paths which in turn witness the existence of a swirl (compare g. 2).While the Types of the jumps have no imminent meaning for this exposition, we encourage the reader to analyse g. 4 and ponder on why Type 0 jumps marked with immediately yield wall-usable crosses, while a single Type or jump does, in itself, not witness the existence of such a cross.Leaving out some details, it turns out that Type and jumps never come alone, and thus their existence witnesses the existence of a wall-usable cross in a local area nonetheless.This relies on the notions of jump-sequences and jump-cycles: starting a path with any jump, then threading it down along the wall-cycle the jump ended at to the next coordinate-vertex, we are guaranteed to nd another jump along which we can extend the path, whence, repeating this construction using Eulerianness, we obtain a cycle alternating between jumps on the wall and sub-paths of wall-cycles.This way we either nd a router (in which case we are done) or some swirl S. As mentioned above, the swirl we nd may in itself still contain crosses: for example the up-paths could pairwise intersect in vertices still being edge-disjoint.We call such swirls tangled.
To get rid of this nuisance we proceed with untangling the swirl as an intermediary step towards the Flat-Swirl Theorem, resulting in a swirl S where the swirl itself is cross-less-it contains no wall-usable cross.This last step turns out to be rather easy: if the starting wall is large we either nd many tangled swirls far apart, or a router by simply applying the previous result to di erent tiles of the wall.Then, either one of the swirls is cross-less, or each of the swirls contains such a wall-usable cross, where many of the swirls and their crosses are pairwise edge-disjoint.Then, this witnesses many disjoint wall-usable crosses spread over W that can again be used to build a router in the same spirit as when starting with jumps.The general scheme to build a -router using disjoint non-planarities is as follows.We try to locate 2 disjoint 'large' bands (see g. 3) in the wall that each cover a wall-usable cross, witnessing the existence of 2 wall-usable crosses in di erent bands: a cross-column.Then the router can be built from the wall-cycles by threading in the direction prescribed by the wall-cycles and pairwise intersecting them in the 2 distinct bands resulting in 2 pairwise intersecting cycles; we skip the details for they are standard in the area of graph structure theory.
Flattening a Swirl.In a second step, we re ne the above analysis proving that either we nd a large router on top of the untangled swirl S, or we nd a large at sub-swirl.To this extent we repeat the analysis as in the previous step-we analyse the di erent paths attaching to S-but with the additional information that we already have a swirl which provides even more structure to build routers.In contrast to the last section the topological gadgets of interest in this step are the swirl-usable crosses.Compared to wall, swirls come with a richer structure: it turns out that almost any path starting and ending in a swirl, and otherwise edge-disjoint from it witnesses the existence of a swirl-usable cross.This way we are able to guarantee that either we nd a router attached to the swirl S, or the swirl contains some sub-swirl that is at.The proof here heavily relies on the Eulerianness of the graph and in particular the Eulerianness of S [ ] − S which consists of the attachments to S that are relevant to nding the crosses.The techniques used share analogies to the techniques we introduce to nd the (untangled) swirl but are far more elegant as we are working on a swirl.
Finally, as a last step, we prove that we can reduce the instance to an equivalent instance such that S [ ] can be Euler-embedded; until here their could be two-cuts or four-cuts in our graph that may attach highly non-planar graphs to the at swirl, which cannot be embdedded but neither be used to cross paths starting and ending in the swirl, similar to ≤ 3-separations in the undirected setting.The reduction relies on a theorem due to Frank, Ibaraki, and Nagamochi [10], proving a version of the aforementioned Two-Paths Theorem tailored to Eulerian digraphs.To state the theorem we rst need the following reductions introduced in [10, Section 3].
1 Let ⊆ ( ) induce a 2-cut such that ∩ = ∅.Let be the tail of the edge from ¯ to and the head of the edge from to ¯ .Then delete and (if ≠ ) add the edge ( , ).Using the above reductions, Frank, Ibaraki, and Nagamochi [10] de ne minimal instances as follows: Let be an Eulerian digraph and ( , 1 , 2 ) an instance of the unordered Eulerian Edge-Disjoint Paths problem; that is, the problem asking whether there exist two edge-disjoint paths connecting to or vice-versa for = { , } and = 1, 2. Then we say that ( , 1 , 2 ) is a minimal instance if none of the reductions from de nition 3.6 are applicable to .The same authors showed that applying any of the reductions in de nition 3.6 results in an equivalent instance.The following is the aforementioned version of the Two-Paths Theorem.
Leveraging the above reductions to our setting in order to reduce our instance + by killing 2-cuts and 4-cuts, we prove the following.Note here that in his dissertation Johnson [15] implicitly proved the existence of such a at swirl in the case that + is internally 6 edge-connected removing the problem of loosely connected attachments and thus circumventing the need of a Two-Paths Theorem.The techniques we use are however very di erent from his and have more of an algorithmic avour analysing the attachments to swirls and wall 'by hand', revealing how a wall (stemming from high treewidth) helps in nding either swirls or routers.Further, the results established in order to nd said ( at) swirl are of independent interest for future work we are pursuing.

Irrelevant Cycles in Routers
The third step of the algorithm relies on a single theorem-the irrelevant cycle theorem for routers-reading as follows.
Theorem 3.9 (Irrelevant Cycle Theorem for Routers).For every function ( ) there exists a function ( ) ≔ ( ; ) satisfying the following.Let + be an instance of the Eulerian Edge-Disjoint Paths problem with | ( )| = ∈ N. Let R ≔ 1 ∪ . . .∪ ( ) ⊂ be a ( )-router in .Then there exists a sub-router R ⊆ R of size ( ) ≤ ( ) which can be found in fpt-time such that each router-cycle ⊂ R is irrelevant to the instance, i.e., ( − ) + is an equivalent instance.
In a nutshell the theorem states that, given a large enough router R in there is a cycle ⊂ R in the router that is irrelevant to the instance, and it can be located in fpt-time on .The proof of this relies on a Theorem due to Frank [9] which states that the Edge-Disjoint Paths problem for Eulerian digraphs can be solved in polynomial time if the demand graph consists of two directed stars; a directed star is a graph where all the edges have the same vertex as a head (or as a tail).Theorem 3.10 (Theorem 2.3 in [9]).If + is an Eulerian digraph, and is the union of two stars, then the directed cut criterion is necessary and su cient for the solvability of the directed Edge-Disjoint Paths problem.(In particular it can be checked in polynomial time on the instance).The graph R + remains Eulerian by construction.We can decide the instance R + in fpt-time using Frank's theorem 3.10 (and Algorithm) and nd the respective paths solving the instance (if it is a YES-instance) in fpt-time.In a nutshell the instance R + asks whether there exist edge-disjoint paths routing from 1 , . . ., to 1 , . . ., that use the router cycles.If the answer is NO, then this means that not all paths can simultaneously use the router, which in turn can be used to apply inductive reasoning.If the instance is a YESinstance, then we are able to show that it remains a YES-instance after deleting part of the router R, that is, after deleting a large sub-router R ′ ∪ * ⊂ R, where * is a designated cycle.Finally, we prove that the deleted sub-router R ′ can be used in to reroute the solution in R omitting and * , and thus producing a solution in omitting a cycle * ⊂ R ; this is proven as the following lemma.
Lemma 3.11.There exists a function ( ) satisfying the following.Let be a graph with designated vertices , ⊂ ( ) such that | | = | | = ∈ N such that any matching of edges connecting vertices in to vertices in results in an Eulerian graph + .Let R ⊂ be a ( )-router in .Further assume that every pushed router-cut of ( , , , R)-think of this as a cut being chosen as close to R as possible--is of order 2 .Then there exists some cycle * ∈ R such that + is a YES-instance if and only if ( − * ) + is a YESinstance for every possible demand graph .In particular we can nd * in fpt-time parameterized by .
Note that the exact knowledge of is not needed, i.e., it su ces for to be pseudo-Eulerian.While the above may sound easy as described here, the proof does in no way follow immediately from theorem 3.10 and requires some ddling and clever choices of sub-routers.

Irrelevant Cycles in Flat-Swirls
The fourth step of the algorithm is the most elaborate part again relying on a single result, namely the irrelevant cycle theorem for at swirls, which, in a nutshell, proves that given a graph + and a large Euler-embedded swirl S in a disc, the most deeply nested cycle of S is irrelevant to the instance.We prove a slightly stronger theorem proving that cycles deeply nested in a large insulation are irrelevant to the instance.An ℎ-insulation in a graph Γ Eulerembedded in some surface Σ is a collection = 2ℎ =1 ⊆ Γ of concentric edge-disjoint cycles embedded in a disc Δ ⊂ Σ such that and +1 have alternating orientation for every 1 ≤ < 2ℎ and such that Δ 1 ⊆ Δ 2 . . .⊆ Δ 2ℎ where Δ ⊂ Δ is a disc bounded by the outline of in Σ, containing but no edge of +1 .Let ′ ⊂ Γ be a subgraph embedded in Σ away from Δ. Then an Eulerian subgraph ′ ⊆ Γ [Δ 1 ] is called ℎ-insulated from ( ′ ).Given a at swirl we may use theorem 3.8 to get an equivalent instance with the at swirl embedded in a disc, providing us with the setting needed for theorem 3.12.We continue with elaborating on the main ideas behind the proof of theorem 3.12, taking up over 70 pages in the full paper, in smaller steps.
Shifting the Paradigm.Readers (now) familiar with the Graph Minor Structure Theorem due to Robertson and Seymour [24] and the main ideas behind the proof that the undirected (vertex-)disjoint paths problem can be solved in fpt-time [23] (which heavily relies on the irrelevant vertex technique [25]) will see the inspiration in our line of argumentation, but also the di erences in the obstacles we had to overcome stemming from the edge directions and the fact that we need to keep our graphs Eulerian.The general line of reasoning uses similar arguments as the respective proof of the undirected version.However, many of our proofs di er from the proofs given in [23,25,26] for many of the arguments do simply not transfer to our setting.
The arguments and ideas we provide heavily exploit the Eulerianness of the graphs and the fact that we are solving the Edge-Disjoint Paths problem rather than the Vertex-Disjoint Paths problem.The latter turns out to be far more impactful than anticipated: for one of the most notable di erences being that we will shift the paradigm from graphs to be seen as a set of vertices with edges being relations on vertices, to graphs to be seen as incidence structures.That is vertices and edges are equals and may (in theory) exist without the need of each other.Of course we were not the rst to view graphs in that way, and by no means do we think we are pioneers in that sense: we call it shifting the paradigm because it deviates from the standard way of viewing graphs and was a very insightful and fruitful step to take.That is, an edge ∈ will be an element for itself where each edge in a (non-partial) graph is adjacent to exactly two vertices which is captured by the tail ⊂ × and head ⊂ × relations.It turns out that switching to incidence graphs facilitates and smoothens a lot of the reasoning for behind the machinery we set up for a proof of theorem 3.12.Note here that the conceptually similar idea of switching to the line-graph-the graph obtained when taking edges ∈ ( ) as vertices and adding an edge if there exists a 2-path between two edges-and trying to leverage the arguments made for the vertex-disjoint case, comes with a lot of obstacles, losing most of the inherent beauty (as far as we were able to reproduce).The key di erence being that, when passing to the line-graph we create unnecessary edges that were of no use in the rst place, for no solution may have ever used a two-path in representing that edge.Note further that, even if the instance has now been reduced to a Vertex-Disjoint Paths problem, many of the arguments made in [25] do not transfer to this setting trivially: we cannot delete vertices, split vertices nor contract edges in the line-graph; operations heavily used to prove the main theorems in [25,26] and in turn the irrelevant vertex theorem.Hence, since we are interested in edges, and edge-disjoint paths, it seemed only natural to take this step, unleashing more potential than anticipated.
A Minimal Counterexample.Let + be an Eulerian digraph and suppose that contains an ℎ-swirl S = 1 ∪ . . .∪ ℎ such that S [ ] can be Euler-embedded in a disc which we found in Step 2. of the algorithm.In order to prove the existence of an irrelevant cycle deeply nested inside the swirl S we assume the contrary and let + be a minimal witness towards our hypothesis together with a -linkage L = { 1 , . . ., } witnessing that + is a YES-instance and thus, by our assumption, it visits all the cycles of S. The main theorem in the analysis of the counterexample states that, given the above, + adheres to a very restrictive structure: + is what we call an ℎ-ower graph (see g. 6).In particular ( ) = (S) ∪ ( ) and every possible edge in + is either an edge of S, , or between two vertices lying on the outer-cycle ℎ .In particular S [ ] = S, and thus S has no attachments to take care of.The most crucial observation is that the linkage L is rigid.De nition 3.13.We call L rigid if + = 1≤ ≤ + and there is no other -linkage L ′ in solving the instance + .This is very restrictive and a key ingredient to most of the proofs towards theorem 3.12.Note that by theorem 3.9 this implies that the graph cannot contain a large router.Using the rigidity we can derive even more restrictive patterns for the paths in L: it turns out that any restriction of some path to the swirl S ⊆ is what we will call a level path.That is, any component ∈ ∩ S is a path starting at the outer-cycle ℎ , then going straight down to some swirl-cycle and from there straight back up to ℎ where it ends; in particular a level path visits every swirl-cycle (up to the level ) exactly twice.This is a rather powerful observation that can be used to prove the irrelevant cycle theorem in the case that the ℎ-ower graph can be Euler-embedded in some xed surface of bounded genus.
Although we could foucs on ℎ-ower graphs for the remainder of the proof-refuting its existence and thus contradicting the minimal counterexample-we present the results in a broader setting.That is, we prove that for ℎ large enough, there exists no rigid linkage in + containing a at ℎ-swirl S. The statements and proofs need a lot of notation and the establishment of some heavy machinery, the gist of which we describe next.
Coastal Maps.The main tool that helps us to dissect the graph + in order to better analyse how a solution L in behaves, is what we call (weak and strong) coastal maps.The exact de nition of coastal maps is rather technical thus we will rely on intuition here: 'admitting a coastal map' means that there exist pseudo-Eulerian graphs Γ, I ⊂ and a surface Σ (possibly with boundary) such that = Γ ∪ I where Γ can be Euler-embedded into Σ with a few vertices (conceptually edges that we call ports) drawn on the boundary bd(Σ) (rather what we call the zone) of Σ, with S ⊆ Γ.Further, each component ′ of I is what we will call an island, that is Γ ∩ ⊂ ( ) is drawn on a single cu ∈ (Σ) of the boundary of Σ-the boundary of Σ consists of disjoint cu s which are homeomorphic to circles-and there exists no large linkage connecting two halves of -in particular starting and ending in Γ ∩ -that is otherwise contained in , i.e., the islands are of bounded depth.Further, the coastal map assigns (sort of) a linear decomposition to each island, guaranteeing some internal linkedness properties taming the behaviour of linkages inside the island; this is too technical to describe in detail, intuitively on may think of this as regrouping an island into a bounded number of chunks arranged in a cyclic order, each chunk attached to some vertex of the same cu , such that (almost) any two neighboring chunks are equally-well connected so we get a good grip on how a rigid solution L may behave inside the islands.
Part of the proof of the existence of a coastal map relies on a structure Theorem for Eulerian directed graphs (actually we only need a structure theorem for the very restricted class of owergraphs).In his dissertation Johnson [15] proved a structure theorem for internally 6 edge-connected Eulerian digraphs that suits our needs.The theorem, restricting and adapting it to terms introduced in this exposition, reads as follows.
Theorem 3.14 (Theorem 17.1 in [15]).Let be a positive integer.There exist integers , , , , and such that the following holds.Let be an internally 6 edge-connected Eulerian digraph.Suppose contains an Eulerian subdigraph which immerses a swirl of size at least .Then either immerses a router of size or there exists a surface Σ and an Eulerian digraph ′ such that: 1. ′ is obtained from by exchanging at most edges.2. ′ Euler-embeds in Σ with at most islands, each of depth at most where every island is surrounded by edge-disjoint cycles of alternating orientation drawn in Σ.
3. The embedding can be chosen so that there is a closed disc disjoint from every island and every changed edge containing an Eulerian subdigraph which immerses a swirl of size .
We use said theorem since giving a rigorous proof ourselves (for ower-graphs) would take up quite some pages without much new insights.However, since the results in the dissertation have not yet been published, we will provide a proof-and hopefully a structure theorem for general Eulerian digraphs-in the future, using di ering techniques matching the constructions and results we provided in this exposition.

Shipping with Coastal Maps
Recall that we assume + to be a graph without large router containing a at ℎ-swirl, and L to be a rigid -linkage (just think of + as an ℎ-ower graph).Using the above one can show that, after cutting through a bounded number of edges in , the graph + admits a (weak) coastal map (note that this does not follow imminently from theorem 3.14 but requires us to chart the islands in order to get the aforementioned linear decompositions).To make the idea of cutting edges more rigorous think of it as follows: let ∈ ( ) be an edge, then since L is rigid there exists ∈ L with ∈ ( ).Since is part of the solution to + , there exists ( , ) ∈ ( ) such that starts in and ends in .Now cut = ( , ) into two edges 1 ≔ ( , ) and 2 ≔ ( , ) by introducing the respective vertices.Then replacing ( , ) ∈ ( ) via ( , ), ( , ) does the trick (since augments we require the number of cuts to be bounded).
Finally, we prove that if + admits a coastal map of bounded depth, then + cannot admit a rigid linkage, a contradiction to the minimal counterexample.The proof reduces the -linkage L to a rigid ( )-linkage L ′ of Γ (a function independent of ℎ).The proof works via induction by cutting through a bounded number of edges in , whose endpoints result in new demand-edges for a new demand graph ′ which by our assumption need not be embedded in Σ and neither be part of any island; this is crucial.Then, after doing some more skilled cuttings (we massage a weak coastal-map into a strong one), if any cu ∈ (Σ) contains 'too' many vertices of and thus too many ports (the edges adjacent to the vertices marking the entries to the islands from the surface), then the linkage can be rerouted inside the island at to omit part of the ports.This uses the aforementioned linear decomposition in the islands.Since L was assumed to be rigid, this is impossible (we cannot nd a di erent linkage solving the same instance) and thus only a bounded (in ) number of vertices may lie on each cu of Σ.This guarantees that our linkage L does only interact with each island a 'bounded number of times'.Therefore, after cutting all the (boundedly many) ports, we get a new graph ′ + ′ where ′ is completely Eulerembedded in Σ together with a rigid ( )-linkage in ′ containing an Euler-embedded ℎ( )-swirl; we killed the islands.
Shipping in the Open Sea.We reduced the problem to the case where we have a graph + together with an ℎ-swirl S such that is Euler-embedded in Σ and admits a rigid -linkage.The last step in the proof of theorem 3.12 is to show that this is impossible; a minimal counterexample is again given by an an ℎower graph as above.We leverage a result due to Cygan, Marx, Pilipczuk, and Pilipczuk [6] implying that given a digraph Eulerembedded in a disc such that there exists a large embedded swirl then the most deeply nested cycle is irrelevant to the instance; this marks the base-case when Σ is a disc.Our proof uses induction on Σ and the structure of the minimal counterexample + : we are able to cut the surface Σ along a closed curve reducing the genus of the surface, and such that intersects the swirl S only in a bounded (say ℓ ( )) number of edges after deleting some edges.This results in a surface of lower genus and an Euler-embedded graph ′ containing a large swirl S ′ , together with a rigid ′ = ( + ℓ ( ))linkage; the claim then follows by induction.The existence of the just mentioned curve heavily relies on the fact that for each path ∈ L it holds that every component ∈ ∩ S is a level path as mentioned above.This allows the following: take a sub-swirl S ′ deeply nested in S. One can show that (using rigidity) there must be some path ∈ L containing some sub-path ⊂ that 'uses' the genus of the surface-closing to a cycle and cutting the surface along reduces its genus-such that has both its endpoints on S ′ .Then has two components 1 , 2 connecting the outer-cycle of S ′ to the outer-cycle of S, where both , being sub-paths of some level path, do not visit any of the swirl-cycles twice.Thus, deleting (the edges of) both paths, cutting Σ along the curve traced by , and cutting straight through S ′ connecting the ends of both paths in order to reduce the genus, results in a graph still containing a large undirected wall and thus a large swirl by our previous work.All in all this augments the number of paths in the linkage roughly by the order of S ′ , i.e., ℓ ( ), concluding the idea of the induction.
It is noteworthy that in his dissertation Johnson [15] proved a similar theorem to what we dubbed Shipping in the Open Sea, that in our setting implies that if is Euler-embedded in Σ and contains a large swirl, then the linkage cannot be rigid.Note also that it is not clear at all how to extend the base-case result for Euler-embedded graphs to the general setting (which we capture by coastal maps).While his proof relies on a classi cation of homotopy classes of curves, our proof is a consequence of preliminary work we have done to analyse the minimal counterexample-this is of independent importance to the proof of theorem 3.12-which is of its own interest, yielding a deeper understanding of the behaviour of rigid linkages in minimal counterexamples (e.g. level paths).
This concludes the high-level discussion of theorem 3.12, gathering the main ideas to a proof of our main theorem 1.2.
stated in [2, Problem 4.5.7] the status of the -Edge-Disjoint-Paths problem on Eulerian digraphs is wide open and could range from xed-parameter tractable with respect to to NP-complete already for = 4.The main result of this paper is to settle this long open problem by showing that the -Edge-Disjoint-Paths problem is xed-parameter tractable parameterized by on the class of all Eulerian digraphs.Theorem 1.2 (Main Theorem).The -Edge-Disjoint-Paths problem in Eulerian digraphs is xed-parameter tractable parameterized by the number of terminal pairs .

Figure 2 :
Figure 2: A 6-swirl with highlighted cycles induced by a tile.
Since each time we enter Step 3 or Step 4 we reduce | | ≔ | ( )| + | ( )| by at least 1, the above algorithm stops after at most | | many recursive steps, each of which run in fpt-time.This concludes the proof.□

Figure 3 :
Figure 3: Part of an embedded wall with wall-coordinates.The green area is a subwall, the blue area forms a band, and the intersection of both marked in dark blue is a tile.

Figure 4 :
Figure 4: Wall with Type 0 jumps marked by , Type and jumps marked by and , and up-paths marked by .

2
Let ⊆ ( ) induce a 2-cut such that | ∩ | = 1.Contract to the terminal ∈ deleting any resulting loops.3 Let induce a 4-cut such that the subgraph [ ] is connected and, | | ≥ 2 and ∩ ( ) = ∅.Then contract [ ] to a single vertex of degree four, and delete possible loops.

Figure 5 :
Figure 5: Constructing by connecting to a 3-router in

Figure 6 :
Figure 6: An example of a possible ℎ-ower graph in the case of = 2 where is highlighted in purple.