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Merging Arcs to Produce Acyclic Phylogenetic Networks and Normal Networks

As phylogenetic networks grow increasingly complicated, systematic methods for simplifying them to reveal properties will become more useful. This paper considers how to modify acyclic phylogenetic networks into other acyclic networks by contracting specific arcs that include a set D. The networks need not be binary, so vertices in the networks may have more than two parents and/or more than two children. In general, in order to make the resulting network acyclic, additional arcs not in D must also be contracted. This paper shows how to choose D so that the resulting acyclic network is “pre-normal”. As a result, removal of all redundant arcs yields a normal network. The set D can be selected based only on the geometry of the network, giving a well-defined normal phylogenetic network depending only on the given network. There are CSD maps relating most of the networks. The resulting network can be visualized as a “wired lift” in the original network, which appears as the original network with each arc drawn in one of three ways.


Introduction
. As a result there is interest in phylogenetic networks, in which some vertices have in-degree two or higher, corresponding to such events (Moret et al. 2004;Solís-Lemus et al. 2016). Overviews of phylogenetic networks may be found in Steel (2016) and Huson et al. (2010).
A phylogenetic X -network is an acyclic directed graph in which the leaves are identified with a particular collection X of species, usually extant species. We assume that a phylogenetic X -network describes gene flow, and each vertex corresponds to a biological species. Such phylogenetic networks can be quite complicated. The focus will be on simplifying such networks by recursively merging the ends of particular arcs in a natural manner. We will then apply the results to study simplification into a normal network.
In this paper, an X -network is a directed graph in which the leaves are identified with members of a particular set X . Our notion of an X -network is broad. Vertices can have in-degree and/or out-degree greater than two, so we are not assuming that the networks are binary. An exact definition is given in Sect. 2, along with other basic notions. In Sect. 3 we describe constructions which do not necessarily yield acyclic networks, and then find conditions that ensure that the results are acyclic. Hence in this paper an X -network need not be acyclic, and we will refer to one that is acyclic as an acyclic X -network. In turn, an acyclic X -network that has no vertices with both out-degree and in-degree equal to one is a phylogenetic X -network.
One measure of the complexity of an acyclic X -network is the number of vertices. In terms of bounds on the number of vertices we have the following comparisons between certain families of networks. The definitions of these families are given in Sect. 2. (The result is from Willson (2010) with slight changes since, in Willson (2010), X contained the root as well as the leaves.) The fact that for normal networks the number of vertices grows at worst quadratically with n indicates that normal networks are potentially a more tractable network type than regular or tree-child. Also indicative of their tractable nature is the fact (Steel 2016) that the number of hybrid vertices is at most n − 2. Yet another indication is that binary normal networks are determined by their caterpillars on three and four leaves (Linz and Semple 2020).
A vertex v of the X -network N = (V , A, ρ, φ) is visible (Francis et al. 2021;Huson et al. 2010) if there exists a leaf φ(x) such that every path in N from the root ρ to φ(x) includes v. In a tree-child network, every vertex is visible (Cardona et al. 2009). Since any normal network is tree-child, every vertex of a normal network is visible, yielding another useful property of normal networks.
As in Pardi and Scornavacca (2015) we take the view that rather than try to deal with networks that are possibly not identifiable, it is desirable to focus instead on networks that are sufficiently tractable to be tested with data. Since every vertex of a normal network is visible, potentially every vertex of a normal network can be so tested, and simplification into uniquely determined normal networks will become useful. This paper relies on results from Willson (2012). This earlier paper focused on networks that were not necessarily acyclic. This current paper extends the results to ensure that the constructed networks are acyclic. If N is a given network and D is a list of certain arcs in N satisfying a weak condition, this paper in Sect. 3 computes the result M D (N ) of merging the arcs in D as well as certain additional arcs required to ensure that M D (N ) is acyclic. Of interest will be the choice of D so as to obtain ultimately a normal network.
In Sect. 5 we study the result R(N ) of removing all "redundant" arcs from N . In Sect. 7 we describe ways to find sets D of arcs of N such that R(M D (N )) is a normal network.
Combining these techniques we describe in Sect. 7 a method, given an X -network N , to construct a normal acyclic X -network Norm(N ) which is a phylogenetic Xnetwork depending only on the geometry of N . The construction makes no arbitrary choices such as between different parents or children.
As phylogenetic X -networks grow increasingly complicated, it will become useful to "simplify" them. Simplification into a normal network may make them easier to interpret since normal networks are potentially tractable.
If N and M are X -networks, a connected surjective digraph map (CSD map) f : N → M is a surjective map f : V (N ) → V (M) with various properties. (See Willson 2012 and Sect. 2 of this paper.) The merging procedure in this paper always yields a CSD map ψ : N → M D (N ). Results in Willson (2012) show that there is then a "wired lift" of M D (N ) into N , from which properties of M D (N ) can be visualized in N . The wired lift is not a subnetwork of N in the usual sense. Section 6 of this paper generalizes the notion of "wired lift". As a result we obtain a wired lift of Norm(N ) into N , even though there is usually no CSD map from N to Norm(N ). The wired lift is visualized by drawing the diagram of N with each arc drawn in one of three different ways. Thus we can visualize the resulting normal network by looking at a redrawn diagram of N . The current author thinks such visualizations can provide a tool for better understanding complicated networks. Section 8 contains two examples of the methods applied to published networks based on biological data. Section 9 contains some discussion. Francis et al. (2021) describe an elegant procedure, given an acyclic X -network N , to find a related, uniquely determined, normal X -network, which I will denote FHS(N ). Its calculation is based on locating the visible vertices of N . The fast program PhyloSketch (Huson and Steel 2020) is available to compute it. The paper (Francis et al. 2021) assumes that non-root vertices have either in-degree one or out-degree one. Nevertheless, visibility of vertices is well-defined for the X -networks defined in this paper and their procedure applies to any acyclic X -network in our sense. I therefore use FHS(N ) to represent the result of this extension of their method. We will occasionally compare FHS(N ) with Norm(N ).

Basic Notions
Let N = (V , A) be a directed graph, where V is a finite set of vertices and A is the set of arcs. An arc (u, v) is regarded as directed from u to v, so we call u a parent of v and v a child of u. We assume N is a simple graph: there are no loops (u, u); and there is at most one arc (a, b) for a = b. We may sometimes denote V (N ) = V or A(N ) = A.
If N = (V , A) is a directed graph, the corresponding undirected graph U nd(N ) = (V , E) is the graph where {u, v} ∈ E iff either (u, v) ∈ A or (v, u) ∈ A. Thus, arcs are replaced by edges and are not directed. In this paper, N will always refer to a directed graph unless otherwise specified.
The in-degree of a vertex v in N , denoted indeg (v) or indeg(v; N ), is the number of arcs (u, v), i.e. the number of parents of v. The out-degree of a vertex v, denoted outdeg (v), is the number of arcs (v, u), i.e., the number of children of v.
We shall not assume that our directed graphs are binary. Thus a vertex v may have A trivial vertex merely subdivides an arc, and we will often systematically suppress trivial vertices.
If u and v are vertices, a path or, for emphasis, a directed path from u to v is a sequence of vertices u = u 0 , u 1 , u 2 , · · · , u n = v such that for all i, 1 ≤ i ≤ n, The length of the path is the number n of arcs. Note that the arcs are uniquely determined by the vertices in the list since N is a simple graph. No two successive vertices can be the same since there are no loops. We say the path contains arc (u k , u k+1 ) for k = 0, · · · , n − 1. In some situations we may focus on a certain part of the path such as u 2 , u 3 , u 4 ; we may refer to such a portion as a segment. (For example, in certain circumstances we might modify the path by replacing a segment u 2 , u 3 , u 4 by a segment u 2 , v 1 , v 2 , u 4 .) The path of length 0 consisting only of u 0 is the trivial path at u 0 . A path u 0 , u 1 , u 2 , · · · , u n is closed if n > 0 and u 0 = u n . A closed path is a cycle.
Let X be a nonempty finite set. In the applications, X is usually a set of extant biological species.  (u, v) with u and v distinct members of V .
(N3) ρ, called the root, is a node with in-degree 0.
(N4) The map φ : X → V is one-to-one. (N5) Each leaf is a vertex with in-degree 1 and hence has a unique parent.
(N6) The image of φ is the set of leaves.
(N7) ρ is the only vertex with in-degree 0.
(N8) For each v ∈ V there is a path from ρ to v.
(N9) For each v ∈ V there is a path from v to some leaf.
An acyclic X -network is an X -network that also satisfies (N10) N has no cycles. Following Steel (2016), we define a phylogenetic X -network to be an acyclic Xnetwork that contains no trivial vertices.
These assumptions are not intended to be the minimal possible; rather, they tell the properties we will utilize the most.
If x ∈ X the unique parent of φ(x) by (N5) will be denoted p(x) or p(x; N ). The arc of form ( p(x), φ(x)) for some x ∈ X will be called the x-arc. If x is not specified, any such arc will be called an X-arc.
Suppose N is an X -network. By (N4) and (N6) we may identify X with the set of leaves.
If there is a directed path from u to v then we write u ≤ v. The trivial path shows If the X -network is acyclic, then ≤ is a partial order; otherwise it is possible that for distinct vertices u and v we have In this situation, N and N are essentially the same and we write N ∼ = N .
Let N be an X -labeled graph. An arc (a, b) is redundant or a short-cut if there exists a path a = u 0 , u 1 , · · · , u n = b, n ≥ 2, that does not contain the arc (a, b). Thus, there is no k ≤ n − 1 such that u k = a and u k+1 = b. Such a path is called a lengthening or a lengthening path of (a, b). Examples will be seen in several figures later, such as Fig. 3. If x ∈ X then the arc ( p(x), φ(x)) cannot be redundant since any such lengthening path would have to satisfy u n−1 = p(x) by (N5).
We shall have need of the following result: Theorem 2.1 Suppose N is an acyclic X -network. Suppose there is a directed path in N from a to b. Then, a directed path in N from a to b of maximal length contains no redundant arc.
Proof Since the vertex set is finite and there are no cycles, there is an upper bound to the length of a path. Suppose a = u 0 , u 1 , · · · , u k = b is a directed path P in N of maximal length k. If the result is false, we may assume that for some i < k, (u i , u i+1 ) is redundant. In that case there is a directed path u i = w 0 , w 1 , · · · , w j = u i+1 with j ≥ 2. We can then lengthen the path P by replacing the segment u i , u i+1 by w 0 , · · · , w j , a contradiction.
There are several types of X -networks which will be of interest: An acyclic X -network N is tree-child (Cardona et al. 2009) if every vertex that is not a leaf has a tree-child.
An acyclic X -network N = (V , A, ρ, φ) (possibly not satisfying (N5)) is regular (Baroni et al. 2004) if (1) the cluster map cl : V → P(X ) is one-to-one, where P(X ) is the power set of X ; (2) N has no redundant arcs; and (2) N contains no redundant arc.
Sometimes there are small differences in the definition of a network. In Baroni et al. (2004) and Willson (2010) the authors do not assume condition (N5). In Baroni et al. (2004) no vertex can have out-degree one. Particularly simple are normal networks in which no vertex has out-degree one, since these are regular (Willson 2010).
Let N and N be acyclic X -networks. One interesting way to compare them is their Robinson-Foulds distance d R F (N , N ) defined as the number of members of Cl(N ) and Cl(N ) which are present in one but not both (an extension of Robinson and Foulds (1981) for trees). It is symmetric and satisfies the triangle inequality. For certain classes of X -networks d R F is a metric. As an example, for fixed X , it is a metric on the collection of regular X -networks (Baroni et al. 2004).
If N is a normal X -network, let S(N ) denote the result of contracting every arc (u, v) such that outdeg(u) = 1. For example, suppose in N , for some x ∈ X , p(x) is hybrid and has out-degree one. Then in S(N ) the arc ( p(x), φ(x)) in N will have been contracted, and S(N ) will not satisfy (N5). Thus in S(N ) a leaf can be hybrid. Moreover, any trivial vertices will have been suppressed.
The following result shows that two normal networks N 1 and N 2 such that d R F (N 1 , N 2 ) = 0 are essentially the same.
Proof (1) For any X -network N , if (u, v) is an arc and outdeg(u) = 1, it is immediate that cl(u) = cl(v). Hence Cl(S(N )) = Cl(N ). Moreover, if N is normal then S(N ) remains normal and hence is a regular network (Willson 2010). ( The result follows from the fact (Baroni et al. 2004) that d R F is a metric on regular X -networks.
The d R F distance has the interesting property that since it is defined for all acyclic X -networks, it can be used to compare how well various networks of various types "approximate" a given network. For example, if N is a complicated acyclic X -network and T and T are X -networks that are rooted trees, then T might be a better approxi- In this paper we will be "simplifying" an acyclic X -network N into a normal Xnetwork N . From this point of view we would prefer that d R F (N , N ) is as small as possible.
Let N = (V , A, ρ, φ) and N = (V , A , ρ , φ ) be X -directed graphs. A connected surjective digraph (CSD) map (Willson 2012) In the latter case we may write ψ (u, v) consists of the vertices of a connected subgraph of N . Thus in the undirected graph U nd (N ) If ψ 1 : N → N and ψ 2 : N → N are CSD maps, then it is proved in Willson (2012) that the composition ψ = ψ 2 • ψ 1 : N → N is also a CSD map. If both maps are leaf-preserving, then it is easy to see that the composition is also leaf-preserving. We will use this fact repeatedly.
Note that in Willson (2012) the term "X -network" refers to what in this paper is an X -directed graph satisfying (N1), (N2), (N3), (N4), (N6), (N7), (N8), and (N9). Thus the networks in Willson (2012) were not required to be acyclic. Of interest in this current paper is the behavior when the final networks are required to be acyclic, as are phylogenetic networks in biology. The CSD maps φ : N → N become more useful to biologists when both N and N are required to be acyclic.

Contraction of Arcs
Here is a summary of this fundamental section: The basic tool used in this paper is that of successively contracting arcs in an X -network. Suppose N is an X -network and D is a subset of its arcs. In this section under weak conditions we describe how to construct an X -network Q D (N ) by merging just the arcs of D. In general Q D (N ) may contain cycles. When D is "strongly closed" we show that Q D (N ) is acyclic. Moreover, any D has a unique "strong closure" K (D) which contains D and is strongly closed. Hence, we are able to define M D (N ) = Q K (D) (N ) as a uniquely determined acyclic X -network that results from contracting the arcs of D and also the other arcs needed for acyclicity. The sections after this one will rely on the iterated use of this construction. The fundamental problem studied in this paper is, roughly, how to choose D so that we can find a normal network from M D (N ).
Let N = (V , A, ρ, φ) be an X -network. Suppose ∼ is an equivalence relation on V . Let [v] denote the equivalence class of v ∈ V . Let P(V , ∼) denote the set of equivalence classes of V under ∼.
The following result is similar to Theorem 3.1 of Willson (2012). We outline the proof here again because some of the definitions have slightly changed, for example, to allow for (N5) and leaf-preserving CSD maps. Each [φ(x)] for x ∈ X is a leaf of N since otherwise there would be an arc ([φ(x)], [u]) for some [u] and hence an arc from some If v is not a leaf of N , then there is a path in N from v to φ(x) for some x ∈ X by (N9) and (N6). Let the path be and v 0 ∈ [ρ] such that there is an arc (u 0 , v 0 ). By (N8) there is a path in N from ρ to u 0 and from there to v 0 via the arc (u 0 , v 0 ). This path from ρ to v 0 satisfies that [ρ] = [v 0 ], so, since ∼ is root-preserving, it follows that each vertex in the path lies in [ρ]. In particular , contradicting the arc ([u], [ρ]). This proves (N3).
For (N7), suppose [u] has in-degree 0. By (N8) there is a path from ρ to u, say This completes the proof of (1). We now prove (2). (C1) is immediate since every vertex of N has the form This proves ψ is leaf-preserving, completing the proof of (2).
We do not claim that N / ∼ is acyclic, even if N is acyclic. We will refer to the map ψ as the projection from N to N / ∼.
Let D be a subset of A. A path u 0 , · · · , u k is called a D-path provided that for Call D root-preserving if whenever ρ = u 0 , · · · , u k = a is a D-path, then every path from ρ to a is a D-path.

Theorem 3.2 Let N = (V , A, ρ, φ) be an X -network. Let D be a subset of A. Then
(1) ∼ D is a connected equivalence relation. ( . This proves (2).
(3) Assume that D is root-preserving. We must show that the equivalence class Assume D contains no X -arc and D is root-preserving. Then the quotient digraph N / ∼ D is an X -network. Moreover, the projection ψ : N → N / ∼ D is a leaf-preserving CSD map.
Proof By Theorem 3.2, ∼ D is a connected leaf-preserving and root-preserving equivalence relation. The conclusions follow from Theorem 3.1.
Henceforth if D contains no X -arc and is root-preserving, we will write Q D (N ) for N / ∼ D . We may call it the quotient X -network of N under D and refer to its formation as contracting or merging the arcs of D. In general, Q D (N ) may contain cycles even if N is acyclic.

Theorem 3.4 Let N = (V , A, ρ, φ) be an X -network. Let D ⊆ A be a subset of arcs.
If D is closed then each equivalence class of ∼ D is convex. Fig. 1 a An acyclic X -network N . Let D = { (7,8), (8,9), (10, 11), (11, 12)}. b Q D (N ). Note that D is not closed proving convexity. Figure 1 shows that the converse of Theorem 3.4 is false. Q D (N ) is acyclic and ∼ D is convex even though D is not closed. Figure 1 also illustrates the fact that often when an arc (u, v) is merged, the number of vertices drops by one, reducing the resolution. In Fig. 1, D contained 4 arcs, and the number of vertices dropped from 13 in (a) to 9 in (b). On the other hand, if D = D ∪ { (7,9)}, then Q D (N ) = Q D (N ) and the merging of (7,9) does not further reduce the number of vertices.
The following theorem shows that from a given D we can construct a uniquely determined strongly closed set K that contains D.
Thus K is the unique minimal strongly closed subset of A containing D.
By construction D n D n+1 . Since A is a finite set, the chain D 0 D 1 D 2 · · · must terminate with some D n , at which point D n is strongly closed. Let K = D n . Then K contains D = D 0 and is strongly closed. Moreover, any strongly closed set C that contains D j for any j < n must necessarily also contain D j+1 by the strong closure property. Hence, C must contain K .
If D is a set of arcs in the X -network N , the strong closure K = K (D) of D is the smallest set K of arcs that contains D and is strongly closed. By Theorem 3.7 K is uniquely determined.
The next theorem is the main result of this section.
. K (D) exists by Theorem 3.7 and is root-preserving by Theorem 3.5. It contains no X -arc since otherwise D would contain an X -arc. Then (1) follows from Theorem 3.6. Note K (D) is closed by Theorem 3.5. Hence (2) follows from Theorem 3.4. Then (3) follows from Theorem 3.3.
Call M D (N ) = Q K (D) (N ) the merged acyclic X -network for D. Note that in general, some arcs not in D need to be merged to produce an acyclic network. We nevertheless call D the merging set for M D (N ).
is strongly closed, it is closed by Theorem 3.6. By closure it follows that (a, b) ∈ K (D).

Deriving an SCD Network from N
This section gives a general construction, given an X -network N , to produce a uniquely determined acyclic X -network called SCD(N ) in which, for almost all arcs (u, v), the clusters are distinct (i.e., cl(u) = cl(v)). The only possible exceptions occur when v is a leaf. For complicated N , SCD(N ) can be very much simpler than N . Moreover, We need to consider the behavior of clusters under contraction of arcs.
. The exception at the end is intended to make the definition consistent with the condition (N5), which often forces p(x) to have out-degree one and therefore cl( p(x)) = cl(φ(x)) = {x}. (In Willson (2012) a network N was called SCD without the exception, but the networks there could fail (N5).) In this section, we show that it is often easy to simplify a network N greatly so as to make it SCD.
Let N = (V , A, ρ, φ) and let ∼ be a connected leaf-preserving and root-preserving equivalence relation on V . Suppose a, b ∈ V . A generalized path or g-path in N from a to b is a sequence of vertices a, u 0 , v 1 , u 1 , v 2 , · · · , u k−1 , v k , b such that a ∼ u 0 , In a g-path one always either utilizes an arc (u i , v i+1 ) ∈ A, or else one stays within an equivalence class (but ignoring the direction of any arcs within the equivalence class). N = (V , A, ρ, φ) be an X -network, let D ⊆ A contain no X -arc, and let M D (N ) = (V , A , ρ , φ ). Let ψ : N → M D (N ) be the projection.

Lemma 4.1 Let
(1) Suppose there is a path in N from v to w. Then, there is a path in M D (N ) from ψ(v) to ψ(w). ( (1). N ). From this the result is clear. (

b) and b is not a leaf}. Then, D is strongly closed and contains no X -arcs.
Proof It is immediate that D contains no X -arcs since such were specifically excluded.
(4) M D (N ) is successively cluster-distinct (SCD). (5) No vertex of M D (N ) (other than possibly p(x) for some x ∈ X ) has out-degree one.
For (5) A , ρ , φ ). Suppose a vertex u ∈ A has out-degree one with unique child c. Then cl(u) = cl(c). Since M D (N ) is SCD, it follows that for some x ∈ X , c = φ(x) and so u = p(x) by (N5).
For (6) Such trivial vertices are a nuisance and it is easy to remove them. Since p(x) is trivial, it has a unique parent u(x). By Theorem 4.3,u(x) satisfies outdeg(u(x)) > 1 and cl(u(x)) = cl( p(x)). Hence, the trivial vertex p(x) can be merged with u(x) and hence removed. We state this as a theorem:
Proof It is immediate that E contains no X -arcs. It is easy to see that E is strongly closed. Hence, SCD(N ) is an acyclic X -network, proving part of (1). There are leaf-preserving CSD maps ψ 1 : N → M D (N ) and ψ 2 : does not have out-degree one in SCD(N ) and [u(x)] is not trivial. Thus, SCD(N ) has no trivial vertices, proving (3). Note cl ([u(x) completing the proof of (1) and proving (5). Then (4) follows from (1) and (3).
A very similar network was described in Willson (2012) by a different approach. Figure 4 gives an example of an X -network N , and Fig. 5 shows SCD(N ). In this case SCD(N ) is a tree, clearly indicating the main features of N and much simpler than N . Vertices in SCD(N ) are labeled by a representative vertex of N with the same cluster.

Removing Redundant Arcs from an X-Network
Our goal in this paper is the construction of normal networks which by definition contain no redundant arcs. A crucial step will be removing from an X -network N all its redundant arcs to form R(N ). This short section studies this process. Unfortunately, the natural map ψ : N → R(N ) is not a CSD map unless N = R(N ), causing complications later in this paper.
If N is an X -network, let R(N ) denote the directed graph obtained from N by removing all redundant arcs. More precisely, if N = (V , A, ρ, φ) then R(N ) = (V , A , ρ, φ) where A is obtained from A by removing all arcs redundant in N . N = (V , A, ρ, φ) is an acyclic X -network. Then R(N ) is an acyclic X -network. is not a CSD map since (6,10) is an arc in R(M) but ψ(6, 10) = ( [6,7], 10) and there is no such arc in R(N ). Indeed, it is easy to see that there is no CSD map f : R(M) → R(N ).

Theorem 5.1 Suppose
In the same figure, one sees easily that there is no CSD map from N to R(N ).

Theorem 5.2 Suppose N is an acyclic X -network. For each v ∈ V cl(v; R(N )) = cl(v; N ). Moreover, Cl(R(N )) = Cl(N ) and d R F (N , R(N )) = 0.
Proof Suppose x ∈ cl(v; N ). There is a path from v to φ(x) in N . By Theorem 2.1 a path from v to φ(x) in N of maximal length contains no redundant arc, hence lies in R(N ). It follows that x ∈ cl(v; R(N )). Conversely, suppose x ∈ cl(v; R(N )). There is a path in R(N ) from v to φ(x), so the same path is a path in N from v to x, proving x ∈ cl(v; N ). Hence cl(v; R(N )) = cl(v; N ). The rest follows easily.

Generalized Wired Lifts
Let N = (V , A, ρ, φ) and N = (V , A , ρ , φ ) be X -networks. Suppose ψ : N → N is a CSD map. In Willson (2012) a wired lift of N into N is described. It provides a method for visualizing N within N . In this section we modify and generalize the notion so that ψ does not quite need to be a CSD map but is only a connected map. This will let us obtain wired lifts from a process that includes both CSD maps and removing redundant arcs. Let N = (V , A, ρ, φ) and N = (V , A , ρ , φ ) be X -networks, and let f : It is immediate that a CSD map is connected, but a connected map need not be CSD. Suppose f 1 : N 1 → N 2 and f 2 : R(N 2 ) → N 3 are CSD maps. Since We shall see in Theorem 6.4 that f is a connected map while in general it is not a CSD map.
Let f : N → N be a connected map, and let 2 V denote the set of subsets of and where E 1 ⊆ A satisfies the following two conditions: u , v ). We will say the arc (u, v) represents (u , v ) or is a pre-arc of (u , v ).
Call the members of E 1 the representative arcs since each represents an arc of A .

Note that the collection of all
Since f is a CSD map, for each (u , v ) ∈ A such a (u, v) exists, and g(u , v ) provides a unique choice of a pre-arc of (u , v ).
There are several situations that give rise to wired lifts. We describe three of them in the next theorem. A fourth will be given in Theorem 6.4.

3) Let R(N ) = (V , A , ρ , φ ) be the result of removing all redundant arcs from N . Let E
Proof (1) and (2) are immediate from the definitions since a CSD map is connected. For (3), note that V (N ) = V (R(N )), so the map f can be regarded as a map f : N → R(N ). This map will not be a CSD map if N has any redundant arcs, but it is a connected map. Then (3) follows.
Given a connected map f : N → N , a wired lift ( f −1 , E 1 ) can be visualized using a diagram of N . An example is shown below in Fig. 7. The diagram is exactly the diagram of N except that each arc may be wide solid, thin solid, or thin dashed. Suppose N = (V , A, ρ, φ). For every arc (u, v) ∈ A such that f (u) = f (v) draw (u, v) a wide solid arrow if (u, v) ∈ E 1 and a thin dashed arrow if (u, v) / ∈ E 1 . For Each arc (u , v ) ∈ A has a corresponding wide solid arc (u, v) ∈ A, justifying the word "lift". The "wires" are the thin solid arcs. Paths in N can be recognized in the wired lift as g-paths using allowed steps, which we will now describe. Let N = (V , A, ρ, φ) and N = (V , A , ρ , φ ) be X -networks, with f : V → V a connected map, and suppose ( f −1 , E 1 ) is a wired lift of f . If u and v are in V , we say there is an allowed step from u to v if either (u, v) ∈ E 1 , or ((u, v) ∈ A and f (u) = f (v)), or ((v, u) ∈ A and f (u) = f (v)). Note that the step either follows a wide solid arc in E 1 forwards or else follows a thin solid arc, possibly forwards, possibly backwards. Dashed arcs cannot be used. (1) In N there is a sequence of vertices a, u 0 , v 1 , (2) There is a sequence of vertices a = u 0 , u 1 , u 2 , · · · , u k−1 , u k = b in N such that, for i such that 0 ≤ i ≤ k − 1, there is an allowed step from u i to u i+1 .
Proof Suppose there is a sequence of type (1).
Thus, there is an allowed step from w i to w i+1 . Hence given a sequence of type (1), there is a sequence of type (2).
Conversely, given a sequence of type (2), if the allowed step from u i to u i+1 then we may replace u j , · · · , u n by simply u j ∼ u n . Thus, there is a sequence of type (1).
We will call a sequence of either type a generalized path or g-path from a to b in ( f −1 , E 1 ). For specification they may be called type (1) or type (2). N = (V , A, ρ, φ) and N = (V , A , ρ , φ ) be X -networks. Let f : N → N be a connected map, and let ( f −1 , E 1 ) be a wired lift of f .
possibly by suppressing multiple successive copies of the same vertex.
(2) Suppose a = w 0 , w 1 , · · · , w k = b is a path in N , f (a) = a , and f (b) = b . Then, there is a g-path in N from a to b.
In Willson (2012) there was a backwards map and, instead of all arcs in ψ −1 (v ), only the arcs in some spanning tree in ψ −1 (v ) containing each vertex in ψ −1 (v ) which lies on an arc in E 1 were included. But this feature is not essential. Figure 7 shows a wired lift that arises from a connected map f : N → N . All the arcs and vertices are from N ; thus if we ignore thickness and dashing and include all arcs with their indicated directions, whether thin, wide, or dashed, the diagram exhibits N . A vertex of N with more than one preimage may be identified with a connected component of thin solid arcs. It is also convenient to identify each vertex v ∈ V by the members of f −1 (v ) inside square brackets. One sees immediately that the vertex f (10) of N satisfies f −1 ( f (10)) = {8, 10, 11, 16} (from the component of thin arcs). We shall designate it [8,10,11,16] or less formally [10], the equivalence class of 10. Similarly f (15) has inverse image f −1 ( f (15)) = {15, 20} and is written [15,20]. Other vertices include [9,18] and [17,21]. Still other vertices have singleton inverse images such as [13] with f −1 ( f (13)) = {13}, but the brackets may be omitted.

Example 2
The dashed arcs are not permitted on g-paths, and wide solid arcs must be followed in their direction. Thin solid arcs can be followed in either direction. Thus,16,10,8,9,15, 1 is a g-path showing that N has a path from f (16) to f (1). The corresponding path in N is formally written [8,10,11,16], [9,18] [9],15,1. There is clearly no path in N from 16 to 1. Similarly the g-path 21,17,4 shows that in N there is a path from [21] to [4]. Thus 4 ∈ cl( f (21); N ).
Suppose N 1 , N 2 , and N 3 are X -networks. Suppose f 1 : N 1 → N 2 and f 2 : R(N 2 ) → N 3 are CSD maps, where f 2 denotes a simplification of R(N 2 ). Let f : f 1 (v 1 )). In general f : N 1 → N 3 is not a CSD map since there is no CSD map from N 2 to R(N 2 ). The following result shows that f is nevertheless a connected map and there is a wired lift of f . Consequently, we are able to visualize simplifications of R(N 2 ).

Theorem 6.4 Suppose for
Proof To see (1), note that f is well defined since N 2 and R(N 2 ) have the same vertex set V 2 . (K1), (K2), and (K3) are immediate. To see The argument for (K5) is the same as that of Theorem 3.3 in Willson (2012), used to prove that the composition of CSD maps is CSD. This completes the proof of (1).
In the situation of Theorem 6.4, to draw the wired lift of f on the diagram of N 1 , it follows that, for every arc (u, v) ∈ A 1 , we draw the arc in one of three ways: (1) a thin solid arc (u, v) ) is an arc in N 2 that is not redundant.

Deriving a Normal Network from an X-Network
This section concerns methods, given an X -network N , to produce an acyclic Xnetwork M D (N ) with desirable properties. Often, an important step may be to remove redundant arcs, thus obtaining R (M D (N )).
In particular, we shall want to find D such that R(M D (N )) is normal. Call a network N pre-normal if R(N ) is normal. Thus, we seek D such that M D (N ) is pre-normal.
Let N be an X -network. A vertex v of N is a pre-normal obstacle or (more simply) an obstacle if (1) v is not a leaf, and (2) every child of the vertex v in R(N ) is hybrid. Thus v may be regarded as an "obstacle" to R(N ) being tree-child. Since R(N ) contains no redundant arcs, this is an "obstacle" to R(N ) being normal, or equivalently to N being pre-normal. We must ignore redundant arcs when deciding our strategies concerning which arcs to merge. To make these decisions we need to have a notion of in-degree and out-degree that does not count redundant arcs.
Suppose v is a vertex of an X -network N . The non-redundant in-degree of v, denoted nrindeg(v), is the number of non-redundant arcs ( p, v); hence it is the number of parents of v by non-redundant arcs. If v = ρ then by Theorem 2.
, is the number of non-redundant arcs (v, c), hence the number of children of v by non-redundant arcs. If v is not a leaf, then by Theorem 2.1 it has a non-redundant child, whence nroutdeg(v) ≥ 1.
It is immediate that v is a pre-normal obstacle iff (1) v is not a leaf, and (2) every nonr-child of v is nonr-hybrid. Figure 8 shows an acyclic X -network with redundant arcs. Note that 8 is an obstacle since both its children 12 and 13 are nonr-hybrids. But 9 is not an obstacle since the only nonr-parent of 10 is 9 and nrindeg(10) = 1. An X -network N is obstacle-free if it contains no pre-normal obstacle.

Theorem 7.1 Suppose N is an acyclic X -network that is obstacle-free. Then R(N ) is a normal X -network and N is pre-normal.
Proof By hypothesis, for every vertex v that is not a leaf, there is a non-redundant arc (v, c) with nrindeg(c) = 1. It follows that in R(N ), c is a tree-child of v. Since R(N ) has no redundant arcs, it follows that R(N ) is normal and N is pre-normal.
Theorem 7.1 further justifies the use of the term "pre-normal obstacle". It is easy to see that a tree-child X -network is always pre-normal, but a pre-normal network need not be tree-child. Theorem 7.1 suggests our strategy for normalization: Given an arbitrary X -network N , when we seek a normal network M, we know by Lemma 2.2 that S(M) will be regular; by (1) in the definition of regular network, S(M) is SCD. We are therefore seeking a network that is very close to being an SCD network, and it is plausible to start with the very general SCD network SCD(N ). We then recursively remove obstacles until there are no obstacles remaining. Next we remove redundant arcs to obtain a normal network. If we seek to obtain a uniquely determined normalization we are careful not to make arbitrary choices about which arcs to merge. Now we show that there are different types of pre-normal obstacles. Let N be an acyclic X -network. Suppose c is an obstacle. An allowable 1-fold parent chain of c is  a path p 1 , c such that ( p 1 , c) is not redundant and p 1 has a nonr-tree-child d = c (so nrindeg(d) = 1, whence necessarily every other parent of d is via a redundant arc). An obstacle c is of type 1 if c has an allowable 1-fold parent chain. If c has type 1 and p 1 , c is an allowable 1-fold parent chain, let Dc( p 1 , c) Suppose c is an obstacle and k > 1 is an integer. An allowable k-fold parent chain for c is a nonr-path p k , p k−1 , · · · , p 1 , p 0 = c such that p k has a nonr-tree-child d distinct from p k−1 . An obstacle c is of type k if (a) c is not of type 1, · · · , k − 1; and (b) c has an allowable k-fold parent chain. In this situation, for this k-fold parent chain write

Theorem 7.2 Let N be an acyclic X -network. Then every pre-normal obstacle c has a unique type.
Proof It is clear that the type, if it exists, is unique. Consider a path from ρ to c which has maximal length k. Write this path as u 0 = ρ, u 1 , · · · , u k = c. By Theorem 2.1 this is a nonr-path. If ρ has a nonr-child d other than u 1 , then this path is an allowable k-fold parent chain of c, so c has type at most k. If, instead, u 1 is the only nonr-child of ρ, then every other child q of ρ satisfies that (ρ, q) is redundant. There is a lengthening nonr-path ρ = v 0 , v 1 , · · · , v m = q by Theorem 2.1, whence v 1 is a nonr-child of ρ; since u 1 is the only such nonr-child, it follows v 1 = u 1 . Indeed, every nonr-path from ρ to any vertex other than ρ or u 1 must begin with ρ, u 1 . If u 1 has a nonr-child d other u 2 , then u 1 , u 2 , · · · , c is an allowable (k − 1)-fold parent chain and c has type ≤ k − 1. Otherwise u 2 is the only nonr-child of u 1 . Thus any nonr-path from ρ to a vertex other than ρ, u 1 , u 2 must begin ρ, u 1 , u 2 . We repeat the argument. If at any stage we have r such that u r has a nonr-child d = u r +1 , then u r , u r +1 , · · · , c is an allowable (k − r )-fold parent chain. Otherwise every nonr-path from ρ to a vertex other than ρ, u 1 , · · · , u r must start with ρ, u 1 , · · · , u r .
If no such r < k occurs, then we find that ρ, u 1 , · · · , u k = c is a nonr-path and every nonr-path from ρ to any vertex other than ρ, u 1 , · · · , c must begin with ρ, u 1 , · · · , c. But c is not a leaf hence must have a nonr-child e. Since c is an obstacle, nrindeg(e) ≥ 2 so e has a nonr-parent q = c. Every nonr-path from ρ to q must start ρ, u 1 , · · · , c, so there is a nonr-path from c to q, hence a nonr-path from c to q to e, showing that (c, e) is redundant, a contradiction. Hence some such r < k must occur, and c has an allowable (k − r )-fold parent chain.
The following result shows a simple way to remove a type 1 obstacle:

Lemma 7.3 Suppose N is an acyclic X -network and c is a type 1 obstacle with allowable 1-fold parent chain p, c, where p has nonr-tree-child d = c. Let D = {( p, c)}. Form M D (N ) and let ψ : N → M D (N ) be the projection. Then ψ(c) is not an obstacle in M D (N ).
Proof Since c is an obstacle, it is not a leaf, so D contains no X -arc. Moreover, D is strongly closed since ( p, c) is not redundant, and Q D (N ) = M D (N ) is an acyclic X -network. Note ψ(c) = [p, c] and in M D (N ) there is an arc ([ p, c], d). If q is any parent of d in N other than p, then (q, d) is redundant since d is a nonr-tree-child of p. Hence by Theorem 2.1 it has a lengthening of maximal length, ending with a non-redundant arc into d. Thus the lengthening must include the nonr-parent p of d and there is a path in N with non-redundant arcs from q to p to d. Since ψ is a CSD map, there is a path in The result above often generalizes to obstacles of type k. The next result assumes for simplicity that D is strongly closed.

Lemma 7.4 Let N be an acyclic X -network with pre-normal obstacle c of type k.
Suppose p k , ...., c is an allowable k-fold parent chain, where p k has nonr-tree-child Proof Note that D contains no X -arcs since c is not a leaf. By Lemma 7.3, when we identify p k and p k−1 , d becomes a nonr-tree-child of [ p k , p k−1 ]. When we next identify [ p k , p k−1 ] with p k−2 , d becomes a nonr-tree-child of [ p k , p k−1 , p k−2 ]. This continues until we conclude that ψ(c) has the nonr-tree-child d.
The following lemma shows that, often, once an obstacle is removed, it does not reappear when subsequent arcs are merged. It will be useful to remove trivial vertices which may have been created in the construction process. Suppose N = (V , A, ρ, φ) is an X -network. Let E = {(u, v) ∈ A : v is a trivial vertex}. Define T(N ) = M E (N ). Note that v will have unique parent u and also a unique child since v is trivial. The next result shows that T(N ) has desirable properties.

Theorem 7.6 Let N = (V , A, ρ, φ) be an acyclic X -network. Then T(N ) is an acyclic X -network. Moreover
(1) T(N ) contains no trivial vertices and hence is a phylogenetic X -network.
Proof Note that E contains no X -arc because a leaf does not have out-degree one.
Since E is clearly strongly closed, T(N ) is an acyclic X -network. (1) and (2) follow as in Theorem 4.4. (3) is obvious since if (u, v) ∈ A and u is trivial, then cl(u) = cl (v).
For (4) we first show that since N is tree-child, N = T(N ) must also be tree-child. It suffices to prove this in the case where N is obtained from N by removing one trivial vertex t with parent q and child c. In N every non-leaf vertex except q obviously still has a tree-child, the same one as in N . We must show that q has a tree-child in N . But t has a tree-child in N which must be c so c has no other nonr-parent than t in N . Hence q in N has child c which has no other nonr-parent and is therefore a tree-child. This proves T(N ) is tree-child.
For (4) we must also prove that T(N ) has no redundant arc. Again we may assume that N is obtained from N by the removal of a single trivial vertex t with parent q and child c. The only possible redundant arc in N is the new arc (q, c). If it is redundant, there is a path in N from q to c other than the arc, hence a path in N from q to c not through t. Such a path of maximal length by the proof of Theorem 2.1 contains no redundant arc, so c has a nonr-parent besides t. This contradicts that N was normal, since t in N has no tree-child.
We now show how, given an X -network N , to compute a uniquely determined normal X -network. We first compute a uniquely determined pre-normal acyclic Xnetwork Prenorm(N ), which we call the pre-normalization of N . The computation uses the procedure PRENORM described below. Briefly, if N is not already a prenormal acyclic X -network, we compute N 1 = SCD(N ). If N 1 contains no obstacles then Prenorm(N ) = N 1 . Otherwise, for each obstacle c we compute its type k and find all the allowable k-fold parent chains for c. Let D(c) be the union of Dc( p k , · · · , c) = {( p k , p k−1 ), · · · , (p 1 , c) for all such allowable chains p k , · · · , c for c. Let D be the union of the D(c) for all the obstacles c. We then compute N 2 = M D (N 1 ). If this has no obstacles then Prenorm(N ) = N 2 . If not, we repeat the process.
Here is a more detailed description of the computation of Prenorm(N ): Procedure PRENORM. Input: An X -network N .
Output: A pre-normal acyclic X -network and an integer i. 1. Let N 0 = N and set i = 0. 2. If N 0 is acyclic and contains no pre-normal obstacle, go to step 9. Otherwise, go to step 3.
Since Prenorm(N ) is a pre-normal acyclic X -network, we remove the redundant arcs to form R(Prenorm(N )), which will be normal. It may, however, contain trivial vertices, so we define Norm(N ) = T(R(Prenorm(N ))), which will be normal and contain no trivial vertices. We call Norm(N ) the normalization of N . The next theorem records its basic properties. (1) N orm(N ) = T(R (Prenorm(N ))) is a normal acyclic X -network containing no trivial vertices, hence a phylogenetic X -network.
(2) The definition of Norm(N ) depends only on the geometry of N .
(4) The composition f = ψ 2 • ψ 1 as maps of vertices from N to Norm(N ) is a connected map. (5) There is a wired lift of Norm(N ) into N which may contain dashed arcs.
We will abbreviate the name of the procedure to VARIANT. Thus while PRENORM uses all allowable k-fold parent chains for each obstacle c of type k, VARIANT would use just one allowable parent chain for each such obstacle. The following theorem shows that the output of VARIANT has interesting properties. The proof is like those of Theorems 7.7 and 7.8 and is omitted.
Apply procedure VARIANT PRENORM to N . Then (1) The procedure terminates and outputs an acyclic X -network N r which is prenormal.
(2) The projection ψ 1 : N → N r is a leaf-preserving CSD map.
(3) There is a wired lift of N r into N that contains no dashed arcs.   [7,11],13) and ( [6,8],12). There are no obstacles, so the height r = 2 and Prenorm(N ) = N 2 . We remove the redundant arcs to find R(M D (N )), which contains the trivial vertex 13. Then we compute T(R(M D (N ))) to remove the trivial vertex by merging the arc (10,13) as in Theorem 7.6 to yield Norm(N ) = T(R(M D (N ))), shown in Fig. 11.
It is easy to compute that d R F (N ,  Figure 13 shows the wired lift of Norm(N ) into an X -network N with a single obstacle 16 of type 3. N is seen by changing all thin solid or dashed arcs to wide solid. N is easily verified to be already SCD, so N 1 = N . One 3-fold parent chain in N is 11, 14, 17, 16; another is 13, 14, 17, 16. The chain 13,14,16 is not Fig. 13 The wired lift of Norm(N ) for an X -network N with a single obstacle 16 of type 3. N is seen if all arcs are instead made wide solid. The wired lift of Prenorm(N ) is seen if the arcs (9,11), (16,22), and (19,22) are all instead made wide solid  (9, 16) and (16, 22), arising from the arcs (9,11) and (16,22) in N . When these arcs are removed from M D (N ) and the resulting trivial vertex 22 is removed, we obtain Norm(N ) = T(R(M D (N ))).

Example 5
As maps of vertex sets ψ 1 : N → M D (N ) and ψ 2 : R(M D (N )) → Norm(N ) can be composed to yield the resulting connected map f = ψ 2 • ψ 1 : N → N orm(N ); it is not a CSD map because the vertex map from M D (N ) to R(M D (N )) is not CSD. The wired lift ( f −1 , E 1 ) of Norm(N ) is shown in Fig. 13. The arcs (9,11) and (16,22) are dashed because they are pre-arcs to the redundant arcs of M D (N ), which are not arcs of Norm(N ), hence are not in E 1 of the wired lift, using Theorem 6.3. Arcs (u,v) such that f (u) = f (v) are thin solid. Hence arcs in the induced subgraph of f −1 ( f (16)) = {11, 13, 14, 16,17} are thin solid. The arc (19, 22) is thin solid because it was merged to remove the trivial vertex 22. Note the g-path 16,17,19,22,4 from 16 to 4; but 16,22,4 is not a g-path.

Examples with Real Data
This section contains two examples from real biological data. Glémin et al. (2019) study pervasive hybridizations of wheat relatives. Their Fig. 5 shows their proposed scenario for the history of diploid Aegilops/Triticum species. Let N be their graph. A wired lift of Norm(N ) is shown in our Fig. 15. The network N is seen if each arc in Fig. 15 is made wide solid. In SCD(N ) we find only a single pre-normal obstacle 21 of type 1. The height of the computation is r = 2. When we compute Prenorm(N ), there is a single redundant arc. Norm(N ) contains 23 vertices and 29 arcs; it is thus simpler than N , which contains 31 vertices (of which eight are hybrid), and 38 arcs. We find d R F (N , Norm(N )) = 1. It is interesting that our dashed arc (21,22) is also dashed in Glémin et al. (2019) to indicate a less likely event. It turns out that in this case FHS(N ) ∼ = Norm(N ). Marcussen et al. (2015) exhibit a network N for the angiosperm genus Viola in their Fig. 4. Our methods find N 1 = SCD(N ) has 2 obstacles. One obstacle is type 1 with two allowable 1-fold parent chains. The other is type 2 with one allowable 2-fold parent chain. Thus for computing Prenorm(N ), D contains 4 arcs. M D (N 1 ) has no obstacles, so Prenorm(N ) = M D (N 1 ) and the height is two. A wired lift of Norm(N ) = T(R(Prenorm(N ))) is shown in Fig. 16. We see that Norm(N ) has 29 vertices (equivalence classes under thin solid arcs). It turns out to have 31 arcs, while the wired lift has 34 wide solid arcs. If ψ : N → Norm(N ) is the connected map, more than one wide solid arc (u, v) can map to the same arc (ψ(u), ψ(v)) of Norm(N ). Thus (14,10) and (17,18) map to the same arc in Norm(N ), as do (28,37) and (28,36), and also (11,12) and (11, 21). Francis et al. (2021) in their Fig. 3. We compute that d R F (N , FHS(N )) = 4 while d R F (N , Norm(N )) = 5. Thus FHS(N ) is a better approximation to N than Norm(N ) but lacks a wired lift.
Both such Norm V (N ) have wired lifts into N by Theorem 7.9. The wired lift of each Norm V (N ) is very similar to that for Norm(N ) with one additional wide solid arc replacing a thin solid arc. For one Norm V (N ) the wired lift is given by making the arc (12,13) in Fig. 16 be wide solid; for the other Norm V (N ), the only change is that the arc (22,13) in Fig. 16 is wide solid.
Further comments concerning this example are given in Sect. 9.

Discussion
Comparison of Norm(N ) and FHS(N ) Let N = (V , A, ρ, φ) be an X -network. It is interesting to contrast Norm(N ) with FHS(N ), defined in Francis et al. (2021). Both are uniquely determined normal phylogenetic X -networks depending only on the  Fig. 16 The wired lift of Norm(N ) for the Viola data N in Marcussen et al. (2015). The entire vertical line labeled 42 represents one vertex with out-degree 7 geometry of N . Both allow vertices of N to have in-degree greater than 2 or out-degree greater than 2, and both apply quite generally. FHS(N ) is fast to compute using Huson and Steel (2020) and very elegant. It works by locating the "visible" vertices of N . A vertex v is visible if there exists x ∈ X such that every path from ρ to φ(x) contains v. This set of visible vertices forms the initial vertex set of FHS(N ). Hence each initial vertex of FHS(N ) can be highlighted in the diagram for N , as is done in Francis et al. (2021). At the end, trivial initial vertices of FHS(N ) are suppressed. In a tangled network like our Fig. 4, the only visible vertices are the root and the leaves, since there is a great multiplicity of possible paths from the root to a given leaf. In such a situation, FHS(N ) does not perform well. For less tangled networks such as Example 7 the computation works well. Perhaps it would be useful in general to compute FHS(SCD(N )).
The arcs of FHS(N ) are harder to interpret than the vertices. In FHS(N ) there is an arc (u, v), where u and v are distinct visible vertices of N , precisely when u ≤ v in N and there is no third visible vertex w such that u ≤ w and w ≤ v. Thus, for example, two different arcs (u, v 1 ) and (u, v 2 ) emerging from the same u could be present because of directed paths in N from u to v 1 and from u to v 2 such that the paths have significant overlap, invisible in FHS(N ).
In contrast, the arcs of Norm(N ) are easy to interpret. The wide solid arcs highlight the arcs N that appear in Norm(N ); the thin dashed arcs indicate redundant arcs in Prenorm(N ) and must be avoided in g-paths; the thin solid arcs tell what arcs must be merged to obtain the normal network Norm(N ). The use of g-paths lets us understand Norm(N ) from just the wired lift.
Software The author has written software using Xcode which implements the calculation of Norm(N ) somewhat interactively. It was essential for the examples based on real data. It computes SCD(N ), M D (N ), R(N ), and T(N ) and locates all obstacles. It finds all allowable 1-fold and 2-fold parent chains, but obstacles of type k ≥ 3 must be handled interactively. The software is far from ready for general use, but it shows that the calculations can be automated.
Future work One can ask whether there are other classes of networks besides normal networks for which a similar construction could be used to simplify a network N into one of this other class. Suppose, given an X -network N , we sought a treechild X -network C. Since a tree child network may contain redundant arcs, we should like a construction that depends only on the geometry of N and yields a CSD map ψ : N → C. At first glance we might think that we could use the CSD map ψ : N → Prenorm(N ); but while Norm(N ) is tree-child, Prenorm(N ) need not be tree-child. The author is currently looking at such problems for tree-child and some other classes of networks.