Quarnet Inference Rules for Level1 Networks
Abstract
An important problem in phylogenetics is the construction of phylogenetic trees. One way to approach this problem, known as the supertree method, involves inferring a phylogenetic tree with leaves consisting of a set X of species from a collection of trees, each having leafset some subset of X. In the 1980s, Colonius and Schulze gave certain inference rules for deciding when a collection of 4leaved trees, one for each 4element subset of X, can be simultaneously displayed by a single supertree with leafset X. Recently, it has become of interest to extend this and related results to phylogenetic networks. These are a generalization of phylogenetic trees which can be used to represent reticulate evolution (where species can come together to form a new species). It has recently been shown that a certain type of phylogenetic network, called a (unrooted) level1 network, can essentially be constructed from 4leaved trees. However, the problem of providing appropriate inference rules for such networks remains unresolved. Here, we show that by considering 4leaved networks, called quarnets, as opposed to 4leaved trees, it is possible to provide such rules. In particular, we show that these rules can be used to characterize when a collection of quarnets, one for each 4element subset of X, can all be simultaneously displayed by a level1 network with leafset X. The rules are an intriguing mixture of tree inference rules, and an inference rule for building up a cyclic ordering of X from orderings on subsets of X of size 4. This opens up several new directions of research for inferring phylogenetic networks from smaller ones, which could yield new algorithms for solving the supernetwork problem in phylogenetics.
Keywords
Inference rules Phylogenetic network Quartet trees Closure Cyclic orderings Level1 network Quarnet Qnet1 Introduction
In case the collection \(\mathcal {Q}\) consists of a quartet tree for every possible subset of X of size 4 (which we denote by \(X \atopwithdelims ()4\)), this problem has an elegant solution that was originally presented by Colonius and Schulze (1981) (see also Bandelt and Dress 1986 for related results). We present full details in Theorem 1, but essentially their result states that, given a collection of quartet trees \(\mathcal {Q}\), one for each element in \(X \atopwithdelims ()4\), there exists (a necessarily unique) binary phylogenetic Xtree displaying every quartet tree in the collection if and only if when the quartet trees abcx and abxd are contained in \(\mathcal {Q}\) then so is the quartet tree abcd. Rules such as abcx plus abxd implies abcd are known as inference rules, and they have been extensively studied in the phylogenetics literature (see e.g. Semple and Steel 2003, Chapter 6.7 and the references therein).
Although phylogenetic trees are extremely useful for representing evolutionary histories, in certain circumstances they can be inadequate. For example, when two viruses recombine to form a new virus (e.g. swine flu), this is not best represented by a tree as it involves species combining together to form a new one rather than splitting apart. In such cases, phylogenetic networks provide a more accurate alternative to trees and there has been much recent work on such structures (see e.g. Steel 2016, Chapter 10 for a recent review).
In this paper, we will consider properties of a particular type of phylogenetic network called a level1 network (Gambette et al. 2012).^{1} For a set X of species, this is a connected graph with leafset X and such that every maximal subgraph with no cutedge is either a vertex or a cycle (see Sect. 2 for more details). Our main results will apply to binary level1 networks, where we also assume that every vertex has degree 1 or 3. We present an example of such a network in Fig. 1. Note that a phylogenetic Xtree is a special example of a level1 network with leafset X. As with phylogenetic Xtrees, it is possible to construct level1 networks from quartets (Gambette et al. 2012). However, it has been pointed out that there are problems with understanding such networks in terms of inference rules (see e.g. Keijsper and Pendavingh 2014, p. 2540).
Here, we circumvent these problems by considering a certain type of subnetwork of level1 network called a quarnet instead of using quartet trees. A quarnet is a 4leaved, binary, level1 network (see e.g. Fig. 1); they are displayed by binary level1 networks in a similar way to quartets (see Sect. 3 for details). As we shall see, quarnets naturally lead to inference rules for level1 networks which can be thought of as a combination of quartet inference and inference rules for building circular orderings of a set. Moreover, in our main result we show that, just as with phylogenetic trees, the quarnet inference rules that we introduce can be used to characterize when a collection of quarnets, one for each element in \({X \atopwithdelims ()4}\), is equal to the set of quarnets displayed by a binary level1 network with leafset X.
We now summarize the contents of the rest of the paper. In the next section, we present some preliminaries concerning phylogenetic trees and level1 networks, as well as their relationship with quartets. Then, in Sect. 3, we prove an analogous theorem to the quartet results of Colonius and Schulze for level1 networks (Theorem 2). In Sect. 4, we use Theorem 2 to provide a characterization for when a set of quartets, one for each element of \(X \atopwithdelims ()4\), can be displayed by a binary level1 network (Theorem 3). In Sect. 5, we then define the closure of a set of quarnets. This can be thought of as the collection of quarnets that is obtained by applying inference rules to a given collection of quarnets until no further quarnets are generated. We show that this has similar properties to the socalled semidyadic closure of a set of quartets (see Theorem 4). We conclude with a brief discussion of some possible further directions.
2 Preliminaries
In this section, we review some definitions as well as results concerning the connection between phylogenetic trees and quartets. From now on, we assume that X is a finite set with \(X\ge 2\).
2.1 Definitions
An unrooted phylogenetic network N (on X) (or network N (on X) for short) is a connected graph (V, E) with \(X \subseteq V\), every vertex has either degree 1 or degree at least 3, and the set of degree1 vertices is X. The elements in X are the leaves of N. We also denote the leafset of N by L(N). The network is called binary if every vertex in N has degree 1 or 3. An interior vertex of N is a vertex that is not a leaf. A cherry in N is a pair of leaves that are adjacent with the same vertex. Two phylogenetic networks N and \(N'\) on X are isomorphic if there exists a graph theoretical isomorphism between N and \(N'\) whose restriction to X is the identity map.
Note that a phylogenetic (X) tree is a network which is also a tree. For any three vertices \(u_1,u_2,u_3\) in such a tree T, their median, denoted by \(\mathop {med}(u_1,u_2,u_3)=\mathop {med}_T(u_1,u_2,u_3)\), is the unique vertex in T that is contained in every path between any two vertices in \(\{u_1, u_2, u_3\}\).
A cutvertex of a network is a vertex whose removal disconnects the network, and a cutedge of a network is an edge whose removal disconnects the network. A cutedge is trivial if one of the connected components induced by removing the cutedge is a vertex (which must necessarily be a leaf). A network is simple if all of the cutedges are trivial (so for instance, note that phylogenetic trees with more than three leaves are not simple networks). A network N is level1 if every maximal subgraph in N that has no cutedge is either a vertex or a cycle. Note that we shall say that a network N on X, where \(X\ge 3\), is of cycle type if it contains a unique cycle of length X, and the number of vertices in N is 2X (so in particular, a network is of cycle type if it is simple, binary, level1 and is not a phylogenetic tree).
2.2 Quartets, Trees and Networks
We now briefly recall some notation and results concerning quartet systems (for more details see Dress et al. 2012, Chapter 3).
Although quartets are often considered as being 4leaved trees, here it is more convenient to consider a quartet Q to be a partition of a subset Y of X of size 4 into two subsets of size 2. The set Y is called the support of Q. If \(Q = \{\{a,b\},\{c,d\}\}\) for \(a,b,c,d \in X\) distinct, we denote Q by abcd. The set of all quartets on X is denoted by \({\mathcal {Q}}(X)\), and any nonempty subset \({\mathcal {Q}}\subseteq {\mathcal {Q}}(X)\) is called a quartet system (on X). Given a quartet system \({\mathcal {Q}}\) on X and a subset \(Y\in {X \atopwithdelims ()4 }\), let \(m(Y)=m_{{\mathcal {Q}}}(Y)\) be the number of quartets in \({\mathcal {Q}}\) whose support is Y. For simplicity, we write \(m(\{a,b,c,d\})\) as m(a, b, c, d). If \(m(Y)\ge 1\) for every subset \(Y\in {X \atopwithdelims ()4 }\), then \({\mathcal {Q}}\) is said to be dense.

thin if no pair of quartets in \({\mathcal {Q}}\) have the same support;

saturated if for all \(\{a,b,c,d,x\}\in {X \atopwithdelims ()5}\) with \(abcd \in {\mathcal {Q}}\), the system \({\mathcal {Q}}\) contains at least one quartet in \(\{axcd, abcx\}\);

transitive if for all \(\{a,b,c,d,x\}\in {X \atopwithdelims ()5}\), if \(\{abcx, abxd\}\subseteq {\mathcal {Q}}\) holds, then abcd is also contained in \({\mathcal {Q}}\).
Lemma 1
Suppose that \({\mathcal {Q}}\) is a quartet system on X. If \({\mathcal {Q}}\) is saturated and thin, then \({\mathcal {Q}}\) is transitive.
Proof
We use a similar argument to that used by Bandelt and Dress (1986, Lemma 1). Suppose \(\{a,b,c,d,x\}\in {X \atopwithdelims ()5}\) with \(\{abcx, abxd\}\subseteq {\mathcal {Q}}\). We need to show \(abcd\in {\mathcal {Q}}\).
Since \({\mathcal {Q}}\) is saturated and abcx is contained in \({\mathcal {Q}}\), we have \(\{abcd, adcx\}\cap {\mathcal {Q}}\not = \emptyset \). Using a similar argument, abdx in \({\mathcal {Q}}\) implies that \(\{abcd, acdx\}\cap {\mathcal {Q}}\not = \emptyset \). Therefore, we must have \(abcd\in {\mathcal {Q}}\) as otherwise \(\{adcx,acdx\}\subset {\mathcal {Q}}\), a contradiction to the assumption that \({\mathcal {Q}}\) is thin. \(\square \)
A quartet abcd on X is displayed by a phylogenetic Xtree T if the path between a and b in T is vertex disjoint from the path between c and d in T. The quartet system displayed by T is denoted by \({\mathcal {Q}}(T)\).
In view of Dress et al. (2012, Theorem 3.7) and the last lemma, we have the following slightly stronger characterisation of quartet systems displayed by a phylogenetic tree, which was stated in Bandelt and Dress (1986, Proposition 2) using slightly different terminology.
Theorem 1
A quartet system \({\mathcal {Q}}\subseteq {\mathcal {Q}}(X)\) is of the form \({\mathcal {Q}}={\mathcal {Q}}(T)\) for a (necessarily unique) phylogenetic Xtree T if and only if \({\mathcal {Q}}\) is thin and saturated.
We now turn our attention to the relationship between quartets and level1 networks.
A split AB of X is a bipartition of X into two nonempty parts A and B (note that since AB is a bipartition, order does not matter, that is, \(AB=BA\)). Such a split is induced by a network N if there exists a cutedge in N whose removal results in two connected components, one with leafset A and the other with leafset B. A quartet abcd is exhibited by a network N if there exists a split AB induced by N such that \(\{a,b\}\subseteq A\) and \(\{c,d\}\subseteq B\).
Note that if a quartet \(abcd \in {\mathcal {Q}}(X)\) is exhibited by N, then it is displayed by N, that is, N contains two disjoint paths, one from a to b, and the other from c to d. However, the converse is not true. For example, quartet abcd is displayed by the network in Fig. 4(iv), but abcd is not exhibited by this network. Given a network N, we let \(\Sigma (N)\) denote the set of quartets exhibited by N, and let \({\mathcal {Q}}(N)\) be the set of quartets displayed by N. In the light of the last remark, clearly we have \(\Sigma (N)\subseteq {\mathcal {Q}}(N)\).
3 Quarnets
In this section, we shall show that an analogue of Theorem 1 holds for quarnets and level1 networks. We begin by formally defining the concept of a quarnet and how quarnets can be obtained from level1 networks.
We now turn to characterizing when a qnet system is displayed by a level1 network. To do this, we introduce some additional concepts concerning qnet systems.
First, a qnet system \({\mathcal {F}}\) on X is consistent (on subsets of X of size three) if for all subsets \(A \in {X \atopwithdelims ()3}\), \(F_A\) is isomorphic to \(F'_A\), for each pair of qnets in \({\mathcal {F}}\) with \(A\subseteq L(F)\cap L(F')\). In addition, a qnet system \({\mathcal {F}}\) on X is minimally dense if for all \(Y \in {X \atopwithdelims ()4}\), there exists precisely one qnet in \({\mathcal {F}}\) with support Y.
Second, we say that a qnet system \({\mathcal {F}}\) on X is cyclically transitive or cyclative if for all subsets \(\{a,b,c,d,x\} \in {X \atopwithdelims ()5}\) with \(\{{a}\oplus {b}\oplus {c}\oplus {d}, {x}\oplus {a}\oplus {c}\oplus {d}\} \subseteq {\mathcal {F}}\), the system \({\mathcal {F}}\) also contains \({a}\oplus {b}\oplus {d}\oplus {x}\). Note that this is closely related to the cyclicordering inference rule given in Bandelt and Dress (1992, Proposition 1).
 (S1)
If \({\mathcal {F}}\) contains \({a}\ominus {b}{c} \ominus {d}\), then \({a}\ominus {b}{c} \ominus {x}\), or \({a}\ominus {b}{c}\oplus {x}\), or \({a}\ominus {x}{c} \ominus {d}\), or \({a}\oplus {x}{c}\ominus {d}\) is contained in \({\mathcal {F}}\).
 (S2)
If \({\mathcal {F}}\) contains \({a}\oplus {b}{c}\ominus {d}\), then \({a}\oplus {b}{c}\ominus {x}\), or \({a}\oplus {b}{c}\oplus {x}\), or \({a}\ominus {x}{c} \ominus {d}\), or \({a}\oplus {x}{c}\ominus {d}\) is contained in \({\mathcal {F}}\).
 (S3)
If \({\mathcal {F}}\) contains \({a}\oplus {b}{c}\oplus {d}\), then \({a}\oplus {b}{c}\ominus {x}\), or \({a}\oplus {b}{c}\oplus {x}\), or \({a}\ominus {x}{c}\oplus {d}\), or \({a}\oplus {x}{c}\oplus {d}\) is contained in \({\mathcal {F}}\).
Lemma 2
 (i)
If \({\mathcal {F}}\) is minimally dense, then \(\Sigma ({\mathcal {F}})\) is thin.
 (ii)
If \({\mathcal {F}}\) is saturated, then \(\Sigma ({\mathcal {F}})\) is saturated.
Proof
For the proof of (i), as \({\mathcal {F}}\) is minimally dense, for each subset Y of X with size four, there exists precisely one qnet F in \({\mathcal {F}}\) whose support is Y. Hence, there exists at most one quartet in \(\Sigma ({\mathcal {F}})\) with support Y.
We now characterize when a minimally dense set of qnets is displayed by a level1 network.
Theorem 2
Let \({\mathcal {F}}\) be a minimally dense qnet system on X with \(X\ge 4\). Then, \({\mathcal {F}}={\mathcal {F}}(N)\) for some (necessarily unique) binary, level1 network N on X if and only if \({\mathcal {F}}\) is consistent, cyclative and saturated.
Proof
Clearly, if \({\mathcal {F}}={\mathcal {F}}(N)\) holds for a binary, level1 network N, then \({\mathcal {F}}(N)\) is consistent, cyclative and saturated.
We now show that the converse holds. Suppose that \({\mathcal {F}}\) is a minimally dense qnet system on X that is consistent, cyclative and saturated. Consider the quartet system \(\Sigma =\Sigma ({\mathcal {F}})\). By Lemma 2, \(\Sigma \) is thin and saturated. Therefore, by Theorem 1, there exists a unique phylogenetic tree T with \({\mathcal {Q}}(T)=\Sigma \).
For each interior vertex v in T, let \({\mathcal {A}}_v\) denote the partition of X induced by deleting v from T so that, in particular, the number of parts in \({\mathcal {A}}_v\) is equal to the degree of v. Note that, for all \(A \in {\mathcal {A}}_v\), if \(a \in A\) and \(b\in XA\), the path in T between a and b must contain v, and if \(a,b \in A\), the path between a and b does not contain v.
We next partition the set of interior vertices of T. Let \(V_1(T)\) be the set of degree3 vertices v in T with the property that there exist three elements, one from each distinct part of \({\mathcal {A}}_v\), so that there exists a qnet F in \({\mathcal {F}}\) whose restriction to these three elements is of cycle type. Let \(V_0(T)\) be the set of degree3 vertices in T not contained in \(V_1(T)\). Lastly, let \(V_2(T)\) be the set of interior vertices in T with degree at least 4.\(\square \)
Claim 1
A degree3 vertex v in T is contained in \(V_1(T)\) if and only if, for each subset Y of X of size three that contains precisely one element from each part of \({\mathcal {A}}_v\), the restriction \(F_{Y}\) is of cycle type for every qnet F in \({\mathcal {F}}\) with \(Y\subset L(F)\).
Proof
Since \({\mathcal {F}}\) is minimally dense, the “if ” direction follows directly from the definition of \(V_1(T)\).
Conversely, let \(Y^*=\{a^*_1,a^*_2,a^*_3\}\) be such that \(a^*_i\), \(1\le i \le 3\), are all contained in distinct parts of \({\mathcal A}_v\) and there exists a qnet \(F^*\) in \({\mathcal {F}}\) such that \(F^*_{Y^*}\) is of cycle type. Now let \(Y=\{a_1,a_2,a_3\}\) with \(a_i\) all contained in distinct parts of \({\mathcal {A}}_v\) and let F be an arbitrary qnet in \({\mathcal {F}}\) with \(Y\subset L(F)\). We shall show that \(F_Y\) is of cycle type by considering the size of the intersection \(Y\cap Y^*\).
First assume that \(Y\cap Y^*=3\), that is, \(Y=Y^*\). Then, as \({\mathcal {F}}\) is consistent, \(F_Y\) is of cycle type since it is isomorphic to \(F^*_{Y^*}\).
Second assume that \(Y\cap Y^*=2\). By swapping the indices, we may further assume that \(a_1=a^*_1\), \(a_2=a^*_2\), and \(a_3\not =a^*_3\). In other words, we have \(Y=\{a^*_1,a^*_2,a_3\}\). Consider \(Y'=\{a^*_1,a^*_2,a_3,a^*_3\}\) and let \(F'\) be the qnet in \({\mathcal {F}}\) with \(L(F')=Y'\). Since \(a_3,a^*_3\) are both contained in \(A_v\), the quartet \(Q'=a^*_1a^*_2a_3a^*_3\) is contained in \({\mathcal {Q}}(T)\). As \(F'_{Y^*}\) is of cycle type, this implies that \(F'\) is either \({a^*_1}\oplus {a^*_2}{a_3}\ominus {a^*_3}\) or \({a^*_1}\oplus {a^*_2}\oplus {a_3}\oplus {a^*_3}\). In both cases, \(F'_{Y}\) is of cycle type, and hence \(F_Y\) is also of cycle type in view of the consistency of \({\mathcal {F}}\).
Next assume that \(Y\cap Y^*=0\). By swapping the indices, we may further assume that, for \(1\le i \le 3\), elements \(a_i\) and \(a^*_i\) are contained in the same part of \({\mathcal {A}}_v\) but \(a_i\not = a^*_i\). Consider the sets \(Y_1=\{a^*_1,a^*_2,a_3\}\) and \(Y_2=\{a^*_1,a_2,a_3\}\), and put \(Y_0=Y^*\) and \(Y_3=Y\). Then, we have \(Y_i\cap Y_{i+1}=2\) for \(0\le i \le 2\). Repeatedly applying the argument used when the size of the intersection is two, it follows that \(F_Y\) is of cycle type, as required.
Lastly, the case \(Y\cap Y^*=1\) can be established using a similar argument to that when the size of the intersection is zero. This completes the proof of the claim. \(\square \)
Although we will not use this fact later, note that it follows from Claim 1 that a vertex v in T is contained in \(V_0(T)\) if and only if, for each subset Y of X of size three whose elements are contained in distinct elements of \(\mathcal {A}_v\), the restriction \(F_Y\) is a tree type for every qnet F in \(\mathcal {F}\) with \(Y\subset L(F)\).
Claim 2
Suppose \(v \in V_2(T)\). Let \(x,y,p,q \in X\) be contained in distinct parts \(A_x,A_y,A_p,A_q\) of \(\mathcal {A}_v\), respectively. Then, the qnet F in \({\mathcal {F}}\) with support \(A = \{x,y,p,q\}\) is of Type IV. Moreover, if F is \({x}\oplus {y}\oplus {p}\oplus {q}\), then, for all \(x'\in A_x\), \(y' \in A_y\), \(p'\in A_p\) and \(q'\in A_q\), the qnet \(F'\) with support \(A' = \{x',y',p',q'\}\) is \({x'}\oplus {y'}\oplus {p'}\oplus {q'}\).
Proof
Suppose F is not of Type IV. Then, \(\Sigma (F)\) contains precisely one quartet, denoted by Q, and \(L(Q)=A\). This implies that \(Q \in \Sigma ({\mathcal {F}})={\mathcal {Q}}(T)\). However, Q is not contained in \({\mathcal {Q}}(T)\) because the path between any pair of distinct elements in A contains v; a contradiction. Thus, F is of Type IV.
Now, suppose \(A\cap A'=3\). Then, we may further assume without loss of generality that \(x=x'\), \(y=y'\), \(p=p'\), and \(q\not = q'\). Hence, \(A'=\{x,y,p,q'\}\). Note that the argument in the last paragraph implies that \(F'\) is of Type IV. If \(F'\) is not isomorphic to \({x}\oplus {y}\oplus {p}\oplus {q'}\), then \(F'\) is isomorphic to either \({x}\oplus {y}\oplus {q'}\oplus {p}\) or \({x}\oplus {p}\oplus {y}\oplus {q'}\). In the first subcase, since \({\mathcal {F}}\) is cyclative and \(\{{x}\oplus {y}\oplus {p}\oplus {q}, {x}\oplus {y}\oplus {q'}\oplus {p}\} \subset {\mathcal {F}}\), the qnet \({p}\oplus {q}\oplus {y}\oplus {q'}\) is contained in \({\mathcal {F}}\). This implies that the quartet \(Q'= pyq q'\) is not contained in \({\mathcal {Q}}(T)\), a contradiction since \(q,q'\) are contained in \(A_q\) while p, y are contained in \(XA_q\). The second subcase follows in a similar way.
Lastly, if \(A\cap A'\le 2\), then note that there exists a list of 4element subsets \(A=A_0,\ldots ,A_t=A'\) for some \(t\ge 1\) such that, for \(0\le i <t\), we have \(A_i \cap A_{i+1}=3\) and the two elements in \((A_iA_{i+1})\cup (A_{i+1}A_i)\) are contained in the same part of \(\mathcal {A}_v\). Claim 2 follows by repeatedly applying the argument in the last paragraph to the list. \(\square \)
Using the last claim, we next establish the following.
Claim 3
For each vertex \(v \in V_2(T)\), there exists a unique circular ordering of the parts \(A^1,\ldots ,A^m\) of \(\mathcal {A}_v\) such that, for each tuple \(A=(a_i,a_j,a_k,a_l) \in A^i \times A^j \times A^k \times A^l\) with \(1\le i<j<k<l \le m\), the qnet in \({\mathcal {F}}\) with support \(\{{a_i,a_j,a_k,a_l}\}\) is isomorphic to \({a_i}\oplus {a_j}\oplus {a_k}\oplus {a_l}\).
Proof
In the light of Claim 2, we can define a quaternary relation  on the parts of \(\mathcal {A}_v\) by setting ABCD, for all distinct parts \(A,B,C,D \in \mathcal {A}_v\), if and only if, for all \(x \in A\), \(y \in B\), \(p \in C\) and \(q \in D\), the qnet with support \(\{x,y,p,q\}\) is \({x}\oplus {p}\oplus {y}\oplus {q}\). Put differently, the distance between x and p in the qnet with support \(\{x,y,p,q\}\) is two, and so is the distance between y and q.
 (BD1):
ABCD implies BACD and CDAB;
 (BD2):
either ABCD, or ACBD, or ADBC (exclusively);
 (BD3):
ACBD and ADCE implies ACBE.
Since the quaternary relation  on \(\mathcal {A}_v\) satisfies the conditions (BD1)–(BD3) as specified in Proposition 1 on page 73 of Bandelt and Dress (1992), it follows that  determines a unique circular ordering of the parts in \(\mathcal {A}_v\) as specified in Claim 3. \(\square \)
Now let \(V'=V_1(T)\cup V_2(T)\), and for each vertex \(u\in V'\), fix a circular ordering of its neighbourhood \(N_u(T)\) induced by the ordering of \(\mathcal {A}_u\) in Claim 3 if \(u\in V_2(T)\), or the necessarily unique circular ordering (clockwise and anticlockwise are treated as the same) of \(N_u(T)\) if \(u\in V_1(T)\) (and hence \(N_u(T) =3\)). Let N be the level1 network obtained from T by blowing up each vertex u in \(V'\) using the given circular ordering of \(N_u(T)\). We next show that \({\mathcal {F}}\subseteq {\mathcal {F}}(N)\). To this end, fix four arbitrary elements a, b, c, d in X and let F be the qnet in \({\mathcal {F}}\) with support \(\{a,b,c,d\}\). We need to show that \(F\in {\mathcal {F}}(N)\). There are four cases depending upon whether F is Type I, II, III, or IV.
First suppose F is of Type I. Without loss of generality, we may assume that \(F={a}\ominus {b}{c} \ominus {d}\). Let \(u=\mathop {med}_T(a,b,c)\). If \(u\in V_1(T)\cup V_2(T)\), then a, b, c are contained in three distinct parts in the partition \({\mathcal {A}}_u\) of X on u. By Claims 1 and 2, it follows that \(F_A\) with \(A=\{a,b,c\}\) is of cycle type, a contradiction. Thus, \(u\in V_0(T)\) and so there exists a cutvertex in N whose removal induces three connected components, containing a, b and c, respectively. Similarly, the median \(v=\mathop {med}_T(a,c,d)\) is contained in \(V_0(T)\). Hence, there exists a cutvertex in N whose removal induces three connected components, containing a, c and d, respectively. Let \(F'\) be the qnet in \({\mathcal {F}}(N)\) whose support is \(\{a,b,c,d\}\). Thus, by inspecting all possible qnets on \(\{a,b,c,d\}\), it follows that \(F'\) is isomorphic to \({a}\ominus {b}{c} \ominus {d}\), and hence \(F\in {\mathcal {F}}(N)\).
Second, suppose that F is of Type II. Without loss of generality, we may assume that \(F={a}\oplus {b}{c}\ominus {d}\). Let \(F'\) be the qnet in \({\mathcal {F}}(N)\) whose support is \(\{a,b,c,d\}\). Let u be the median of a, c, d in T. Then, by an argument similar to the one used in the last paragraph, it follows that there exists a cutvertex in N (and hence also a cutvertex in \(F'\)) whose removal results in three connected components, containing a, c and d, respectively. On the other hand, let v be the median of \(A=\{a,b,c\}\) in T. Then, a, b, c are contained in three distinct parts of \({\mathcal {A}}_v\). Since \(F_A\) is of cycle type, by Claim 2 it follows that \(v\in V_1(T)\cup V_2(T)\), which implies that \(F'_A\) is also of cycle type. Thus, by inspecting all possible qnets on \(\{a,b,c,d\}\), it follows that \(F'\) is isomorphic to \({a}\oplus {b}{c}\ominus {d}\), and hence \(F\in {\mathcal {F}}(N)\).
Next, suppose that F is of Type III. Without loss of generality, we may assume that \(F={a}\oplus {b}{c}\oplus {d}\). Let \(F'\) be the qnet in \({\mathcal {F}}(N)\) whose support is \(\{a,b,c,d\}\). Let u be the median of \(A=\{a,b,c\}\) in T and v be the median of \(B=\{a,c,d\}\) in T. Since the quartet abcd is contained in \({\mathcal {Q}}(T)\), we know that u and v are distinct. Hence, there exists a cutedge whose deletion puts a and b in one component and c and d in the other connected component. By an argument similar to that used for analysing when F is of Type II, it follows that \(F'_A\) and \(F'_B\) are both of cycle type. Hence, by inspecting all possible qnets on \(\{a,b,c,d\}\), the qnet \(F'\) is isomorphic to \({a}\oplus {b}{c}\oplus {d}\), and hence \(F\in {\mathcal {F}}(N)\).
Lastly, suppose that F is of Type IV. Without loss of generality, we may assume that \(F={a}\oplus {b}\oplus {c}\oplus {d}\). Let \(F'\) be the qnet in \({\mathcal {F}}(N)\) whose support is \(A=\{a,b,c,d\}\). Hence, there exists no quartet in \({\mathcal {Q}}({\mathcal {F}})\) whose support is A. Therefore, \(\mathop {med}_T(a,b,c)=\mathop {med}_T(a,b,d)=\mathop {med}_T(a,c,d)=\mathop {med}_T(b,c,d)\). Denoting this median by u, it follows that u is necessarily contained in \(V_2(T)\), and hence \(N_T(u)\) contains \(m \ge 4\) vertices. Now let \((v_1,v_2,\ldots ,v_m)\) be the unique circular ordering of vertices \(N_T(u)\) induced by the circular ordering \(A^1,\ldots ,A^m\) of \(\mathcal {A}_u\) in Claim 3. Without loss of generality, we may assume that \(a\in A^1\). Then, there exists \(1<j<k<l\le m\) such that \((b,c,d)\in A^j \times A^k \times A^l\). By the construction of N (which locally is the blowup at u with respect to the circular ordering), it follows that \(F'\) is isomorphic to F, and hence \(F\in {\mathcal {F}}(N)\).
This shows that \({\mathcal {F}}\subseteq {\mathcal {F}}(N)\). Since \({\mathcal {F}}\) and \({\mathcal {F}}(N)\) are both minimally dense, we have \({\mathcal {F}}={\mathcal {F}}(N)\). Finally, the uniqueness statement concerning N is a direct consequence of the uniqueness of T and the unique way in which N is constructed from T.
4 A Characterization of Level1 Quartet Systems
We now use Theorem 2 to characterize when a quartet system is equal to the set of quartets displayed by a binary level1 network. This characterization is given as Theorem 3. Let \({\mathcal {Q}}\) be a quartet system on X. A quartet Q in \({\mathcal {Q}}\) is distinguished if Q is the only quartet in \({\mathcal {Q}}\) with support equal to the leafset of Q. Moreover, a network N is called 3cycle free if it does not contain any cycle consisting of three vertices.
Theorem 3
 (D1)
For all \(Y\in {X \atopwithdelims ()4}\), we have \(m_{\mathcal {Q}}(Y)=1\) or \(m_{\mathcal {Q}}(Y)=2\).
 (D2)
If \(\{abcd, adbc, axcd, acxd\}\subseteq {\mathcal {Q}}\), then \(\{abdx,bdax\}\subseteq {\mathcal {Q}}\), for \(a,b,c,d\in X\) distinct.
 (D3)
If abcd is a distinguished quartet in \({\mathcal {Q}}\), then, for each \(x\in X\{a,b,c,d\}\) where \(a,b,c,d\in X\) are distinct, either axcd or abcx is a distinguished quartet in \({\mathcal {Q}}\).
Proof
It is easily checked that, if \({\mathcal {Q}}={\mathcal {Q}}(N)\) holds for some binary level1 network N, then (D1)–(D3) holds. Conversely, let \({\mathcal {Q}}\) be a dense quartet system satisfying (D1)–(D3). Let \({\mathcal {Q}}_1 \subseteq {\mathcal {Q}}\) be the set consisting of the distinguished quartets contained in \({\mathcal {Q}}\). We first associate a phylogenetic Xtree T to \({\mathcal {Q}}_1\). If \({\mathcal {Q}}_1=\emptyset \), then we let T denote the phylogenetic Xtree which contains precisely one vertex that is not a leaf (i.e. a “star tree”). If \({\mathcal {Q}}_1 \ne \emptyset \), then let \(Q=abcd\) be some quartet contained in \({\mathcal {Q}}_1\), \(a,b,c,d\in X\). Suppose that there exists some \(x \in X  \{a,b,c,d\}\). Then, by (D3), either \(axcd \in {\mathcal {Q}}_1\) or \(abcx \in {\mathcal {Q}}_1\). It follows that \(\bigcup _{Q \in {\mathcal {Q}}_1} L(Q) = X\). Moreover, as \({\mathcal {Q}}_1\) is clearly thin and by (D3) \({\mathcal {Q}}_1\) is saturated, it follows by Theorem 1, that there exists a phylogenetic Xtree T with \({\mathcal {Q}}(T)={\mathcal {Q}}_1\).
Now we construct a qnet system \({\mathcal {F}}\) as follows. Let \(\Pi _1\) be the subset of \({X \atopwithdelims ()4}\) consisting of those Y with \(m_{\mathcal {Q}}(Y)=1\), and \(\Pi _2= {X \atopwithdelims ()4} \setminus \Pi _1\). To each \(\pi =\{a,b,c,d\} \in \Pi _1\) we associate a qnet \(F(\pi )\) as follows. Swapping the labels of the elements in \(\pi \) if necessary, we may assume that \(Q=abcd\) is the (necessarily unique) quartet in \({\mathcal {Q}}_1\) with leafset \(\pi \). Now let \(v_1\) and \(v_1'\) be the median of \(\{a,b,c\}\) in Q and T, respectively. Similarly, let \(v_2\) and \(v_2'\) be the median of \(\{a,c,d\}\) in Q and T, respectively. Then, \(F(\pi )\) is the qnet on \(\{a,b,c,d\}\) obtained from Q by performing a blowup on each of \(v_i\), where \(i\in \{1,2\}\), if and only if the degree of \(v'_i\) in T is at least four.
We also associate a qnet \(F(\pi )\) to each \(\pi =\{a,b,c,d\} \in \Pi _2\) as follows. Swapping the labels of the elements in \(\pi \) if necessary, we may assume that the quartets in \({\mathcal {Q}}\) with leafset \(\{a,b,c,d\}\) are abcd and adbc. We then define \(F(\pi )\) to be the qnet \({a}\oplus {b}\oplus {c}\oplus {d}\).
Now, let \(\mathcal {F}=\{F(\pi ): \pi \in \left( {\begin{array}{c}X\\ 4\end{array}}\right) \}\). By construction \({\mathcal {F}}\) is minimally dense. Moreover, \({\mathcal {Q}}({\mathcal {F}})={\mathcal {Q}}\), and \({\mathcal {F}}\) is cyclative in view of (D2).
Next, we shall show that \({\mathcal {F}}\) is consistent. Fix a subset \(\{a,b,c\} \in {X \atopwithdelims ()3}\) and consider its median v in T. By construction, it suffices to establish the claim that the degree of v is three in T if and only if, for each \(d\in X\{a,b,c\}\), the set \(\pi =\{a,b,c,d\}\) is not contained in \(\Pi _2\).
To see that this claim holds first note that if v has degree three, then each of the three components of \(T\{v\}\) contains precisely one element in \(\{a,b,c\}\). Without loss of generality, we may assume that element d is contained in the connected component containing element c. But this implies that abcd is a quartet in \({\mathcal {Q}}(T)\), and hence \(\{a,b,c,d\} \in \Pi _1\). On the other hand, if v has degree at least four, then there exists an element \(x\in X\{a,b,c\}\) such that x, a, b, c belong to four different connected components of \(T\{v\}\). Therefore, \({\mathcal {Q}}(T)\) and \(\{abcx,acbx,axbc\}\) are disjoint. This implies that \(\pi =\{a,b,c,x\}\) is not contained in \(\Pi _1\), and so it is contained in \(\Pi _2\). This establishes the claim.
Next, we show that \({\mathcal {F}}\) is saturated. We shall show that (S2) holds; the fact that \({\mathcal {F}}\) satisfies (S1) and (S3) can be established by a similar argument. Let \(\{a,b,c,d\} \in {X \atopwithdelims ()4}\) be a set that satisfies the condition in (S2), that is, \({a}\oplus {b}{c}\ominus {d}\) is contained in \({\mathcal {F}}\). Then, abcd is a quartet in \({\mathcal {Q}}_1={\mathcal {Q}}(T)\). Furthermore, put \(u=\mathop {med}_T(a,b,c)\) and \(v=\mathop {med}_T(a,c,d)\), then the degree of u is at least four and the degree of v is three. Now, fix an element \(x\in X\{a,b,c,d\}\). If x and a are in the same connected component resulting from deleting v from T, then axcd is a quartet in \({\mathcal {Q}}_1\). Since the median of a, c, d in T has degree three, by construction either \({a}\ominus {x}{c} \ominus {d}\) or \({a}\oplus {x}{c}\ominus {d}\) (but not both) is contained in \({\mathcal {F}}\). Otherwise, abcx is a quartet in \({\mathcal {Q}}_1\). Since the median u of a, b, c in T has degree greater than three, by construction we can conclude that either \({a}\oplus {b}{c}\ominus {x}\) or \({a}\oplus {b}{c}\oplus {x}\) is contained in \({\mathcal {F}}\) (but not both). This completes the verification of (S2).
It follows that \({\mathcal {F}}\) is minimally dense, cyclative, consistent and saturated. By Theorem 2, there exists a unique binary level1 network N on X such that \({\mathcal {F}}(N)={\mathcal {F}}\). By construction, it also follows that \({\mathcal {Q}}(N)={\mathcal {Q}}({\mathcal {F}}(N))={\mathcal {Q}}({\mathcal {F}})={\mathcal {Q}}\). The uniqueness statement in the theorem follows from the uniqueness of N and the fact that \({\mathcal {Q}}(N)={\mathcal {Q}}(N')\) for two binary level1 networks N and \(N'\) if and only if N and \(N'\) on X differ only by 3cycles (see e.g. Keijsper and Pendavingh 2014, Lemma 2). \(\square \)
5 Quarnet Inference Rules and Closure
 (CL1):
\(\{a*b  c \diamond d, b\diamond c  d \circ e\} \vdash a *b  c \diamond e\) for all \(*, \diamond ,\circ \in \{\ominus , \oplus \}\);
 (CL2):
\(\{a \oplus b  c *d, {a}\oplus {c}\oplus {e}\oplus {b}\} \vdash a\oplus e  c *d\) and \(\{a \oplus b  c *d, {a}\oplus {c}\oplus {b}\oplus {e}\} \vdash a\oplus e  c *d\) and \(\{a \oplus b  c *d, {a}\oplus {e}\oplus {c}\oplus {b}\} \vdash a\oplus e  c *d\,\,\) for all \(*\in \{\ominus , \oplus \}\);
 (CL3):
\( \{{a}\oplus {b}\oplus {c}\oplus {d},{e}\oplus {a}\oplus {c}\oplus {d}\} \vdash {a}\oplus {b}\oplus {d}\oplus {e}. \)
We remark in passing that the qnet system \(\{a*b  c \diamond d, b\diamond c  d \circ e\,:\, *, \diamond ,\circ \in \{\ominus , \oplus \} \} \cup \{ a \oplus b  c *d, {a}\oplus {c}\oplus {e}\oplus {b}, {a}\oplus {c}\oplus {b}\oplus {e}, {a}\oplus {e}\oplus {c}\oplus {b}\,:\, *\in \{\ominus , \oplus \} \} \cup \{{a}\oplus {b}\oplus {c}\oplus {d},{e}\oplus {a}\oplus {c}\oplus {d}\}\) implies that inference rules (CL1)–(CL3) are independent from one another.
Using Theorem 2, it is straightforward to show that the above three rules are well defined. That is, given three qnets \(F_1\), \(F_2\) and F such that \(\{F_1,F_2\} \vdash F\) holds for one of the above three rules, then every binary level1 network that displays \(\{F_1,F_2\}\) must display F.
For a qnet system \({\mathcal {F}}\), we define the set \(\mathrm{cl}_2({\mathcal {F}})\) to be the minimal qnet system (under setinclusion) that contains \({\mathcal {F}}\) such that if \(\mathrm{cl}_2({\mathcal {F}}) \vdash F\) holds under (CL1)–(CL3), then \(F\in \mathrm{cl}_2({\mathcal {F}})\) holds. We call \(\mathrm{cl}_2({\mathcal {F}})\) the closure of \({\mathcal {F}}\).
The following key proposition is analogous to that for semidyadic closure for quartet systems (cf. Meacham 1983; Huber et al. 2005, Proposition 2.1). It follows from the fact that the closure of a qnet system \({\mathcal {F}}\) can clearly be obtained from \({\mathcal {F}}\) by repeatedly applying the qnet rules (CL1)–(CL3) until the sequence of sets so obtained stabilizes. Note that this process must clearly terminate in polynomial time.
Proposition 1
Let \({\mathcal {F}}\) be a qnet system and let N be a binary, level1 network. Then, N displays \({\mathcal {F}}\) if and only if N displays \(\mathrm{cl}_2({\mathcal {F}})\).
We now show that \(\mathrm{cl}_2({\mathcal {F}})\) behaves in a similar way to the semidyadic closure of a quartet system (cf. Semple and Steel 2003, Exercise 19, p. 143).
Theorem 4
 (i)
\({\mathcal {F}}={\mathcal {F}}(N)\) holds for a (necessarily unique) binary, level1 network N on X;
 (ii)
\(\mathrm{cl}_2({\mathcal {F}})={\mathcal {F}}\);
 (iii)
For every 3element subset \({\mathcal {F}}'\) of \({\mathcal {F}}\), the subset \({\mathcal {F}}'\) is displayed by some binary level1 network on X.
Proof
The fact that (i) implies (ii) and (i) implies (iii) are straightforward. We complete the proof by showing that (ii) implies (i) and (iii) implies (i).
For the proof of (ii) implies (i), suppose that \(\mathrm{cl}_2({\mathcal {F}})={\mathcal {F}}\). Note first that by (CL3) \({\mathcal {F}}\) is cyclative. Moreover, \({\mathcal {F}}\) is minimally dense and consistent by assumption. Hence, by Theorem 2, it suffices to show that \({\mathcal {F}}\) is saturated. To this end, let w, x, y, z, t be five pairwise distinct elements in X such that \(F=w*xy \diamond z\) is contained in \({\mathcal {F}}\) with \(*,\diamond \in \{\oplus ,\ominus \}\) and \((*,\diamond )\not =(\ominus ,\oplus )\). We need to show that \({\mathcal {F}}\) satisfies (S1)–(S3).
For \(p \in \{w,x,y,z\}\), let \(F_p\) be the qnet on \(\{w,x,y,z,t\}\{p\}\) that is contained in \({\mathcal {F}}\) (which must exist as \({\mathcal {F}}\) is minimally dense). First assume that there exists some element p in \(\{w,x,y,z\}\) such that the qnet \(F_p\) is of Type IV. Without loss of generality, assume \(p=w\) (the other cases can be established in a similar manner). Since \(F_w\) is of Type IV, by the consistency of \({\mathcal {F}}\) we have \(F=y\oplus zw*x\). Now, applying (CL2) with \(a=y\), \(b=z\), \(c=w\), \(d=x\), \(e=t\) implies that \(y\oplus tw *x \in \mathrm{cl}_2({\mathcal {F}})={\mathcal {F}}\), by (ii). Therefore, \({\mathcal {F}}\) satisfies (S2) and (S3) (corresponding, respectively, to taking \(*=\ominus \) and \(*=\oplus \)). It follows that in the remainder of the proof we can assume that none of the qnets in \(\{F_w,F_x,F_y,F_z\}\) is of Type IV.
For convenience, in the following, we will use the convention that when we apply (CL1), we will write a 5tuple and assume that the ith element in the 5tuple will correspond to the ith element in the tuple (a, b, c, d, e) of elements used in (CL1) for \(1\le i \le 5\).
To show that \({\mathcal {F}}\) satisfies (S1), suppose that \(F={w}\ominus {x}{y} \ominus {z}\). Note first that if \(F_x=w\ominus yz *t\), then applying (CL1) to (x, w, y, z, t) implies \(x\ominus w y \ominus t \in \mathrm{cl}_2({\mathcal {F}})={\mathcal {F}}\), and hence (S1) holds. Similarly, if \(F_z=w\ominus yx *t\), then applying (CL1) to (z, y, w, x, t) implies \(z\ominus y w \ominus t \in {\mathcal {F}}\), and hence (S1) holds. Therefore, if (S1) does not hold, then, by consistency, we may assume \(F_x=w\ominus zy *t\) and \(F_z=x\ominus yw *t\) with \(*\in \{\ominus , \oplus \}\). Considering \(F_x\) and \(F_z\), and applying (CL1) to (x, y, t, w, z) implies \(x\ominus y  t *z \in {\mathcal {F}}\). On the other hand, considering F and \(F_z\) and applying (CL1) to (z, y, x, w, t) implies that \(z\ominus y  x \ominus t \in {\mathcal {F}}\), a contradiction to the fact that \({\mathcal {F}}\) is minimally dense. Thus, \({\mathcal {F}}\) satisfies (S1).
Using an argument similar to the one that we used to show that \({\mathcal {F}}\) satisfies (S1), it is straightforward to deduce that \({\mathcal {F}}\) satisfies (S2) and (S3).
We next prove that (iii) implies (i). Since \({\mathcal {F}}\) is minimally dense and consistent by assumption, it follows by Theorem 2 that it suffices to show that \({\mathcal {F}}\) is cyclative and saturated.
First, we show that \({\mathcal {F}}\) is cyclative. If not, then there exist five elements \(Y=\{w,x,y,z,t\}\) such that \(F_1={w}\oplus {x}\oplus {y}\oplus {z}\) and \(F_2={t}\oplus {w}\oplus {y}\oplus {z}\) are contained in \({\mathcal {F}}\) but \(F={w}\oplus {x}\oplus {z}\oplus {t}\) is not contained in \({\mathcal {F}}\). Let \(F'\) be the (necessarily unique) qnet in \({\mathcal {F}}\) whose leafset is \(\{w,x,z,t\}\). Then, \(F'\not = F\). Consider the set \({\mathcal {F}}'=\{F',F_1,F_2\}\). The assumption (iii) implies that \({\mathcal {F}}'\) is displayed by a binary level1 network N on X. Consider \(N'=N_Y\). Then, \({\mathcal {F}}'\subseteq {\mathcal {F}}(N')\). By Theorem 2, \({\mathcal {F}}(N')\) is minimally dense and cyclative. Since \(\{F_1,F_2\}\subseteq {\mathcal {F}}(N')\), it follows that \(F\in {\mathcal {F}}(N')\), a contradiction in view of \(F' \in {\mathcal {F}}(N')\).
Lastly, consider the subset \({\mathcal {F}}'=\{F,F_1,F_2\}\) of \({\mathcal {F}}\). Then, as assumption (iii) holds it follows that \({\mathcal {F}}'\) is displayed by a binary level1 network N on X. Consider \(N'=N_Y\). Then, \({\mathcal {F}}'\subseteq {\mathcal {F}}(N')\). By Theorem 2, \({\mathcal {F}}(N')\) is minimally dense and saturated. Using the fact that \({\mathcal {F}}(N')\) is saturated, it follows that \({\mathcal {F}}^*\cap {\mathcal {F}}(N') \not = \emptyset \) as \(F \in {\mathcal {F}}(N')\). Therefore, \({\mathcal {F}}(N')\) contains either two distinct qnets on A or two distinct qnets on B, a contradiction to the fact that \({\mathcal {F}}(N')\) is minimally dense. Thus, (iii) implies (i), thereby completing the proof of the theorem. \(\square \)
Note that it follows from Theorem 4 that we can decide whether or not a given minimally dense set of qnets \({\mathcal {F}}\) is displayed by a level1 binary phylogenetic network on \(n\ge 2\) leaves in \(O(n^5)\) time. This follows since we can compute \(\mathrm{cl}_2({\mathcal {F}})\) in \(O(n^5)\) time. It would be interesting to see if this time bound can be improved upon.
6 Discussion
We have shown that by considering quarnets we can define natural inference rules, as well as the concept of quarnet closure. With quartets, there are various types of inference rules, which imply alternative definitions of closure for quartet systems (see e.g. Bryant and Steel 1995; Semple and Steel 2003). It would thus be of interest to explore whether there are other types of inference rules for quarnets and, if so, what their properties are. In this paper, we have focused on understanding the closure for a minimally dense set of quarnets. For real data, there can be cases where it may be necessary to consider nonminimally dense sets (e.g. in case there is missing data). Hence, it could be useful to develop results for such situations. However, it should be noted that understanding the closure of a nonminimally dense set quartets is already quite challenging (for example, as opposed to the minimally dense case, deciding whether or not an arbitrary set of quartets can be displayed by a phylogenetic tree is NPcomplete) (Steel 1992).
In many applications, biologists prefer to use weighted phylogenetic trees and networks to model their data, where nonnegative numbers are assigned to edges of the tree or network to, for example, represent evolutionary distance. The problem of considering when a dense set of weighted quartets can be represented by a weighted phylogenetic tree has been considered in Dress and Erdös (2003), Grünewald et al. (2008). Given the results in this paper, it could therefore be of interest to consider how weighted level1 networks may be inferred from dense sets of weighted quarnets. In applications, it can also be useful to consider rooted networks, which are essentially leaflabelled, directed acyclic graphs. Edges in such networks have a direction which represents the fact that species evolve through time from a common ancestor (represented in graph theoretical terms by a root vertex). For such networks, the concept of level1 networks can be defined in a similar way to the unrooted case, and algorithms are known for deciding when minimally dense collections of 3leaved, rooted level1 phylogenetic networks (which are known as trinets) can be displayed by a single phylogenetic network (Huber and Moulton 2013; Huber et al. 2017). It would thus be of interest to consider inference rules for trinets. Moreover, for both the rooted and unrooted case, it could be worth exploring whether there are inference rules for more complicated networks (e.g. networks with level higher than one, as defined in e.g. Gambette et al. 2012). Although results in Iersel and Moulton (2017) indicate that such inference rules might exist, if they do, then we expect that these will probably be quite complicated.
Footnotes
Notes
Acknowledgements
The authors thank the anonymous referees for their helpful suggestions. In addition, KTH, VM, and CS thank the London Mathematical Society for its support. CS was also supported by the New Zealand Marsden Fund.
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