Vines and MAT-labeled graphs

Abstract

The present paper explores a connection between two concepts arising from different fields of mathematics. The first concept, called vine, is a graphical model for dependent random variables. This concept first appeared in a work of Joe (1994), and the formal definition was given later by Cooke (1997). Vines have nowadays become an active research area whose applications can be found in probability theory and uncertainty analysis. The second concept, called MAT-freeness, is a combinatorial property in the theory of freeness of logarithmic derivation modules of hyperplane arrangements. This concept was first studied by Abe-Barakat-Cuntz-Hoge-Terao (2016), and soon afterwards investigated further by Cuntz-Mücksch (2020).

In the particular case of graphic arrangements, the last two authors (2023) recently proved that the MAT-freeness is completely characterized by the existence of certain edge-labeled graphs, called MAT-labeled graphs. In this paper, we first introduce a poset characterization of a vine. Then we show that, interestingly, there exists an explicit equivalence between the categories of locally regular vines and MAT-labeled graphs. In particular, we obtain an equivalence between the categories of regular vines and MAT-labeled complete graphs.

Several applications will be mentioned to illustrate the interaction between the two concepts. Notably, we give an affirmative answer to a question of Cuntz-Mücksch that MAT-freeness can be characterized by a generalization of the root poset in the case of graphic arrangements.

1. Introduction

1.1. Motivation

The starting point of our paper is a question of Cuntz-Mücksch [10] (Question 1.3) in the theory of free hyperplane arrangements.

Let V be a finite dimensional vector space. A hyperplane in V is a $1$ -codimensional linear subspace of V. Let $ \{x_{1}, \dots , x_{\ell }\} $ be a basis for the dual space $V^{\ast } $ . Any hyperplane in V can be described by a linear equation of the form $a_1x_1+\cdots +a_\ell x_\ell =0$ where at least one of the $a_{i}$ ’s is nonzero.

A hyperplane arrangement $\mathcal {A}$ is a finite set of hyperplanes in V. The intersection lattice of $\mathcal {A}$ is the set of all intersections of hyperplanes in $\mathcal {A}$ , which is often referred to as the combinatorics of $\mathcal {A}$ . An arrangement is said to be free if its module of logarithmic derivations is a free module. For basic definitions and properties of free arrangements, we refer the interested reader to [22, 26]. Freeness is an algebraic property of hyperplane arrangements which has been a major topic of research since the 1970s. A central question in the theory is to study the freeness of an arrangement by combinatorial structures, especially by the intersection lattice of the arrangement.

Among others, MAT-freeness is an important concept which was first used by Abe-Barakat-Cuntz-Hoge-Terao [1] to settle the conjecture of Sommers-Tymoczko [24] on the freeness of ideal subarrangements of Weyl arrangements. This concept is formally defined later by Cuntz-Mücksch [10], and we will use their definition throughout. For a hyperplane $H \in \mathcal {A}$ , define the restriction ${\mathcal {A}}^{H}$ of ${\mathcal {A}}$ to H by

$$ \begin{align*}{\mathcal{A}}^{H}:= \{ K \cap H \mid K \in \mathcal{A} \setminus\{H\}\}.\end{align*} $$

Definition 1.1 (MAT-partition and MAT-free arrangement [10]).

Let $\mathcal {A}$ be a nonempty arrangement. An ordered partition (disjoint union of nonempty subsets) $\pi = (\pi _{1}, \dots , \pi _{n}) $ of $\mathcal {A}$ is called an MAT-partition if the following three conditions hold for every $1 \le k \le n $ .

1. (MP1) The hyperplanes in $\pi _k$ are linearly independent.

2. (MP2) There does not exist $ H^{\prime } \in \mathcal {A}_{k-1} $ such that $ \bigcap _{H \in \pi _{k}}H \subseteq H^{\prime } $ , where $ \mathcal {A}_{k-1} := \pi _{1} \sqcup \dots \sqcup \pi _{k-1} $ (disjoint union) and $ \mathcal {A}_{0} := \emptyset $ is the empty arrangement.

3. (MP3) For each $ H \in \pi _{k} $ , $ |\mathcal {A}_{k-1}| - |(\mathcal {A}_{k-1} \cup \{H\})^{H}| = k-1 $ .

An arrangement is called MAT-free if it is empty or admits an MAT-partition.

As the name suggests, any MAT-free arrangement is a free arrangement. This follows from the remarkable Multiple Addition Theorem by Abe-Barakat-Cuntz-Hoge-Terao [1, Theorem 3.1] (justifying the abbreviation MAT). MAT-freeness is a helpful combinatorial tool (as it depends only on the intersection lattice) to examine the freeness of arrangements. One of its most famous applications we mentioned earlier is a proof that the ideal subarrangements of Weyl arrangements are free. The MAT-freeness has received increasing attention in recent years; see [2, 3, 9, 20] for some other applications.

Let $V = \Bbb R^\ell $ with the standard inner product $(\cdot ,\cdot )$ . Let $\Phi $ be an irreducible (crystallographic) root system in V, with a fixed positive system $\Phi ^+ \subseteq \Phi $ and the associated set of simple roots $\Delta := \{\alpha _1,\ldots ,\alpha _\ell \}$ . For $\alpha \in \Phi $ , define $H_{\alpha } :=\{x\in V \mid (\alpha ,x)=0\}.$ For $\Sigma \subseteq \Phi ^+$ , the Weyl subarrangement $\mathcal {A}_{\Sigma }$ is defined by $\mathcal {A}_{\Sigma }:= \{H_{\alpha } \mid \alpha \in \Sigma \}$ . In particular, $\mathcal {A}_{\Phi ^+}$ is called the Weyl arrangement.

We can make $\Phi ^+$ into a poset (partially ordered set) by defining a partial order $\le $ on $\Phi ^+$ as follows: $\beta _1 \le \beta _2$ if $\beta _2-\beta _1 \in \sum _{i=1}^\ell \mathbb {Z}_{\ge 0}\alpha _i$ . The poset $(\Phi ^+, \le )$ is called the root poset of $\Phi $ . For an ideal ${\mathcal {I}}$ (Definition 3.4) of the root poset $\Phi ^+$ , the corresponding Weyl subarrangement $\mathcal {A}_{{\mathcal {I}}}$ is called the ideal subarrangement.

Theorem 1.2 [1, Theorem 1.1].

Any ideal subarrangement $\mathcal {A}_{{\mathcal {I}}}$ is MAT-free, and hence free.

The ideal subarrangements form a significant subclass of MAT-free arrangements. However, there are many MAT-free arrangements (or MAT-partitions of a given MAT-free arrangement) that do not arise from ideal subarrangements (Example 7.2). One may wonder if the hyperplanes in an arbitrary MAT-free arrangement satisfy some poset structure similar to the root poset? This question was asked by Cuntz-Mücksch [10] and is the main motivation of our work.

Question 1.3 [10, Problem 47].

Given an MAT-free arrangement $\mathcal {A}$ , can we characterize all possible MAT-partitions of $\mathcal {A}$ by a poset structure generalizing the classical root poset?

Cuntz-Mücksch’s question is difficult in general as the number of different MAT-partitions of a given MAT-free arrangement might be very large. Also, the definition of an MAT-partition itself does not reveal a natural choice of the desirable partial order. In the present paper, we pursue this question along graphic arrangements, a well-behaved class of arrangements in which both freeness and MAT-freeness are completely characterized by combinatorial properties of graphs.

Let G be a simple graph (i.e., no loops and no multiple edges) with vertex (or node) set $N_G = \{v_{1}, \dots , v_{\ell }\}$ and edge set $E_G$ . The graphic arrangement $\mathcal {A}_G$ is an arrangement in an $\ell $ -dimensional vector space V defined by

$$ \begin{align*}\mathcal{A}_G:= \{x_i - x_j=0 \mid \{v_i,v_j\} \in E_G \}.\end{align*} $$

A graph is chordal if it does not contain an induced cycle of length greater than three. A chordal graph is strongly chordal if it does not contain a sun graph as an induced subgraph (Definition 2.2).

Theorem 1.4 [25], [12, Theorem 3.3].

The graphic arrangement $\mathcal {A}_G$ is free if and only if G is chordal.

Theorem 1.5 [27, Theorem 2.10].

The graphic arrangement $\mathcal {A}_G$ is MAT-free if and only if G is strongly chordal.

While the definition of an MAT-free arrangement may seem technical at first glance, Theorem 1.5 enables us to view MAT-freeness as a rather natural property. Furthermore, the correspondence between MAT-freeness and strong chordality establishes a nice analog¹ of the classical correspondence between freeness and chordality.

The good thing about graphs is that MAT-partition of a graphic arrangement can be rephrased in terms of a special edge-labeling of graphs, the so-called MAT-labeling (Definition 2.4). A graph together with such a labeling is called an MAT-labeled graph. To approach Question 1.3 for graphic arrangements, the first question would be how many non-isomorphic MAT-labelings can a (strongly chordal) graph have? A computation aided by computer for complete graphs on up to $8$ vertices gives us the sequence $1,1 ,1 , 2 , 6,40, 560, 17024$ . Surprisingly, we found out that this sequence coincides with the number of equivalence classes of (graphical) regular vines (or R-vines) in dimension up to $8$ given in [17, Table 10.4]. This observation is indeed compelling as it leads us to the notion of the node poset of a graphical vine (Definitions 3.6 and 3.7), which is a perfect candidate for the poset structure we are looking for.

1.2. Main result

In this paper, we first introduce a poset realization of graphical (R-)vines (Definitions 3.12 and 3.14). Our aim is to convert the important terms and properties of graphical vines into the language of posets in which considerable power and development of poset theory would be brought to bear. We also introduce the notion of a locally regular vine (or LR-vine) (Definition 3.16). Roughly speaking, an LR-vine is a vine that ‘locally’ looks like an R-vine. It is worth mentioning that any ideal of an R-vine, or m-vine (Definition 3.19), gets characterized by an LR-vine (Theorem 6.13).

Having introduced the concepts, we define the category $\mathsf {MG}$ of MAT-labeled graphs and the category $\mathsf {LRV}$ of LR-vines (Definitions 6.2 and 6.3). Our main result is that these categories are equivalent (Theorem 6.10). In particular, we obtain the equivalence between the category of MAT-labeled complete graphs and the category of R-vines (Corollary 6.11). The correspondences are summarized in Table 1.

Table 1 Correspondence between concepts in vine and graph theories.

To prove the equivalence between $\mathsf {MG}$ and $\mathsf {LRV}$ , we construct two functors $\Psi \colon \mathsf {MG} \longrightarrow \mathsf {LRV} $ and $\Omega \colon \mathsf {LRV}\longrightarrow \mathsf {MG}$ . The former amounts to constructing an LR-vine from a given MAT-labeled graph which will be presented in Definition 4.11 and Theorem 4.13. The proof is direct and largely dependent upon the notion of MAT-perfect elimination ordering (Definition 2.16) developed in an earlier work of the last two authors [27]. The argument on the functor $\Omega $ is, however, more complicated. We need to show some new properties of LR-vines in §5.1 before giving the construction in Definition 5.16 and Theorem 5.17.

It is known that graphic arrangements are equivalent to Weyl subarrangements when the root system is of type A. It would be interesting to extend our main result to the root systems of other types. However, it is quite challenging since a complete characterization of either MAT-freeness or freeness of Weyl subarrangements is unknown except for type A.

1.3. Applications

From the view point of category theory, the equivalence establishes a strong similarity between the categories and allows many properties and structures to be translated from one to the other. We obtain three main applications from LR-vines to MAT-labeled graphs. First, our main theorem 6.10 gives a new poset characterization of the MAT-free graphic arrangements compared with the characterization by strong chordality in Theorem 1.5. In particular, LR-vine is an answer for Question 1.3 in the case of graphic arrangements (see §7.1.1). We find it interesting that although the class of MAT-free arrangements strictly contains that of ideal subarrangements in general, any MAT-free graphic arrangement is characterized by being an ideal of a poset structure. Second, an explicit formula for the number of non-isomorphic MAT-labelings of complete graphs is obtained as it equals the number of equivalence classes of regular vines (see §7.1.2). Third, the notion of upper truncation (Definition 4.9) of an LR-vine gives rise to a nontrivial graph operation which produces a new MAT-labeled graph from a given one (see §7.1.3).

A vine is a graphical tool for representing the joint distribution of random variables. The first construction of a vine was given by Joe [14], and the formal definition was given and refined further by Cooke, Bedford and Kurowicka [7, 5, 15]. Vines have been studied extensively and proved to have various applications in probability theory and related areas. We refer the reader to [17] for a comprehensive handbook of vines. Our main result gives a new appearance and applications of vines in the arrangement theory. In the present paper, we do not pursue the probabilistic or applied aspects of vines (neither does the proof of the main result) but emphasize and develop more on the theoretical aspects. There are several new combinatorial properties of vines presented throughout, and we hope that they will be useful for the future research on vines. For instance, we give an alternative way to associate an m-vine to a strongly chordal graph compared with the work of Zhu-Kurowicka [28] (see §7.2.1), an extension of the notion of sampling order [8] from R-vine to LR-vine (see §7.2.2), and a conjectural formula for the number of ideals in a C-vine (see §7.2.3).

2. MAT-labelings of graphs

2.1. Graphs

In this subsection, we recall some basic definitions and notions of graphs. All graphs in this paper are undirected, finite and simple. Let $ G = (N_{G}, E_{G}) $ be a graph with the set $N_G$ of vertices (or nodes) and the set $E_G$ of edges (unordered pairs of vertices). In this paper, a vertex and a node in a graph are synonyms. The former will be used more often for graphs, while the latter will be used for an element in a poset.

For $S \subseteq N_{G}$ , denote by $G[S] = (S, E_{G[S]})$ where $E_{G[S]} =\{ \{u,v\} \in E_G \mid u, v \in S\}$ the (vertex-)induced subgraph of S. Denote by $ G\setminus S$ the induced subgraph $G[N_G\setminus S]$ . In particular, $G\setminus v:=G \setminus \{v\}$ when v is a vertex of G. For $ F \subseteq E_{G} $ , define the subgraph $ G\setminus F := (N_{G}, E_{G} \setminus F) $ . In particular, $G\setminus e:=G \setminus \{e\} $ when e is an edge of G.

A complete graph $K_{n} $ ( $n \ge 0$ ) is a graph with vertex set $N_{K_{n}} = \{u_{1}, \dots , u_{n}\} $ and edge set

$$ \begin{align*} E_{K_{n}} = \{\{u_{i},u_{j}\} \mid 1 \leq i < j \leq n\}. \end{align*} $$

In other words, a complete graph is a graph such that each pair of vertices is an edge. Here, the order-zero graph $K_0$ (i.e., the graph having no vertices is also considered to be a complete graph). A clique C of G is a subset of $N_{G}$ such that the induced subgraph $G[C]$ is a complete graph.

For each $v \in N_{G}$ , the neighborhood $ \operatorname {Nbd}_{G}(v) $ of v in G is defined by $ \operatorname {Nbd}_{G}(v) :=\{ u \in N_{G} \mid \{u,v\} \in E_G\}$ . The degree of $ v $ in $ G $ is defined by $ \deg _{G}(v) :=|\operatorname {Nbd}_{G}(v) |$ . A leaf is a vertex of degree  $1$ . A vertex is called simplicial if its neighborhood is a clique. An ordering $ (v_{1}, \dots , v_{\ell }) $ of vertices of a graph $ G $ is called a perfect elimination ordering (PEO) if $ v_{i} $ is simplicial in the induced subgraph $ G[\{v_{1}, \dots , v_{i}\}] $ for each $1 \le i \le \ell $ .

A maximal clique is a clique that it is not a subset of any other clique. A largest clique is a clique that has the largest possible number of vertices. Denote by $\mathcal {K}(G)$ the set of all maximal cliques of G. In particular, $ |\mathcal {K}(G)| =0$ or $1 $ if and only if G is a complete graph. The clique number of G, denoted $\omega (G)$ , is the number of vertices in a largest clique of $ G $ .

A walk W in a graph G is a sequence of edges $(e_1, e_2,\ldots , e_{p})$ of G for which there is a sequence of vertices $(v_1, v_2, \ldots , v_{p+1})$ of G such that $e_i = \{v_i, v_{i + 1}\}$ for $1 \le i \le p$ . The vertices $v_1$ and $v_{p+1} $ are said to be connected by the walk W, called the initial and final vertices of W, respectively. The length of a walk is the number of edges in the walk (hence, the length of W is $p \ge 0$ ). Throughout the paper, a walk W is denoted by its vertex sequence $W=(v_1, v_2, \ldots , v_{p+1})$ . If $W_1=(v_1, v_2, \ldots , v_{p+1})$ and $W_2=(v_{p+1}, v_{p+2},\ldots , v_{n})$ are walks, then the concatenation of $W_1$ and $W_2$ is the walk $(v_1, v_2, \ldots , v_{n})$ .

A path $P=(v_1, v_2, \ldots , v_{p+1})$ is a walk with no repeated vertices except possibly the initial and final vertex. A subpath of P is a path of the form $(v_i, v_{i+1}, \ldots , v_{j}) $ for some $1 \le i \le j \le p+1$ . The following fact is well-known.

Lemma 2.1. Given two vertices $a,b $ in a graph G, every walk connecting a and $b $ contains a path connecting a and $b $ .

A graph G is called connected if any two vertices are connected by a walk (hence by a path by Lemma 2.1 above).

A $\mathbf {p}$ -cycle $C_{p}=(v_1, v_2, \ldots , v_{p})$ for $p\ge 3$ is a path with $v_{p}=v_1$ . The $3$ -cycle is also called a triangle. A chord of a cycle is an edge connecting two non-consecutive vertices of the cycle. A forest is a graph containing no cycles. A tree is a connected forest. In a forest (resp. tree), any two distinct vertices are connected by at most (resp. exactly) one path.

An $\mathbf {n}$ -sun $ S_{n} $ ( $n \ge 3$ ) is a graph with vertex set $N_{S_{n}} = \{u_{1}, \dots , u_{n}\} \cup \{v_{1}, \dots , v_{n}\} $ and edge set

$$ \begin{align*} E_{S_{n}} = \{\{u_{i},u_{j}\} \mid 1 \leq i < j \leq n\} \cup \{\{v_{i}, u_{j}\} \mid 1 \leq i \leq n, j \in \{i, i+1\} \}, \end{align*} $$

where we let $ u_{n+1} = u_{1} $ .

Definition 2.2 (Strongly chordal graph).

A simple graph is strongly chordal if it is $C_p$ -free² for $p\ge 4$ and $S_n$ -free for $n\ge 3$ .

The following property of a strongly chordal graph is a special case of [27, Lemma 5.9].

Lemma 2.3. Let $ G $ be a strongly chordal graph with $ |\mathcal {K}(G)| \geq 2 $ . Then there exist distinct maximal cliques $ X_{0}, Y_{0} \in \mathcal {K}(G) $ such that $ X_{0} \cap Y_{0} \supseteq X_{0} \cap Y $ for all $ Y \in \mathcal {K}(G) \setminus \{X_{0}\} $ .

2.2. MAT-labeled graphs

In this subsection, we recall some preliminary definitions and facts of MAT-labeled graphs following [27]. An edge-labeled graph is pair $ (G,\lambda ) $ , where $ G $ is a simple graph and $ \lambda \colon E_{G} \longrightarrow \mathbb {Z}_{>0} $ is a map, called (edge-)labeling. The following definition of an MAT-labeling is equivalent to the original one in [27, Definition 4.2].

Definition 2.4 (MAT-labeling).

Let $ (G,\lambda ) $ be an edge-labeled graph. For $ k \in \mathbb {Z}_{>0} $ , let $ \pi _{k}:= \lambda ^{-1}(k) \subseteq E_{G}$ denote the set of edges of label k. Define $ \pi _{\le k} := \pi _{1} \sqcup \dots \sqcup \pi _{k} $ and $ \pi _{<1}:= \varnothing $ . The labeling $ \lambda $ is an MAT-labeling if the following two conditions hold for every $ k \in \mathbb {Z}_{>0} $ .

1. (ML1) Any edge $e \in \pi _{\le k} $ does not form a cycle with edges in $\pi _k$ .

2. (ML2) Every edge $e \in \pi _{k} $ forms exactly $ k-1 $ triangles with edges in $ \pi _{<k} $ .

Given an edge $e \in \pi _{k} $ , a conditioning vertex of e is a vertex that together with the endvertices of e forms two edges both of label $<k$ . Condition (ML2) above can be rephrased as every edge $e $ of label k has exactly $k-1$ conditioning vertices.

Definition 2.5 (MAT-labeled (complete) graph).

An edge-labeled graph $ (G,\lambda ) $ is an MAT-labeled graph if $ \lambda $ is an MAT-labeling of G. In particular, an MAT-labeled graph $ (G,\lambda ) $ is an MAT-labeled complete graph if G is a complete graph.

MAT-partition of a graphic arrangement is nothing but MAT-labeling of the underlying graph [27, Proposition 4.3]. Thus, MAT-free graphic arrangement and MAT-labeled graph are essentially the same object. The following properties of an MAT-labeled graph are deduced thanks to the relation with freeness.

Lemma 2.6 [27, Proposition 4.8].

Let $(G,\lambda )$ be an MAT-labeled graph with clique number $ \omega (G)$ . Then the following statements hold.

1. (1) The largest label of edges in $(G,\lambda )$ is equal to $ \omega (G)-1 $ .

2. (2) There exists a bijection between the set of largest cliques of $ G $ and $ \pi _{n}$ , where $n = \omega (G)-1$ via the relation: For each $ e \in \pi _{n}$ , there exists a unique largest clique of G containing the endvertices of e.

Lemma 2.7 [27, Proposition 4.4].

If $ \lambda $ is an MAT-labeling of the complete graph $ K_{\ell } $ , then $ |\pi _{k}| = \ell -k $ for all $1 \le k \le \ell -1$ .

Now we present some results on the restrictions of MAT-labelings.

Definition 2.8 (MAT-labeled subgraph).

Let $(G,\lambda )$ be an MAT-labeled graph. An edge-labeled graph $(G', \lambda ')$ is an MAT-labeled subgraph of $(G, \lambda )$ , written $(G', \lambda ') \le (G,\lambda )$ , if $G'$ is a subgraph of G, $\lambda ' = \lambda |_{E_{G'}}$ (i.e., $\lambda ' $ is the restriction of $\lambda $ on $E_{G'}$ ), and $(G', \lambda ')$ itself is an MAT-labeled graph.

Lemma 2.9 [27, Lemma 4.7].

Let $(G,\lambda )$ be an MAT-labeled graph and let $ F_{1}, F_{2} \subseteq E_{G} $ . If $ \lambda |_{F_{1}} $ and $ \lambda |_{F_{2}} $ are MAT-labelings of the subgraphs $(N_{G}, F_1)$ and $(N_{G}, F_2)$ , respectively, then $ \lambda |_{F_{1} \cup F_{2}} $ is an MAT-labeling of $(N_{G}, F_{1} \cup F_{2})$ .

Lemma 2.10 [27, Lemma 4.9].

Let $(G,\lambda )$ be an MAT-labeled graph. Let X denote the intersection of some maximal cliques of G. Then $(G[X], \lambda |_{E_{G[X]}} ) \le (G, \lambda )$ .

The following is an immediate consequence of Lemmas 2.9 and 2.10 above.

Corollary 2.11. Let $ (G, \lambda ) $ be an MAT-labeled graph. Let $\mathcal {B} \subseteq \mathcal {K}(G) $ be a set of some maximal cliques of G. Let $ G_{\mathcal {B}}$ be the subgraph of $ G $ with vertex set $ N_{G_{\mathcal {B}}}:= \cup _{Y \in \mathcal {B}}Y$ and edge set $ E_{G_{\mathcal {B}}}:= \cup _{Y \in \mathcal {B}} E_{G[Y]}$ . Then $(G_{\mathcal {B}}, \lambda |_{E_{G_{\mathcal {B}}}}) \le (G, \lambda ) $ .

Notation 2.12. For simplicity of notation, if $\lambda \colon E_{G} \longrightarrow \mathbb {Z}_{>0} $ is a labeling and $\{u, v\} \in E_G$ is an edge, we write $\lambda (u,v):=\lambda (\{u,v\})$ for the label of $\{u, v\} $ .

The following analogs of simplicial vertex and perfect elimination ordering of chordal graphs are important concepts in the study of MAT-labeled graphs.

Definition 2.13 (MAT-simplicial vertex).

Given an edge-labeled graph $ (G,\lambda ) $ , a vertex $ v \in N_{G} $ is MAT-simplicial if the following conditions hold.

1. (MS1) $ v $ is a simplicial vertex of $ G $ .

2. (MS2) The edges of G incident on v are labeled by labels from $1$ to $\deg _{G}(v)$ (i.e. $ \{\lambda (u,v) \in \mathbb {Z}_{>0} \mid u \in \operatorname {Nbd}_{G}(v) \} = \{1,2, \dots , \deg _{G}(v)\} $ ).

3. (MS3) For any distinct vertices $ u_{1}, u_{2} \in \operatorname {Nbd}_{G}(v) $ , $ \lambda (u_{1}, u_{2}) < \max \{\lambda (u_{1}, v), \lambda (u_{2}, v)\} $ .

Lemma 2.14 [27, Lemma 5.2].

If $ (G, \lambda ) $ is an MAT-labeled complete graph, then the endvertices of the edge of largest label are MAT-simplicial.

Lemma 2.15 [27, Proposition 5.3].

Let $ (G, \lambda ) $ be an edge-labeled graph having at least $2$ vertices. Suppose that $ v $ is an MAT-simplicial vertex of $ (G,\lambda ) $ . Then $ \lambda $ is an MAT-labeling of $ G $ if and only if $ \lambda |_{E_{G\setminus v}} $ is an MAT-labeling of $ G\setminus v $ .

Definition 2.16 (MAT-PEO).

Let $ (G,\lambda ) $ be an edge-labeled graph on $\ell $ vertices. An ordering $ (v_{1}, \dots , v_{\ell }) $ of vertices in $ G $ is an MAT-perfect elimination ordering (MAT-PEO) of $ (G,\lambda ) $ if $ v_{i} $ is MAT-simplicial in $ (G_{i}, \lambda _{i}) $ for each $1 \le i \le \ell $ , where $ G_{i}:= G[\{v_{1}, \dots , v_{i}\}] $ and $ \lambda _{i}:= \lambda |_{E_{G_{i}}} $ .

Theorem 2.17 [27, Theorem 5.5].

An edge-labeled graph $ (G, \lambda ) $ is an MAT-labeled graph if and only if there exists an MAT-PEO of $ (G, \lambda ) $ .

Remark 2.18. It is known that a graph is chordal if and only if it has a perfect elimination ordering [13]. Theorem 2.17 is an analog of this classical result for MAT-labeled graphs.

The method of merging regular vines was given in [8, 28]. We have a very similar³ method for merging MAT-labeled complete graphs.

Lemma 2.19 (Merging MAT-labeled complete graphs [27, Lemma 5.7]).

Let $ (G_1, \lambda _1) $ and $ (G_2, \lambda _2) $ be MAT-labeled complete graphs. Denote the common complete graph $G_1[N_{G_1} \cap N_{G_2}] = G_2[N_{G_1} \cap N_{G_2}]$ by $G'$ . Assume that there exists an MAT-labeling $\lambda '$ of $G'$ such that $(G',\lambda ')\le (G_1, \lambda _1) $ and $(G',\lambda ')\le (G_2, \lambda _2) $ . Let G denote the complete graph with vertex set $N_{G_1} \cup N_{G_2}$ . Then there exists an MAT-labeling $ \lambda $ of $ G $ such that $ (G_1, \lambda _1) \le (G, \lambda ) $ and $ (G_2, \lambda _2) \le (G, \lambda ) $ .

Proposition 2.20. Let $ (G, \lambda ) $ be an MAT-labeled graph and K be the complete graph with vertex set $N_G$ . Then there exists an MAT-labeling $\widetilde \lambda $ of K such that $ (G, \lambda ) \le (K,\widetilde \lambda ) $ .

Proof. We argue by induction on the number $ |\mathcal {K}(G)| $ of maximal cliques of G. If $ |\mathcal {K}(G)| =0$ or $1 $ , then the assertion follows since G is a complete graph. Suppose $ |\mathcal {K}(G)| \geq 2 $ .

Note that by Theorem 1.5, G is a strongly chordal graph. By Lemma 2.3, there exist distinct $ X_{0}, Y_{0} \in \mathcal {K}(G) $ such that $ X_{0} \cap Y_{0} \supseteq X_{0} \cap Y $ for all $ Y \in \mathcal {K}(G) \setminus \{X_{0}\} $ . Denote $\mathcal {B} := \mathcal {K}(G) \setminus \{X_{0}\} $ . Let $ G_{\mathcal {B}}$ be the subgraph of $ G $ with vertex set $ N_{G_{\mathcal {B}}}:= \cup _{Y \in \mathcal {B}}Y$ and edge set $ E_{G_{\mathcal {B}}}:= \cup _{Y \in \mathcal {B}} E_{G[Y]}$ . Thus, $ G_{\mathcal {B}}$ has $ |\mathcal {K}(G)| -1$ maximal cliques. Moreover, $(G_{\mathcal {B}}, \lambda |_{E_{G_{\mathcal {B}}}})$ and $(G[X_0], \lambda |_{E_{G[X_0]}})$ are MAT-labeled graphs by Corollary 2.11. By the induction hypothesis, there exists an MAT-labeling $ \lambda _{\mathcal {B}}$ of the complete graph $K_{\mathcal {B}}$ with vertex set $N_{G_{\mathcal {B}}}$ such that $ (G_{\mathcal {B}}, \lambda |_{E_{G_{\mathcal {B}}}}) \le (K_{\mathcal {B}},\lambda _{\mathcal {B}}) $ .

Now we are in the setting of Lemma 2.19 with $ (G_1, \lambda _1)=(G[X_0], \lambda |_{E_{G[X_0]}})$ and $ (G_2, \lambda _2) =(K_{\mathcal {B}},\lambda _{\mathcal {B}}) $ . Indeed, first note that

$$ \begin{align*} N_{G_1} \cap N_{G_2} = X_{0} \cap \left(\bigcup_{Y \in \mathcal{B}}Y\right) = \bigcup_{Y \in \mathcal{B}}(X_{0} \cap Y) = X_{0} \cap Y_{0}. \end{align*} $$

Hence, $G_1[N_{G_1} \cap N_{G_2}] = G_2[N_{G_1} \cap N_{G_2}] =G[X_{0} \cap Y_{0}]$ . Denote this common complete graph by $G'$ . By Lemma 2.10, the restriction $\lambda ':=\lambda |_{E_{G'}}$ is an MAT-labeling $G'$ . Hence, $(G',\lambda ')\le (G_1, \lambda _1) $ and $(G',\lambda ')\le (G_2, \lambda _2) $ .

Therefore, by Lemma 2.19, there exists an MAT-labeling $ \widetilde \lambda $ of the complete graph K with vertex set $N_{G_1} \cup N_{G_2}=N_{G}$ such that $ (G_1, \lambda _1) \le (K,\widetilde \lambda )$ and $ (G_2, \lambda _2) \le (K,\widetilde \lambda )$ . In particular, $ (G, \lambda ) \le (K,\widetilde \lambda ) $ .

The following ‘gluing trick’ plays an important role in the construction of an MAT-labeling for a given strongly chordal graph in [27, §5.2].

Lemma 2.21 (Gluing MAT-labeled graphs [27, Theorem 5.8]).

Let $ (G_1, \lambda _1) $ and $ (G_2, \lambda _2) $ be MAT-labeled graphs such that $G_1[N_{G_1} \cap N_{G_2}] = G_2[N_{G_1} \cap N_{G_2}]$ . Suppose that this common subgraph, denoted $G'$ , is a complete graph. Suppose further that there exists an MAT-labeling $ \lambda ' $ of $ G' $ such that $ (G', \lambda ') \le (G_1, \lambda _1)$ and $ (G', \lambda ') \le (G_2, \lambda _2) $ . Define an edge-labeled graph $ (G, \lambda ) $ by $ N_{G} = N_{G_1} \cup N_{G_2}$ , $E_{G} = E_{G_1} \cup E_{G_2} $ , $\lambda |_{E_{G_1}} = \lambda _{1}, \lambda |_{E_{G_2}} = \lambda _{2}$ . Then $ (G, \lambda ) $ is an MAT-labeled graph.

We close this subsection by introducing the notion of principal cliques in an MAT-labeled graph. This notion will be useful for the construction of an LR-vine from a given MAT-labeled graph in Definition 4.11.

Lemma 2.22 (Principal clique).

Let $(G,\lambda )$ be an MAT-labeled graph. Let $e=\{i,j\} \in \pi _k$ be an edge in G of label k and $h_1, \ldots , h_{k-1}$ be the conditioning vertices of e. Then the set $K_e:=\{i,j, h_1, \ldots , h_{k-1}\}$ is a clique of G. Moreover, $(G[K_e], \lambda |_{E_{G[K_e]}} ) \le (G, \lambda )$ , and all the edges in $G[K_e] \setminus e$ have label $<k$ . We call $K_e$ the principal clique generated by $\mathbf {e}$ .

Proof. Let $G'$ denote the graph obtained from G by removing all edges of labels $>k$ . By definition, $(G', \lambda ') \le (G,\lambda )$ , where $\lambda ' := \lambda |_{E_{G'}}$ . Since e is an edge of largest label in $(G', \lambda ')$ , by Lemma 2.6, there exists a unique largest clique C of $G'$ containing the endvertices of e. Note also that C does not contain the endvertices of any edge of label k apart from e. Moreover, $(G[C], \lambda |_{E_{G[C]}} ) \le (G', \lambda ') $ by Lemma 2.10. Thus, the number of conditioning vertices of e is $k-1$ in both G and $G[C]$ . Hence, $C=K_e$ .

The converse of Lemma 2.22 is also true.

Lemma 2.23. If C is a clique in an MAT-labeled graph $ (G, \lambda ) $ such that $(G[C], \lambda |_{E_{G[C]}} ) \le (G, \lambda ) $ , then C is a principal clique. In particular, any maximal clique is principal.

Proof. Let $G':= G[C]$ and $ \lambda ':=\lambda |_{E_{G[C]}}$ . By the assumption, $(G', \lambda ')$ is an MAT-labeled complete graph. Thus, $(G', \lambda ')$ has a unique edge of maximal label by Lemma 2.7. Hence, C is a principal clique in $(G', \lambda ')$ (generated by this unique edge) and hence in $(G,\lambda )$ . The consequence follows directly from Lemma 2.10.

Lemma 2.24. Let $(G,\lambda )$ be an MAT-labeled graph. Let $e=\{i,j\} $ be an edge in G and $K_e$ the principal clique generated by e. Then both $C_1:=K_e\setminus \{i\}$ and $C_2:=K_e\setminus \{j\}$ are principal cliques in $(G,\lambda )$ . Moreover, if C is a principal clique in $(G,\lambda )$ such that $C \subsetneq K_e$ , then either $C \subseteq C_1$ or $C \subseteq C_2$ .

Proof. Let $G':= G[K_e]$ and $ \lambda ':=\lambda |_{E_{G[K_e]}}$ . By Lemma 2.22, $(G', \lambda ')$ is an MAT-labeled complete graph in which e is the unique edge of largest label. By Lemma 2.14, the endvertices i and j of e are MAT-simplicial. By Lemma 2.15, $(G'[C_1], \lambda '|_{E_{G'[C_1]}})$ and $(G'[C_2], \lambda '|_{E_{G'[C_2]}})$ are MAT-labeled complete graphs. By Lemma 2.23, $C_1 $ and $C_2 $ are principal cliques in $(G,\lambda )$ .

Let C be a principal clique in $(G,\lambda )$ such that $C \subsetneq K_e$ . Then $C = K_f$ for some edge f in $G[K_e]$ . Moreover, $ \lambda (f) = |C| -1 < |K_e| - 1 = \lambda (e).$ Thus, e is not an edge of $G[C]$ by Lemma 2.22. Hence, C cannot contain both i and j. It follows that either $C \subseteq C_1$ or $C \subseteq C_2$ .

3. Vines: graphical and poset definitions

3.1. Posets

In this subsection, we recall some basic definitions and notions of posets. All posets $\mathcal {P}=(\mathcal {P}, \le _{\mathcal {P}})$ in this paper are finite. For a poset $\mathcal {P}$ , an element $a \in \mathcal {P}$ is called maximal (resp. minimal) if there is no other element $b \in \mathcal {P}$ such that $a < b$ (resp. $a> b$ ). Denote by $\max (\mathcal {P})$ (resp. $\min (\mathcal {P})$ ) the set of all maximal (resp. minimal) elements in $\mathcal {P}$ .

Definition 3.1 (Join).

Let $\mathcal {P}$ be a poset and let $x,y \in \mathcal {P}$ . An element $v \in \mathcal {P}$ is called the join of x and y, denoted $ x\vee y$ , if the following two conditions are satisfied:

1. (1) $x \le v$ and $y \le v$ .

2. (2) For any $w \in \mathcal {P}$ , if $x \le w$ and $y \le w$ , then $v \le w$ .

The join $ x\vee y$ is unique if it exists.

Definition 3.2 (Induced subposet).

A poset $(\mathcal {Q},\le _{\mathcal {Q}})$ is an induced subposet of a poset $(\mathcal {P},\le _{\mathcal {P}})$ if $\mathcal {Q}\subseteq \mathcal {P}$ and for any $a,b \in \mathcal {Q}$ , it holds that $a \le _{\mathcal {Q}} b$ if and only if $a \le _{\mathcal {P}} b.$

For $x, y \in \mathcal {P}$ , by y covers x, we mean $x<y$ and $x\le z<y$ implies $x = z$ .

Definition 3.3 (Graded poset).

A finite poset $\mathcal {P}$ is graded if there exists a rank function $\operatorname {rk}=\operatorname {rk}_{\mathcal {P}}: \mathcal {P} \longrightarrow \mathbb {Z}_{\ge 0}$ satisfying the following three properties:

1. (1) For any $x,y \in \mathcal {P}$ , if $x < y$ , then $\operatorname {rk}(x) < \operatorname {rk}(y)$ .

2. (2) If y covers x, then $\operatorname {rk}(x) =\operatorname {rk}(y)-1$ .

3. (3) All minimal elements of $\mathcal {P}$ have the same rank. In this paper, we assume⁴ $\operatorname {rk}(x)=1$ for all $x \in \min (\mathcal {P})$ .

Equivalently, for every $x \in \mathcal {P}$ , all maximal chains among those with x as greatest element have the same length.

The dimension ⁵ $\dim (\mathcal {P})$ of $\mathcal {P}$ is defined as $\dim (\mathcal {P}):= |\min (\mathcal {P})|$ . The rank $\operatorname {rk}(\mathcal {P})$ of a graded poset $\mathcal {P}$ with rank function $\operatorname {rk}$ is defined as

$$ \begin{align*}\operatorname{rk}(\mathcal{P}):=\max \{\operatorname{rk}(x) \mid x \in \mathcal{P}\}.\end{align*} $$

Definition 3.4 (Ideal, principal ideal).

Let $\mathcal {P}$ be a poset. An (order) ideal ${\mathcal {I}}$ of $\mathcal {P}$ is a downward-closed subset (i.e., for every $x \in \mathcal {P}$ and $y \in {\mathcal {I}}$ , $x \le y$ implies that $x \in {\mathcal {I}}$ ). For $a \in \mathcal {P}$ , the ideal

$$ \begin{align*}\mathcal{P}_{\le a} : = \{ x \in \mathcal{P} \mid x \le a\}\end{align*} $$

is called the principal ideal of $\mathcal {P}$ generated by a.

Definition 3.5 (Poset homomorphism).

Let $\mathcal {P}$ and $ \mathcal {P}'$ be posets. A (poset) homomorphism $\varphi : \mathcal {P} \longrightarrow \mathcal {P}'$ is an order-preserving map, i.e. $x \le y$ implies $\varphi (x) \le \varphi (y)$ for all $x, y \in \mathcal {P}$ .

We call $\varphi $ a join-preserving homomorphism if for any $x, y \in \mathcal {P}$ such that the join $ x\vee y$ exists, then $ \varphi (x) \vee \varphi (y)$ exists and $\varphi ( x\vee y) = \varphi (x) \vee \varphi (y)$ .

We call $\varphi $ an isomorphism if $\varphi $ is bijective and its inverse is a homomorphism. The posets $\mathcal {P}$ and $ \mathcal {P}'$ are said to be isomorphic, written $\mathcal {P} \simeq \mathcal {P}'$ if there exists an isomorphism $\varphi : \mathcal {P} \longrightarrow \mathcal {P}'$ .

When $\mathcal {P}=(\mathcal {P}, \operatorname {rk})$ and $\mathcal {P}'=(\mathcal {P}', \operatorname {rk}')$ are graded posets, a homomorphism $\varphi : \mathcal {P} \longrightarrow \mathcal {P}'$ is called rank-preserving if $\operatorname {rk}'( \varphi (x) ) = \operatorname {rk}(x)$ for all $x \in \mathcal {P} $ .

A rank-preserving homomorphism $\varphi : \mathcal {P} \longrightarrow \mathcal {P}'$ sends a minimal element to a minimal element (i.e., $\varphi (\min (\mathcal {P})) \subseteq \min (\mathcal {P}')$ ). Any isomorphism between two graded posets is a homomorphism preserving rank and join.

3.2. Vines (graphical definition)

First we recall the graphical definition of a vine following [5, Definition 4.1].

Definition 3.6 (Graphical definition of vine).

Let $1\le n \le \ell $ be positive integers. A (graphical) vine ${\mathcal {V}}$ on $\ell $ elements $[\ell ]= \{1,\ldots ,\ell \} $ (or more generally, on an $\ell $ -element set called $N_1$ ) is an ordered n-tuple ${\mathcal {V}}=(F_1,F_2,\ldots ,F_{n})$ such that

1. (1) $F_1$ is a forest with nodes $N_1 = [\ell ]$ and a set of edges denoted $E_1$ ,

2. (2) for $2\le i \le n$ , $F_i $ is a forest with nodes $N_i = E_{{i-1}}$ and edge set $E_i$ .

We call $F_i$ the $\mathbf {i}$ -th associated forest of ${\mathcal {V}}$ . A graphical vine is uniquely determined by its associated forests. Denote by $N({\mathcal {V}})=N_1 \cup \cdots \cup N_n$ the set of nodes (of the associated forests) of ${\mathcal {V}}$ . We call the numbers n and $\ell $ the rank and dimension of ${\mathcal {V}}$ , respectively.

If node u is an element of node v (i.e., $u \in v$ ), we say that u is a child of v. If v is reachable from u via the membership relation: $u \in u_1 \in \cdots \in v$ , we say that u is a descendant of v.

Definition 3.7 (Node poset).

Let ${\mathcal {V}}$ be a graphical vine with node set $N({\mathcal {V}})$ . The node poset $\mathcal {P}=\mathcal {P}({\mathcal {V}})$ of ${\mathcal {V}}$ is the poset $(N({\mathcal {V}}), \le )$ defined as follows: For any $u,v \in N({\mathcal {V}})$ ,

$$ \begin{align*}u \le v \quad\text{if}\quad u \text{ is a descendant of } v.\end{align*} $$

Remark 3.8. We emphasize that a graphical vine is uniquely determined by its node poset. The terminology ‘rank’ of a vine has motivation from poset theory. If a vine ${\mathcal {V}}$ is an ordered n-tuple, then $\mathcal {P}=\mathcal {P}({\mathcal {V}})$ is a graded poset with rank function $\operatorname {rk}(v) = i$ for $v \in N_i$ ( $1 \le i \le n$ ). Thus, this number n equals the rank of $\mathcal {P}$ . In addition, the dimension of ${\mathcal {V}}$ equals the number of minimal elements in $\mathcal {P}$ , or the dimension of $\mathcal {P}$ .

Now we introduce the notion of an induced subvine, or more generally, a subvine of a vine following [8, §5], [28, §2.2.1].

Definition 3.9 (Subvine, induced subvine).

Let ${\mathcal {V}}=(F_1,F_2,\ldots ,F_{n})$ be a graphical vine.

1. 1. An ordered p-tuple ${\mathcal {V}}'=(F^{\prime }_1,F^{\prime }_2,\ldots ,F^{\prime }_{p})$ for $p \le n$ is called a subvine of ${\mathcal {V}}$ if $F^{\prime }_i$ is a subgraph of $F_i$ for each $1\le i \le p$ and ${\mathcal {V}}'$ itself is a vine.

2. 2. Given a subset $S \subseteq N_1$ , define a vine ${\mathcal {V}}[S] = (F^{\prime }_1,F^{\prime }_2,\ldots ,F^{\prime }_{p})$ on the set S as follows:

   1. (1) $F^{\prime }_1 = F_1[S]$ with the edge set $E^{\prime }_1 \subseteq E_1 =N_2$ ,

   2. (2) for $2\le i \le p$ , $F^{\prime }_i = F_i[E^{\prime }_{i-1}]$ with the edge set $E^{\prime }_i \subseteq E_{i}=N_{i+1}$ .

   We call ${\mathcal {V}}[S]$ the subvine of $\mathbf {{\mathcal {V}}}$ induced by $\mathbf {S}$ .

Remark 3.10. Any induced subvine is obviously a subvine, but the converse is not necessarily true. For example, let ${\mathcal {V}}=(F_1,F_2)$ be a vine of dimension $2$ with $N_1=\{1,2\}$ , $N_2=E_1 = \{ \{1,2\}\}$ , $E_2=\varnothing $ . The subvine ${\mathcal {V}}'=(F^{\prime }_1)$ defined by $N^{\prime }_1=\{1,2\}$ , $E^{\prime }_1=\varnothing $ is not an induced subvine of ${\mathcal {V}}$ .

3.3. Vines (poset definition)

Assumption & Notation 3.11. From now on, unless otherwise stated, we assume that $\mathcal {P}$ is a finite graded poset with a rank function $\operatorname {rk}: \mathcal {P} \longrightarrow \mathbb {Z}_{>0}$ . Denote $n:=\operatorname {rk}(\mathcal {P})$ and $\ell :=\dim (\mathcal {P})$ . For $v\in \mathcal {P}$ , denote by ${\mathcal {E}}(v)$ the set of elements covered by v. For $i \ge 0$ , define $\mathcal {P}_i := \{ v \in \mathcal {P} \mid \operatorname {rk}(v) =i\}$ and ${\mathcal {E}}(\mathcal {P}_i) := \{ {\mathcal {E}}(v) \mid v \in \mathcal {P}_i \}$ . If $\mathcal {P}$ is an $\ell $ -dimensional poset, we assume $\mathcal {P}_1=\min (\mathcal {P})=[\ell ]$ .

As noted earlier in Remark 3.8, we may think of a graphical vine and its node poset essentially as the same object. It is thus natural to look for a characterization of the node poset of a vine. We give below such a characterization obtained immediately from Definition 3.6.

Definition & Proposition 3.12 (Poset definition of vine).

A finite graded poset $\mathcal {P}$ is the node poset of a graphical vine if and only if $\mathcal {P}$ satisfies the following conditions:

1. (1) Every non-minimal node covers exactly two other nodes.

2. (2) For each $1\le i \le n = \operatorname {rk}(\mathcal {P})$ , the graph $F_i = (N_i, E_i)$ with node set $N_i:=\mathcal {P}_{i} $ and edge set $E_i:={\mathcal {E}}(\mathcal {P}_{i+1}) $ is a forest.

Assumption & Notation 3.13. From now on, unless otherwise stated, by a vine $\mathcal {P}$ we mean a finite graded poset satisfying the two conditions in 3.12. We will also retain the notion i-th associated forest $F_i =(\mathcal {P}_i, {\mathcal {E}}(\mathcal {P}_{i+1}))$ ( $1\le i \le n$ ) of $\mathcal {P}$ . If v is a node in a vine $\mathcal {P}$ and ${\mathcal {E}}(v) =\{a,b\}$ , we will often abuse notation and write $v=\{a,b\}$ . This notation is compatible with the notation of node/edge in the graphical definition of a vine.

The main reason why we choose the poset definition of a vine is because many terms and properties of a (graphical) vine have natural meanings in the language of posets. For example, subvine corresponds to ideal (Lemma 3.18), conditioned set corresponds to join (Lemma 5.1), and m-vine corresponds to LR-vine or local regularity of vine (Theorem 6.13).

Under this consideration, the following poset definition of a regular vine is equivalent to the well-known graphical definition of it in the literature (e.g., [5, Definition 4.1]).

Definition 3.14 (R-vine).

A vine $\mathcal {P}$ is a regular vine, or an R-vine for short, if $\mathcal {P}$ satisfies the following conditions:

1. (1) $\operatorname {rk}(\mathcal {P}) =\dim (\mathcal {P})$ (i.e., $n=\ell $ ).

2. (2) Each associated forest $F_i =(\mathcal {P}_i, {\mathcal {E}}(\mathcal {P}_{i+1}))$ is a tree ( $1\le i \le n$ ).

3. (3) Proximity: For any distinct nodes $a,b \in \mathcal {P}_i$ for $i\ge 2$ , if a and b are covered by a common node, then a and b cover a common node.

Remark 3.15. If $\mathcal {P}$ is an R-vine of rank n, then $|\mathcal {P}_i| = n+1-i$ for each $1 \le i \le n$ . In particular, $|\mathcal {P}| = n(n+1)/2$ .

Next, we introduce the notion of a locally regular vine.

Definition 3.16 (LR-vine).

A vine $\mathcal {P}$ is a locally regular vine, or an LR-vine for short, if every principal ideal of $\mathcal {P}$ is an R-vine.

Remark 3.17. Intuitively, an LR-vine is a vine that ‘locally’ looks like an R-vine. In particular, any R-vine is an LR-vine (see Proposition 4.4). Any ideal of a vine (resp. an LR-vine) is itself a vine (resp. an LR-vine).

Lemma 3.18. Let ${\mathcal {V}}$ be a graphical vine with the node poset $\mathcal {P}({\mathcal {V}})$ . A subset ${\mathcal {I}}$ is an ideal of $\mathcal {P}({\mathcal {V}})$ if and only if ${\mathcal {I}} = \mathcal {P}({\mathcal {V}}')$ where ${\mathcal {V}}'$ is a subvine of ${\mathcal {V}}$ uniquely determined by ${\mathcal {I}}$ .

As a result, there is a one-to-one correspondence between the subvines of a graphical vine and the ideals of its node poset.

Proof. Let ${\mathcal {V}}'$ be a subvine of ${\mathcal {V}}$ . Since ${\mathcal {V}}'$ itself is a vine, if v is a node in $\mathcal {P}({\mathcal {V}}')$ , then both children, hence all descendants of v, are also nodes in $\mathcal {P}({\mathcal {V}}')$ . Hence, $\mathcal {P}({\mathcal {V}}')$ is an ideal of $\mathcal {P}({\mathcal {V}})$ . Conversely, let ${\mathcal {I}}$ be an ideal of the vine $\mathcal {P}({\mathcal {V}})$ . By Remark 3.17, ${\mathcal {I}}$ itself is a vine. By Remark 3.8, ${\mathcal {I}}$ uniquely determines a vine ${\mathcal {V}}'$ which is a subvine of ${\mathcal {V}}$ and satisfies ${\mathcal {I}} = \mathcal {P}({\mathcal {V}}')$ .

We close this section by recalling the definition of an m-saturated vine from [16, Definition 4.2].

Definition 3.19 (M-vine).

A vine $\mathcal {P}$ is called an m-saturated vine, or an m-vine for short, if $\mathcal {P}$ is an ideal of an R-vine.

By Remark 3.17, any m-vine is an LR-vine. We will see in Theorem 6.13 that the converse also holds true.

4. From MAT-labeled graphs to LR-vines

4.1. Some known properties of vines

We begin by defining some statistics on the nodes of a vine. They play an important role in probabilistic applications of vines (e.g., [4, Theorem 3]).

Definition 4.1 (k-fold union, complete union).

Let $\mathcal {P}$ be a vine of rank n. For any node $v_i \in \mathcal {P}_i$ ( $1 \le i \le n$ ) and integer k with $0 \le k \le i-1$ , the $\mathbf {k}$ -fold union of $v_i$ is the subset $U_{v_i} (k) \subseteq \mathcal {P}_{i-k}$ defined by

$$ \begin{align*}U_{v_i} (k) :=\{ x \in \mathcal{P}_{i-k} \mid x \le v_i \}.\end{align*} $$

The complete union $U_{v_i}$ of $v_i \in \mathcal {P}_i$ is defined as the $(i-1)$ -fold union of $v_i$ ; that is,

$$ \begin{align*}U_{v_i} := U_{v_i} (i-1) \subseteq \mathcal{P}_1.\end{align*} $$

Definition 4.2 (Conditioned set, conditioning set).

Let $\mathcal {P}$ be a vine of rank n. Let $v_i = \{a, b\}\in \mathcal {P}_i$ ( $2 \le i \le n$ ) with $a,b \in \mathcal {P}_{i-1}$ (see notation in 3.13). The conditioning set $D_{v_i}$ associated with $v_i$ is defined by

$$ \begin{align*}D_{v_i} := U_a \cap U_b,\end{align*} $$

and the conditioned set $C_{v_i}$ associated with $v_i$ is defined by

$$ \begin{align*}C_{v_i} := U_a \,\triangle\, U_b,\end{align*} $$

where $\triangle $ denotes the symmetric difference.

It is easily seen that the nodes of an LR-vine satisfy the proximity condition. The following properties were proved for an R-vine in [5, 15, 16]. The arguments therein apply to a vine satisfying proximity condition as well since we only need the proximity of principal ideals.

Lemma 4.3. Let $\mathcal {P}$ be a vine of rank n and $v_i \in \mathcal {P}_i$ ( $2 \le i \le n$ ). Suppose that the proximity condition holds. The following hold:

1. (a) $|U_{v_i} (k) |= k + 1$ for $0 \le k \le i-1$ . In particular, $|U_{v_i} |= i = \operatorname {rk}(v_i)$ .

2. (b) $|D_{v_i} |= i- 2$ and $|C_{v_i}|=2$ .

We show below that local regularity and proximity of a vine are actually equivalent.

Proposition 4.4. A vine $\mathcal {P}$ is locally regular if and only if the proximity condition holds for the nodes of $\mathcal {P}$ .

Proof. It remains to show proximity implies local regularity. Let $v \in \mathcal {P}$ . We need to show that the principal ideal $\mathcal {P}_{\le v}$ itself is an R-vine. Write $\mathcal {P}_{\le v}=(T_1,T_2,\ldots ,T_{p})$ , where $p= \operatorname {rk}(v) \le n$ . By Lemma 4.3(a), the rank p and dimension $|U_{v}|$ of $\mathcal {P}_{\le v}$ are equal. Also by Lemma 4.3(a), each forest $T_{p-k}$ ( $0\le k \le p-1$ ) has $k+1$ nodes and k edges. Thus, these forests must be trees. Clearly, proximity of a vine is preserved under taking ideals. It follows that $\mathcal {P}_{\le v}$ is an R-vine.

Lemma 4.5. Let $\mathcal {P}=(F_1, \ldots ,F_{n})$ be an LR-vine and ${\mathcal {V}}$ be the graphical vine defined by $\mathcal {P}$ . Then for every $a \in \mathcal {P}$ , the ideal $\mathcal {P}_{\le a}$ coincides with the node poset of the induced subvine ${\mathcal {V}}[U_a]$ .

Proof. Write ${\mathcal {V}}[U_a]=(F^{\prime }_1, \ldots ,F^{\prime }_{q})$ and $\mathcal {P}_{\le a}=(T_1,\ldots ,T_{p})$ , where $T_i$ ’s all are trees. Note that $F^{\prime }_1 = F_1[U_a]$ is a forest with at most $|U_a|-1$ edges. However, the tree $T_1$ is a subgraph of $F_1$ with node set $U_a$ that has exactly $|U_a|-1$ edges. Hence, $F^{\prime }_1 =T_1$ . A repeated application of this argument yields $p=q$ and $F^{\prime }_k = T_k$ for all $1\le k \le p$ . Therefore, $\mathcal {P}_{\le a}$ is the node poset of ${\mathcal {V}}[U_a]$ .

Corollary 4.6. Let $\mathcal {P}$ be an LR-vine and $a,b$ be nodes in $\mathcal {P}$ . If $U_a \subseteq U_b$ , then $a \le b$ . In particular, if $U_a = U_b$ , then $a = b$ .

Proof. Let ${\mathcal {V}}$ be the graphical vine defined by $\mathcal {P}$ . If $U_a \subseteq U_b$ , then ${\mathcal {V}}[U_a]$ is a subvine of ${\mathcal {V}}[U_b]$ . By Lemma 4.5, $\mathcal {P}_{\le a}\subseteq \mathcal {P}_{\le b}$ . Hence, $a \le b$ . If $U_a = U_b$ , then by the first assertion, $a \le b$ and $b \le a$ . Thus, $a = b$ .

Remark 4.7. By definition, an R-vine has a unique maximal element. Thus, an ideal ${\mathcal {I}}$ of an LR-vine $\mathcal {P}$ is regular if and only if ${\mathcal {I}}$ is a principal ideal.

If two posets $\mathcal {P}$ and $ \mathcal {P}'$ are isomorphic and $\mathcal {P}$ is an (L)R-vine, then $\mathcal {P}'$ is also an (L)R-vine. The result below enables us to represent a node in an LR-vine by its complete union (see 4.17 for an example).

Proposition 4.8. Let $\mathcal {P}$ be an LR-vine. Let $ \widehat {\mathcal {P}}$ be the poset consisting of the complete unions of the nodes in $\mathcal {P}$ (i.e., $ \widehat {\mathcal {P}} =\{ U_a \mid a \in \mathcal {P}\}$ with partial order given by set inclusion). Define a map

$$ \begin{align*}\eta_{\mathcal{P}}\colon \mathcal{P} \longrightarrow \widehat{\mathcal{P}} \quad \text{via} \quad a \mapsto U_a.\end{align*} $$

Then $\eta _{\mathcal {P}}$ is a poset isomorphism; hence, $\mathcal {P} \simeq \widehat {\mathcal {P}} $ .

Proof. Clearly, $\eta _{\mathcal {P}}$ is a surjective homomorphism. By Corollary 4.6, for any $a,b \in \mathcal {P}$ , if $U_a= U_b$ , then $a = b$ . Thus, $\eta _{\mathcal {P}}$ is injective and hence bijective. Again by Corollary 4.6, for any $a,b \in \mathcal {P}$ , $U_a \subseteq U_b$ if and only if $a \le b$ . Thus, the inverse of $\eta _{\mathcal {P}}$ is a poset homomorphism. We conclude that $\eta _{\mathcal {P}}$ is an isomorphism.

Given a vine, it is important to know which induced subposet is again a vine. This motivates the following notion of truncation of a vine [6].

Definition 4.9 (Truncation).

Let $(\mathcal {P},\operatorname {rk})$ be a finite graded poset of rank n and let $1\le k \le n$ . The induced subposet $\overline {\mathcal {P}}_{\le k} := \{ x \in \mathcal {P} \mid \operatorname {rk}(x) \le k \} = \bigcup _{i=1}^k \mathcal {P}_i$ with the rank function $\overline {\operatorname {rk}} = \operatorname {rk}$ is called the ${k}$ -lower truncation of $\mathcal {P}$ .

Likewise, the induced subposet $\overline {\mathcal {P}}_{\ge k}:= \{ x \in \mathcal {P} \mid \operatorname {rk}(x) \ge k \} = \bigcup _{i=k}^{n} \mathcal {P}_i$ with the rank function $\overline {\operatorname {rk}}(v) = \operatorname {rk}(v) - k+1$ for all $v \in \overline {\mathcal {P}}_{\ge k}$ is called the ${k}$ -upper truncation of $\mathcal {P}$ .

An induced subposet $\mathcal {Q}$ of $\mathcal {P}$ is called a lower (resp. an upper) truncation if $\mathcal {Q} = (\overline {\mathcal {P}}_{\le k},\overline {\operatorname {rk}} ) $ (resp. $\mathcal {Q} = (\overline {\mathcal {P}}_{\ge k},\overline {\operatorname {rk}} ) $ ) for some k. A truncation $\mathcal {Q}$ of $\mathcal {P}$ is called proper if $\mathcal {Q} \ne \mathcal {P}$ .

Remark 4.10. Any lower truncation of a vine is an ideal. Hence, by Remark 3.17, any lower truncation of a vine (resp. an LR-vine) is itself a vine (resp. an LR-vine). However, a proper lower truncation of an R-vine of rank $>1$ is not an R-vine. See Figure 5 for an example of a lower truncation.

A proper upper truncation of a vine of rank $>1$ is not an ideal. However, proximity is preserved under taking either upper or lower truncation. Hence, by Proposition 4.4, any upper truncation of an LR-vine (resp. a vine) is an LR-vine (resp. a vine). Unlike the lower truncation case, any upper truncation of an R-vine is an R-vine by Remark 3.15.

The discussion above indicates that LR-vines are closed under either upper or lower truncation, while R-vines are only closed under upper truncation. We will see in §7.2.2 that these classes are also closed under ‘vertical’ truncation, or more precisely, marginalization.

4.2. Construct an LR-vine from a given MAT-labeled graph

Definition 4.11. Let $(G,\lambda )$ be an MAT-labeled graph with $N_G =[\ell ] $ and clique number $\omega (G)$ . Define a finite graded poset $\mathcal {P}=(\mathcal {P}, \le _{\mathcal {P}}, \operatorname {rk}_{\mathcal {P}})$ from $(G,\lambda )$ as follows:

1. (1) $\mathcal {P}$ consists of the sets $\{i\}$ for $1 \le i \le \ell $ and all the principal cliques in $(G,\lambda )$ (Lemma 2.22).

2. (2) For $u,v \in \mathcal {P}$ , $u \le _{\mathcal {P}} v$ if u is a subset of v.

3. (3) $\operatorname {rk}_{\mathcal {P}}(v) = |v|$ for all $v \in \mathcal {P}$ .

Remark 4.12. It is easily seen that $\min (\mathcal {P}) =\{ \{i\} \mid 1 \le i \le \ell \}$ . The poset $\mathcal {P}$ is graded by $\operatorname {rk}_{\mathcal {P}}$ because by Lemma 2.24, for every edge $e=\{i,j\}$ in G, the principal clique $K_e$ generated by e covers exactly two principal cliques $K_e\setminus \{i\}$ and $K_e\setminus \{j\}$ . Note also that $\operatorname {rk}(\mathcal {P})=\omega (G)$ by Lemma 2.6.

Theorem 4.13. The poset $\mathcal {P}=(\mathcal {P}, \le _{\mathcal {P}}, \operatorname {rk}_{\mathcal {P}})$ from Definition 4.11 is an LR-vine. In particular, if $(G,\lambda )$ is an MAT-labeled complete graph, then $\mathcal {P}$ is an R-vine.

Proof. First we prove the first assertion. We argue by induction on the number $\ell $ of vertices of G. The assertion is clearly true when $\ell =1$ . Suppose $\ell \ge 2$ . By Theorem 2.17, there exists an MAT-PEO $(a_1,\ldots , a_\ell )$ of $(G,\lambda )$ . Denote $G' := G \setminus a_\ell $ and $\lambda ' := \lambda |_{E_{G'}} $ . By Lemma 2.15, $(G',\lambda ')$ is an MAT-labeled graph. Let $\mathcal {P}'$ be the poset defined by $(G',\lambda ')$ using Definition 4.11. By the induction hypothesis, $\mathcal {P}'$ is an LR-vine.

Denote $d:= \deg _G(a_\ell )$ . If $d=0$ (i.e., $a_\ell $ is an isolated vertex), then $\mathcal {P}= \mathcal {P}' \cup \{\{a_\ell \}\}$ is clearly an LR-vine. Suppose $d \ge 1$ . Write $\operatorname {Nbd}_G(a_\ell ) :=\{b_q \mid 1 \le q \le d\} \subseteq N_{G'}$ so that $\{a_\ell , b_q\} \in \pi _q$ . For $1 \le q \le d$ , define the following subgraph $H_q$ of G:

$$ \begin{align*}H_q := G \setminus \{ \{a_\ell, b_{q+1}\}, \ldots,\{a_\ell, b_{d}\} \}.\end{align*} $$

It is not hard to check that $a_\ell $ is an MAT-simplicial vertex in $(H_q, \lambda |_{E_{H_q}})$ for each $1 \le q \le d$ . By Lemma 2.15, each $(H_q, \lambda |_{E_{H_q}})$ is an MAT-labeled graph since $G' = H_q \setminus a_\ell $ .

For each $1 \le q \le d$ , let $\mathcal {R}_q$ be the poset defined by $(H_q, \lambda |_{E_{H_q}})$ from Definition 4.11. We may write

$$ \begin{align*}\mathcal{R}_q = \mathcal{P}' \cup \{v_0,\ldots,v_q\} = \mathcal{P} \setminus \{ v_{q+1}, \ldots, v_d\},\end{align*} $$

where $v_0 := \{a_\ell \}$ and each $v_q$ is the principal clique generated by $\{a_\ell , b_q\}$ . We claim that for each $1 \le q \le d$ , the poset $\mathcal {R}_q$ is an LR-vine. In particular, the case $q=d$ whence $H_d=G$ and $\mathcal {R}_d =\mathcal {P}$ yields the first assertion of Theorem 4.13.

We argue by induction on q. The case $q=1$ is simple. The new non-minimal node $v_1 = \{a_\ell , b_1\}$ covers exactly two nodes $v_0=\{a_\ell \}$ and $\{b_1\}$ . Also, the new minimal node $v_0$ is only covered by $v_1$ since $a_\ell \notin v$ for all $v \in \mathcal {P}'$ . The first associated forest of $\mathcal {R}_1$ is given by that of $\mathcal {P}'$ with $a_\ell $ added so that $a_\ell $ is only connected to $b_1$ . The second associated forest of $\mathcal {R}_1$ is given by that of $\mathcal {P}'$ with an isolated vertex $v_1$ added. The remaining associated forests of $\mathcal {R}_1$ are the same as those of $\mathcal {P}'$ . The proximity condition holds trivially. Hence, $\mathcal {R}_{1}$ is an LR-vine by Proposition 4.4.

Suppose that the claim is true for some $1 \le q <d$ . First note that $H_{q+1}\setminus \{a_\ell , b_{q+1}\} = H_q$ and $\mathcal {R}_{q+1}\setminus \{v_{q+1}\} = \mathcal {R}_q$ . Since $a_\ell $ is an MAT-simplicial vertex in $(H_q, \lambda |_{E_{H_q}})$ , by (MS3), the edges in the complete subgraph $H_q[v_q]$ of $H_q$ induced by the vertices in $v_q$ have label $\le q-1$ except $\{a_\ell , b_q\} \in \pi _q$ . Similarly, any edge of the form $\{b_{q+1},h\}$ where h is a vertex in $v_q\setminus \{a_\ell \}$ has label $\le q$ . Therefore, $v_q\setminus \{a_\ell \}$ consists of the q conditioning vertices of $\{a_\ell , b_{q+1}\}$ in $H_{q+1}$ . Hence, $v_{q+1} = v_q \cup \{b_{q+1}\}$ .

By Lemma 2.24, $u:=v_{q+1} \setminus \{a_\ell \}$ is a principal clique (of cardinality $q+1$ ) in $(G',\lambda ')$ and hence a vertex in $\mathcal {P}'$ . Let c be the vertex in u such that c is largest with respect to the MAT-PEO $(a_1,\ldots , a_{\ell -1})$ of $(G',\lambda ')$ . Since the edges in the complete subgraph $G'[u]$ have label $\le q$ , by (MS2), there exists among these edges an edge $e $ incident on c of label q. By the preceding paragraph, this edge e must be $\{b_{q+1},h^*\}$ for some vertex $h^*$ in $v_q\setminus \{a_\ell \}$ . Moreover, such an edge e is unique which is guaranteed by (ML1). Therefore, u is generated by e. In particular, u covers $v_q\setminus \{a_\ell \}$ in $\mathcal {P}'$ by Lemma 2.24. (See Figure 1 for a pictorial illustration of the proof.)

Figure 1 A pictorial illustration of the proof of Theorem 4.13.

Now $v_{q+1}$ covers exactly two nodes $v_q$ and u in $\mathcal {R}_{q+1}$ . Also, $v_q$ is covered only by $v_{q+1}$ since $a_\ell \notin v$ for all $v \in \mathcal {P}'$ . The $(q+1)$ -th associated forest of $\mathcal {R}_{q+1}$ is given by that of $\mathcal {R}_{q}$ with $v_q$ added so that $v_q$ is only connected to u. The $(q+2)$ -th associated forest of $\mathcal {R}_{q+1}$ is given by that of $\mathcal {R}_{q}$ with an isolated vertex $v_{q+1}$ added. The remaining associated forests of $\mathcal {R}_{q+1}$ are the same as those of $\mathcal {R}_{q}$ . This follows that $\mathcal {R}_{q+1}$ is a vine. Furthermore, both $v_q \in \mathcal {R}_{q}$ and $u \in \mathcal {P}'$ cover $v_q\setminus \{a_\ell \}$ . Therefore, the proximity condition holds in $\mathcal {R}_{q+1}$ . Hence, $\mathcal {R}_{q+1}$ is an LR-vine by Proposition 4.4.

Finally, we show the second assertion of Theorem 4.13. If $(G,\lambda )$ is an MAT-labeled complete graph, then $\operatorname {rk}(\mathcal {P}) =\dim (\mathcal {P})=\ell $ . The proofs for the proximity of $\mathcal {P}$ and the fact the associated forests of $\mathcal {P}$ are trees run essentially along the same line as the proof of the first assertion.

Corollary 4.14. Given an MAT-labeled graph $(G,\lambda ) $ , let $\mathcal {P}$ denote the LR-vine defined by $(G,\lambda )$ from Definition 4.11. Then a node $v \in \mathcal {P}$ has complete union $U_v = \{ \{a\} \mid a \in v\}$ . Moreover, if $v=K_e \in \mathcal {P}$ is a non-minimal node where $e=\{i,j\} \in E_G$ , then v has conditioned set $C_{v}=\{\{i\},\{j\}\}$ .

Proof. Use Remark 4.12 and argue by an induction on $\operatorname {rk}(v)$ .

We close this section by giving an example to illustrate the construction in Definition 4.11 and Theorem 4.13. First we need a definition.

Definition 4.15 (D-vine).

An R-vine is called a D-Vine if each associated tree has the smallest possible number of vertices of degree $1$ . Equivalently, each associated tree is a path graph.

Remark 4.16. Let $\Phi $ be an irreducible root system in $\mathbb {R}^\ell $ with a fixed positive system $\Phi ^+ \subseteq \Phi $ and the associated set of simple roots $\Delta = \{\alpha _1,\ldots ,\alpha _\ell \}$ . Suppose that $\Phi $ is of type $A_\ell $ and the Dynkin diagram of $\Phi $ is the path graph $\alpha _1-\alpha _2-\cdots -\alpha _\ell $ . Then the positive roots of $\Phi $ are given by

$$ \begin{align*}\Phi^+ = \left\{ \sum_{i\le k \le j } \alpha_k \,\middle\vert\, 1\le i\le j \le m\right\}.\end{align*} $$

It is not hard to show that the D-vine $\mathcal {P}$ with the first associated tree $ 1-2-\cdots -\ell $ is isomorphic to the root poset $\mathcal {R} (A_\ell )$ of type $A_\ell $ under the following isomorphism:

$$ \begin{align*}\mu \colon \mathcal{P} \longrightarrow \mathcal{R} (A_\ell) \quad \text{via} \quad v \mapsto \sum_{k \in U_v} \alpha_k .\end{align*} $$

Example 4.17. Figure 2 depicts a $4$ -dimensional D-vine (middle) that can be constructed in three ways. First, it is the node poset of a graphical vine on $[4]$ (left) under the isomorphism in Proposition 4.8. Second, it is the poset defined an MAT-labeled complete graph (right) using Definition 4.11. Third, it is the root poset of type $A_4$ by Remark 4.16. The elements in the poset are written without set symbol for simplicity. The conditioned set of a non-minimal node is given to the left of the ‘ $\mid $ ’ sign, while the conditioning set appears on the right. For example, the top node $\{ 1,2,3,4\} $ (or the largest clique generated by $\{v_1,v_4\}$ ) is written by .

Figure 2 An MAT-labeled complete graph on $4$ vertices (right), the D-vine (middle) ( $=$ type A root poset) defined by the graph using Definition 4.11, and the corresponding graphical vine (left).

Remark 4.18. Max Wakefield let us know an interesting observation that the D-vine is isomorphic to the intersection lattice (with bottom element removed) of the Pascal arrangement introduced in [19]. We leave a possible generalization of the main result in [19] to an R-vine for future research.

5. From LR-vines to MAT-labeled graphs

Constructing an MAT-labeled graph from a given LR-vine needs more effort.

5.1. Some new properties of LR-vines

The following lemma provides the key ingredient of our construction.

Lemma 5.1 (Joining path).

Let $\mathcal {P}=(F_1,\ldots ,F_{n})$ be an $\ell $ -dimensional LR-vine. Let $i, j \in \min (\mathcal {P})=[\ell ]$ be distinct minimal nodes. Let $v \in \mathcal {P}_{r}$ be a non-minimal node ( $2 \le r \le n$ ). The following are equivalent:

1. (1) $v =i \vee j$ (i.e., v is the join of i and j).

2. (2) $C_v = \{i,j\}$ (i.e., $\{i,j\}$ is the conditioned set of v).

3. (3) There exist r paths (uniquely determined by i and j) $P_k=P_k(i,j) \in E_k$ in the forests $F_k$ ( $1 \le k \le r$ ) satisfying the following three conditions:

   1. (a) $P_1=(a_{1,1}, a_{1,2}, \ldots , a_{1,p_1}), p_1 \ge 2$ is a (unique) path connecting nodes $a_{1,1}:=i$ and $a_{1,p_1}:=j$ in $F_1$ .

   2. (b) For $2 \le k \le r$ , $P_k=(a_{k,1}, a_{k,2},\ldots , a_{k,p_k}), p_k \ge 1$ is a (unique) path connecting nodes $a_{k,1}:=\{ a_{k-1,1}, a_{k-1,2}\}$ and $a_{k,p_k}:=\{ a_{k-1,p_{k-1}-1}, a_{k-1,p_{k-1}}\}$ in $F_k$ .

   3. (c) $P_{r}=v$ .

In this case, we call the path $P_k(i,j) \in E_k$ ( $1 \le k \le r$ ) the $\mathrm{k}$ -joining path of the pair $\{i,j\}$ (or $P_k(v)$ the k-joining path of v).

Before giving the proof of Lemma 5.1, we address some remarks.

Remark 5.2. If $\mathcal {P}=(T_1,\ldots ,T_{\ell })$ is an R-vine, then for any distinct $i, j \in [\ell ]$ , the joining paths of $ \{i,j\}$ always exist since $T_k$ is a tree for each $1 \le k \le \ell $ and $T_{\ell }$ has only one node.

The implication $(2) \Leftarrow (3)$ was stated in the proof of [11, Lemma 3.9] for R-vines. Unfortunately, the proof was not complete. We give below a complete proof that works for an arbitrary LR-vine.

Proof of Lemma 5.1.

First we prove $(2) \Leftarrow (3)$ . By definition, $P_{r-1}=\{a_{r-1,1}, a_{r-1,p_{r-1}}\}=v$ . We need to show $C_{v} = U_{a_{r-1,1}} \,\triangle \, U_{a_{r-1,p_{r-1}}} = \{i, j\}$ . Since $|C_{v}|=2$ by Lemma 4.3(b), it is enough to show that $i \le a_{r-1,1} , i \not \le a_{r-1,p_{r-1}}$ and $j \le a_{r-1,p_{r-1}}, j \not \le a_{r-1,1}$ . This follows once we prove that for each $1 \le k \le r $ , the following two statements hold:

1. (S1) $i\le a_{k,1}$ and $i\not \le b$ , where b is any node in $F_k$ such that there exists a (unique) path in $F_k$ connecting b and some non-initial node $a_{k,t}$ ( $2 \le t \le p_k$ ) of $P_k$ but not passing through its initial node $a_{k,1}$ .

2. (S2) $j\le a_{k,p_k}$ and $j \not \le c$ , where c is any node in $F_k$ such that there exists a (unique) path in $F_k$ connecting c and some non-final node $a_{k,t}$ ( $1 \le t \le p_k-1$ ) of $P_k$ but not passing through its final node $a_{k,p_k}$ .

Since the roles of i and j are symmetric, it suffices to prove Statement (S1). First we need the following crucial property of the paths $P_k$ ’s.

Claim 5.3. Fix $1 \le k \le r - 1$ . Suppose $P_k$ is given by $P_k =(\alpha _{1}, \alpha _{2},\ldots , \alpha _{p})$ for $p \ge 1$ in terms of its node sequence. Then the node set of $P_{k+1}$ consists of all the edges $\{\alpha _{t}, \alpha _{t+1}\}$ for $1 \le t \le p-1$ of $P_k$ , and some edges of the form $\{\alpha _{s+1}, \mu \}$ for $1 \le s \le p-2$ and $\mu \in \operatorname {Nbd}_{F_k}(\alpha _{s+1})\setminus \{\alpha _{s},\alpha _{s+2}\}$ .

Proof of Claim 5.3.

By definition, the initial and final nodes in $P_{k+1}$ are $\{\alpha _{1}, \alpha _{2}\}$ and $\{\alpha _{p-1}, \alpha _{p}\}$ , respectively. The claim is clearly true for $p\le 3$ . Suppose $p \ge 4$ . By the proximity condition, the length of $P_{k+1}$ is at least two. Again by the proximity, the node adjacent to the initial node in $P_{k+1}$ must have the form either $\{\alpha _{1}, \beta \}$ , where $\beta \in \operatorname {Nbd}_{F_k}(\alpha _{1}) \setminus \{\alpha _2\}$ , or $\{\alpha _{2}, \gamma \}$ , where $\gamma \in \operatorname {Nbd}_{F_k}(\alpha _{2})\setminus \{\alpha _1\}$ .

The former cannot occur; otherwise, arguing on the proximity yields two different paths in $F_k$ connecting $\alpha _{1}$ and $\alpha _{p}$ ; one is $P_k$ passing through $\alpha _{2}$ , the other passing through some $\beta ' \in \operatorname {Nbd}_{F_k}(\alpha _{1}) \setminus \{\alpha _2\}$ . Thus, the latter occurs, and $P_{k+1}$ possibly continues to pass through some node of the same form as $\{\alpha _{2}, \gamma \}$ . The following two conditions hold: The path $P_{k+1}$ must (i) reach the node $\{\alpha _{2}, \alpha _{3}\}$ , and after that, (ii) does not pass through any further node of this form. If either (i) or (ii) fails, then there are two different paths in $F_k$ connecting $\alpha _{2}$ and $\alpha _{p}$ . Hence, $P_{k+1}$ passes through some node of the form $\{\alpha _{3}, \delta \}$ , where $\delta \in \operatorname {Nbd}_{F_k}(\alpha _{3})\setminus \{\alpha _2\}$ until it reaches $\{\alpha _{3}, \alpha _{4}\}$ and so on.

A repeated application of the argument above completes the proof of the claim.

$$ \begin{align*} P_{k}: &\, \alpha_{1} \rightarrow \alpha_{2} \rightarrow \alpha_{3} \rightarrow \cdots \rightarrow \alpha_{p},\\ P_{k+1}: &\, \{\alpha_{1}, \alpha_{2}\} \rightarrow \{\alpha_{2}, \gamma\} \rightarrow \cdots \rightarrow \{\alpha_{2}, \alpha_{3}\} \rightarrow \cdots \rightarrow \{\alpha_{3}, \alpha_{4}\} \rightarrow \cdots \rightarrow \{\alpha_{p-1}, \alpha_{p}\}.\\[-36pt] \end{align*} $$

Now we return to the proof of (S1). The first part is easy since by definition, $a_{1,1}=i$ and $a_{k,1} \le a_{k+1,1}$ for all $1 \le k \le r - 1$ . We argue the second part by induction on k. The statement is clearly true for $k=1$ . Suppose it is true for any $1 \le k <r$ . Let $b=\{b_1,b_2\}$ be an arbitrary node in $F_{k+1}$ such that there exists a path in $F_{k+1}$ connecting b and some non-initial node of $P_{k+1}$ but not passing through its initial node. By the relation of the paths $P_{k}$ and $P_{k+1}$ proved in Claim 5.3, there exists a path in $F_{k}$ connecting $b_s$ ( $s=1$ or $2$ ) and some non-initial node of $P_{k}$ but not passing through its initial node. By the induction hypothesis, we must have $i \not \le b_1$ and $i\not \le b_2$ . It follows that $i\not \le b$ . This completes the proof of (S1) and hence the proof of $(2) \Leftarrow (3)$ .

To prove $(2) \Rightarrow (3)$ , the following fact is useful.

Remark 5.4. Suppose $\mathcal {P}$ is an R-vine. By Remark 5.2, for any $1 \le i\ne j \le \ell $ , the joining paths of $ \{i,j\}$ always exist. Thus, by $(2) \Leftarrow (3)$ , there exists a non-minimal node $v \in \mathcal {P}$ such that $C_v = \{i,j\}$ . Moreover, by Remark 3.15, the number of non-minimal nodes in $\mathcal {P}$ is equal to $\ell (\ell -1)/2$ . Therefore, every pair of distinct elements in $[\ell ]$ occurs exactly once as the conditioned set of a non-minimal node.

Now we give the proof of $(2) \Rightarrow (3)$ . Write $\mathcal {P}_{\le v}=(T_1,T_2,\ldots ,T_{r})$ . By definition, $\mathcal {P}_{\le v}$ is an R-vine. If $C_v = \{i,j\}$ , then i and j are nodes in $T_1$ . By Remark 5.2 and $(2) \Leftarrow (3)$ , there exist $r'$ joining paths of $ \{i,j\}$ and hence a node $v'$ in $\mathcal {P}_{\le v}$ such that $C_{v'} = \{i,j\} =C_v $ . Hence, by Remark 5.4, $v=v'$ and $r=r'$ .

Next, we show $(1) \Leftarrow (3)$ . By Statements (S1) and (S2), $i \le v$ and $j \le v$ . Let $u\in \mathcal {P}_s$ for $1 \le s \le n$ be a node such that $i \le u$ and $j \le u$ . In particular, i and j are minimal nodes in the R-vine $\mathcal {P}_{\le u}$ . By Remark 5.2, there exist s joining paths of $ \{i,j\}$ in $\mathcal {P}_{\le u}$ . By the uniqueness of the paths in the associated forests $F_k$ ’s, the joining paths of $ \{i,j\}$ in $\mathcal {P}_{\le u}$ and $\mathcal {P}_{\le v}$ must be the same. Thus, $r=s$ and $v \le u$ . Hence, v is the join of i and j.

Finally, we show $(1) \Rightarrow (3)$ . Suppose $v =i \vee j$ . Hence, there exist $s'$ joining paths of $ \{i,j\}$ and hence a node $u'$ in $\mathcal {P}_{\le v}$ such that $i \le u'$ and $j \le u'$ . By the definition of a join, we must have $v=u'$ and $s'=r$ .

Remark 5.5. The missing piece of the proof of [11, Lemma 3.9] is the fact that $i \not \le a$ for any non-initial node a in $P_k$ does not automatically imply that $i \not \le b$ for any non-initial node b in $P_{k+1}$ .

Example 5.6. Let $\mathcal {P}$ be the $4$ -dimensional D-vine in Figure 2. Let $i=1$ and $j=4$ be minimal nodes. The joining paths of $1$ and $4$ are given by $P_1 = (1,2,3,4)$ , $P_2 = (12,23,34)$ , , . The join of $1$ and $4$ is . The conditioned set of is $\{1,4\}$ . These calculations are consistent with Lemma 5.1.

The first two corollaries below are immediate consequences of Lemma 5.1 and Claim 5.3, respectively.

Corollary 5.7. Let $u,v$ be two non-minimal nodes in an LR-vine $\mathcal {P}$ . If $C_u = C_v$ , then $u = v$ .

Corollary 5.8. Let v be a non-minimal node in an LR-vine $\mathcal {P}$ . Then all the nodes of the joining paths of v are in $\mathcal {P}_{\le v}$ .

Corollary 5.9. Let $\mathcal {P} $ be an LR-vine. Let v be a non-minimal node in $\mathcal {P}$ with conditioned set $C_v =\{i,j\}$ for $i, j \in [\ell ]$ . If k is in the conditioning set of v, then the joins $u= i \vee k$ and $w=j \vee k$ exist in $\mathcal {P}$ . Moreover, $v>u$ and $v>w$ .

Proof. If $k \in D_v$ , then k is a minimal node of the R-vine $\mathcal {P}_{\le v}$ . By Remark 5.2, the joining paths of $ \{i,k\}$ and $ \{k,j\}$ always exist. Thus, by Lemma 5.1, the joins $u= i \vee k$ and $w=j \vee k$ exist in $\mathcal {P}_{\le v}$ and hence in $\mathcal {P}$ . Note that $v \ne u$ and $v \ne w$ since $C_v \ne C_u$ and $C_v \ne C_w$ . Thus, $v>u$ and $v>w$ .

We give some further properties of the joining paths of a non-minimal node in an LR-vine.

Lemma 5.10. Let $\mathcal {P} =(F_1,\ldots ,F_{n})$ be an LR-vine and let $u,v$ be two distinct non-minimal nodes in $\mathcal {P}$ . Suppose that there exists a number k ( $1\le k\le \operatorname {rk}(u)$ ) such that the k-joining path $P_k(u)$ is a proper subpath of the k-joining path $P_k(v)$ . Then $P_q(u)$ is a proper subpath of $P_q(v)$ for all $k\le q\le \operatorname {rk}(u)$ . In particular, $v>u$ .

Proof. Write $P_k(v)=(a_{k,1}, a_{k,2},\ldots , a_{k,p_k}), p_k \ge 2$ . Since $P_k(u)$ is a proper subpath of $P_k(v)$ , we may write $P_k(u)=(a_{k,s},\ldots , a_{k,t})$ , where $1\le s\le t\le p_k$ and $(s,t)\neq (1,p_k)$ . Note that $P_{k+1}(u)$ is the unique path connecting $\{a_{k,s},a_{k,s+1}\}$ and $\{a_{k,t-1},a_{k,t}\}$ in the forest $F_{k+1}$ . Moreover, by Claim 5.3, $P_{k+1}(v)$ passes through $\{a_{k,s},a_{k,s+1}\}$ and $\{a_{k,t-1},a_{k,t}\}$ . Therefore, $P_{k+1}(u)$ must be a proper subpath of $P_{k+1}(v)$ . Applying this argument repeatedly, we may show that $P_q(u)$ is a proper subpath of $P_q(v)$ for all $k\le q\le \operatorname {rk}(u)$ . In particular, the case $q=\operatorname {rk}(u)$ yields $v>u$ .

Before giving the next property in Lemma 5.13, we need a technical lemma on paths in a forest.

Lemma 5.11. Let F be a forest. Let $i_1,i_2, \ldots , i_m$ for $m \ge 3$ be mutually distinct nodes in F. For each $1 \le s \le m$ , suppose that there exists a (unique) path $P_{s,s+1}$ in F connecting $i_s$ and $i_{s+1}$ . Here, we take the index modulo m. Denote by $e^{\prime }_{s,s+1}$ and $e^{\prime \prime }_{s,s+1}$ the edges in $P_{s,s+1}$ incident on $i_s$ and $i_{s+1}$ , respectively. Suppose that there exists $1 \le t \le m$ such that $e^{\prime \prime }_{t,t+1} \ne e^{\prime }_{t+1,t+2}$ . Then among the paths $P_{s,s+1}$ ’s for $s \notin \{t,t+1\}$ there exist two paths $P_{a,a+1}$ and $P_{b,b+1}$ (not necessarily distinct) both of length $\ge 2$ containing $e^{\prime \prime }_{t,t+1}$ and $e^{\prime }_{t+1,t+2}$ , respectively.

Proof. Let T denote the subgraph of F induced by the vertices of the paths $P_{s,s+1}$ ’s for all $1 \le s \le m$ . Note that T is a connected subgraph of F; hence, T is a tree.

If $m=3$ , then the path $P_{t,t+2}$ of length $\ge 2$ contains both $e^{\prime \prime }_{t,t+1}$ and $e^{\prime }_{t+1,t+2}$ . Suppose $m>3$ . By the assumption, the concatenation of $P_{t,t+1}$ and $P_{t+1,t+2}$ , denoted P, is the unique path in T connecting $i_{t}$ and $i_{t+2} $ . Moreover, there exists a walk in T connecting $i_{t}$ and $i_{t+2} $ whose edge set is the union of the edge sets of $P_{t+2,t+3}$ , $P_{t+3,t+4}, \ldots , P_{t-1,t} $ . By Lemma 2.1, this walk must contain the path P. In particular, the edge $e^{\prime \prime }_{t,t+1}$ (resp. $e^{\prime }_{t+1,t+2}$ ) is contained in a path $P_{a,a+1}$ (resp. $P_{b,b+1}$ ) for some $a, b\notin \{t,t+1\}$ . Clearly, both $P_{a,a+1}$ and $P_{b,b+1}$ have lengths $\ge 2$ .

Notation 5.12. Let $\mathcal {P}$ be an LR-vine. In what follows, for two distinct minimal nodes $i ,j \in \min (\mathcal {P})$ , if the join $i \vee j$ exists in $\mathcal {P}$ , we denote $v_{i,j} := i \vee j \in \mathcal {P}$ . Sometimes, two minimal nodes in $\mathcal {P}$ are denoted by $i_s,i_t$ , in which case, we write $v_{s,t} :=v_{i_s,i_t}$ . Note that by Lemma 5.1, the nodes $v_{i,j}$ ’s are mutually distinct (i.e., if $\{i,j\} \ne \{i',j'\}$ , then $v_{i,j} \ne v_{i',j'}$ ).

Lemma 5.13. Let $\mathcal {P} =(F_1,\ldots ,F_{n})$ be an LR-vine of rank n. Let $i_1,i_2, \ldots , i_m \in \min (\mathcal {P}) $ for $m \ge 3$ be mutually distinct minimal nodes in $\mathcal {P}$ . Suppose that the join $v_{s,s+1} $ exists in $\mathcal {P}$ for each $1 \le s \le m$ . Here again, we take the index modulo m. Then there exist a node $i_t$ and a join $v_{a,a+1} $ for $a,t \in [m]$ such that $a \notin \{t-1,t\}$ and $i_t$ belongs to the conditioning set of $v_{a,a+1}$ .

Proof. By Lemma 5.1, for each $1 \le s \le m$ , there exist the k-joining paths $P_k(v_{s,s+1})$ ’s of $v_{s,s+1} $ for $1 \le k \le \operatorname {rk}(v_{s,s+1})$ . In particular, $P_1(v_{s,s+1})$ is the unique path in $F_1$ connecting $i_s$ and $i_{s+1}$ . For each $1 \le s \le m$ , denote by $e^{\prime }_{s,s+1}$ and $e^{\prime \prime }_{s,s+1}$ the edges in $P_1(v_{s,s+1})$ incident on $i_s$ and $i_{s+1}$ , respectively. If there exists $t \in [m]$ such that $e^{\prime \prime }_{t,t+1} \ne e^{\prime }_{t+1,t+2}$ , then by Lemma 5.11, $e^{\prime \prime }_{t,t+1}$ is an edge of a path $P_1(v_{a,a+1})$ of length $\ge 2$ for some $a\in [m]\setminus \{t,t+1\}$ . By Claim 5.3, $e^{\prime \prime }_{t,t+1}$ is a node in $P_2(v_{a,a+1})$ . By Corollary 5.8, $i_{t+1} < e^{\prime \prime }_{t,t+1} \le v_{a,a+1}$ . Hence, $i_{t+1}$ belongs to the conditioning set of $v_{a,a+1}$ .

Now consider the case $e^{\prime \prime }_{s-1,s} = e^{\prime }_{s,s+1}$ for each $1 \le s \le m$ . Denote this common edge by $j_{s}$ . Note that these edges become nodes in $F_2$ . By definition, $P_2(v_{s,s+1})$ is the path in $F_2$ connecting $j_{s}$ and $j_{s+1}$ for each $1 \le s \le m$ . Denote by $f^{\prime }_{s,s+1}$ and $f^{\prime \prime }_{s,s+1}$ the edges in $P_2(v_{s,s+1})$ incident on $j_{s}$ and $j_{s+1}$ , respectively. By a similar argument as in the preceding paragraph with the aid of Lemma 5.11, we only need to consider the case $f^{\prime \prime }_{s-1,s} = f^{\prime }_{s,s+1}$ for all $1 \le s \le m$ .

A repeated application of this argument leads us to the situation that for every k, there exist mutually distinct nodes $h_1,h_2, \ldots , h_m$ in $F_k$ such that the k-joining path $P_k(v_{s,s+1})$ connects $h_{s}$ and $h_{s+1}$ for each $1 \le s \le m$ . Furthermore, if $g^{\prime }_{s,s+1}$ and $g^{\prime \prime }_{s,s+1}$ are the edges in $P_k(v_{s,s+1})$ incident on $h_{s}$ and $h_{s+1}$ , respectively, then $g^{\prime \prime }_{s-1,s} = g^{\prime }_{s,s+1}$ for each $1 \le s \le m$ .

However, let $q \in [m]$ such that $\operatorname {rk}(v_{q,q+1}) =\min \{\operatorname {rk}(v_{s,s+1}) \mid 1 \le s \le m\}$ and let $k=\operatorname {rk}(v_{q,q+1})-1$ . Then the path $P_k(v_{q,q+1})$ has length $1$ (or simply an edge in $F_k$ ). The situation in the paragraph above implies that $P_k(v_{q,q+1})$ is a proper subpath of $P_k(v_{q-1,q})$ . By Lemma 5.10, $i_{q+1} < P_k(v_{q,q+1}) \le v_{q-1,q}$ . Hence, $i_{q+1}$ belongs to the conditioning set of $ v_{q-1,q}$ .

We have a stronger statement when $m=3$ in Lemma 5.13. First we address a remark.

Remark 5.14. Let F be a forest. Let $i_1,i_2$ and $i_3$ be three mutually distinct nodes in F. Suppose there exist the paths $P_{1,2}, P_{2,3}, P_{3,1}$ in F connecting $i_1$ and $i_2$ , $i_2$ and $i_3$ , $i_3$ and $i_1$ , respectively. Let T denote the subgraph (tree) of F induced by the vertices of the paths $P_{1,2}, P_{2,3}, P_{3,1}$ . One may show that there does not exist a path among $P_{{1,2}}$ , $P_{2,3}$ and $P_{3,1}$ being the concatenation of the other two paths if and only if $i_1,i_2, i_3$ are leaves in T. In particular, it is not the case if one of the paths has length $1$ .

Lemma 5.15. Let $\mathcal {P} =(F_1,\ldots ,F_{n})$ be an LR-vine of rank n. Let $a_1,b_1, c_1 \in \min (\mathcal {P}) $ be mutually distinct minimal nodes in $\mathcal {P}$ . Suppose that the joins $u=a_1 \vee b_1$ , $v=b_1 \vee c_1$ and $w=a_1 \vee c_1$ exist in $\mathcal {P}$ . Then there exists a number $k (1\le k\le n$ ) such that one of the three k-joining paths $P_k(u)$ , $P_k(v)$ and $P_k(w)$ is the concatenation of the other two paths.

As a consequence, there exists a node among three nodes $u,v$ and w strictly greater than the other two.

Proof. The proof of the first assertion is similar to the proof of Lemma 5.13 with the use of Remark 5.14 in place of Lemma 5.11. The second assertion follows from the first and Lemma 5.10.

5.2. Construct an MAT-labeled graph from a given LR-vine

Definition 5.16. Let $\mathcal {P}$ be an LR-vine of dimension $\ell $ and rank n. Define an edge-labeled graph $(G,\lambda ) $ from $\mathcal {P}$ as follows:

1. (1) The vertex set $N_{G}$ is given by the set of minimal nodes (i.e., $N_G :=\min (\mathcal {P})=[\ell ]$ ).

2. (2) The edge set $E_{G}$ is given by the conditioned sets of non-minimal nodes; that is,

   $$ \begin{align*} E_G\ &{:=}\ \{C_v \mid v \in \mathcal{P}\setminus \min(\mathcal{P})\} \\ &\kern1pt{=}\ \{ \{i,j\} \subseteq [\ell] \mid i\ne j \text{ and the join} v_{i,j} = i \vee j \text{ exists in } \mathcal{P} \}. \end{align*} $$
   (The second expression of $E_G$ above follows from Lemma 5.1.)

3. (3) The labeling $\lambda : E_G \longrightarrow \mathbb {Z}_{>0}$ is defined by

   $$ \begin{align*}\lambda(i,j):= \operatorname{rk}_{\mathcal{P}}(v_{i,j} )-1.\end{align*} $$

Theorem 5.17. The edge-labeled graph $(G,\lambda )$ from Definition 5.16 is an MAT-labeled graph. In particular, if $\mathcal {P}$ is an R-vine, then $(G,\lambda )$ is an MAT-labeled complete graph.

Proof. The first assertion follows from Lemmas 5.18 and 5.19 below. The second assertion follows from the first and Remark 5.4.

Lemma 5.18. $(G, \lambda )$ satisfies (ML1).

Proof. Suppose to the contrary that there exist $1 \le k \le n-1$ and an m-cycle $C_m$ for $m \ge 3$ with edges $\{i_1,i_2\}$ , $\{i_2,i_3\},\ldots ,\{i_m,i_1\}$ such that $ \lambda (i_1,i_2) < k$ and $ \lambda (i_{s},i_{s+1})=k$ for $2 \le s \le m$ . Here, we take the index modulo m. We choose the smallest such m.

If $m=3$ , then by Lemma 5.15, there exists a node among the nodes $v_{1,2}$ , $v_{2,3}$ , $v_{3,1}$ strictly greater than the other two. This is a contradiction since there are two nodes of the same rank $k+1$ , while the remaining node has rank $< k+1$ . We may assume $m>3$ .

By Lemma 5.13, there exist a minimal node $i_t$ and a join $v_{a,a+1}= i_a \vee i_{a+1}$ in $\mathcal {P}$ for $a,t \in [m]$ such that $t \notin \{a,a+1\}$ and $i_t$ belongs to the conditioning set of $v_{a,a+1}$ . By Corollary 5.9, the joins $v_{a,t}$ , $v_{t,a+1}$ exist in $\mathcal {P}$ , and both are strictly smaller than $v_{a,a+1}$ . Hence, the edges $\{i_a,i_t\}$ , $\{i_t,i_{a+1}\}$ exist in $(G, \lambda )$ and both have labels $<k$ . Therefore, there exists a cycle in $(G, \lambda )$ of length $<m$ with one edge of label $<k$ and the other edges of the same label k. This contradicts the minimality of m. Thus, for every $ k \in \mathbb {Z}_{>0} $ , an edge $e \in \pi _{< k} $ does not form a cycle with edges in $\pi _k$ .

Now suppose $(G, \lambda )$ contains an m-cycle $C_m$ for $m \ge 3$ with all edges of the same label k for some $1 \le k \le n-1$ . By Lemma 5.15, we may assume $m>3$ . By a similar argument as in the preceding paragraph with the aid of Lemma 5.13, the cycle $C_m$ has a chord of label $<k$ . This contradicts the conclusion of the preceding paragraph.

Lemma 5.19. Fix $1 \le k \le n-1$ and let $\{i,j\} \in \pi _k$ . Then the conditioning set of $v_{i,j} =i \vee j$ coincides with the set of conditioning vertices of $\{i,j\}$ . In particular, $(G, \lambda )$ satisfies (ML2).

Proof. Note that $v_{i,j} \in \mathcal {P}_{k+1}$ . By Lemma 4.3(b), the conditioning set $D_{v_{i,j}}$ of $v_{i,j}$ contains $k-1$ elements. We may write it as $D_{v_{i,j}} = \{h_t \mid 1\le t\le k-1\}$ . By Corollary 5.9, the joins $v_{i,h_t}$ , $v_{h_t,j}$ for $1\le h_t\le k-1 $ exist in $\mathcal {P}$ , and all are strictly smaller than $v_{i,j}$ . In particular, the edges $\{i,h_t\}$ , $\{h_t,j\}$ have label $< k$ for all $h_t $ . This implies that $D_{v_{i,j}}$ is contained in the set of conditioning vertices of $\{i,j\}$ .

Now let $h \in [\ell ] \setminus U_{v_{i,j}}$ (i.e., $h \not \le v_{i,j} $ ). We show that if the joins $v_{i,h}$ and $v_{h,j}$ both exist, then either $\{i, h\}$ or $\{h, j\}$ has label $> k$ . It cannot happen that $v_{i,h} < v_{i,j}$ or $v_{h,j} < v_{i,j}$ ; otherwise, $h \le v_{i,j}$ . Hence, by Lemma 5.15, either $v_{i,h}> v_{i,j}$ or $v_{h,j}> v_{i,j}$ .

Thus, the conditioning set of $v_{i,j}$ coincides with the set of conditioning vertices of $\{i,j\} \in \pi _k$ . Since this is true for every k, we conclude that $(G, \lambda )$ satisfies (ML2).

6. Equivalences of categories

For basic definitions and notations of category theory, we refer the reader to [18, Chapter 1]. In this section, we will define the categories of MAT-labeled graphs and LR-vines, and construct an explicit equivalence between these two categories.

6.1. Equivalence of MAT-labeled graphs and LR-vines

Definition 6.1 (Label-preserving homomorphism).

Let $(G,\lambda )$ and $(G', \lambda ')$ be edge-labeled graphs. A label-preserving homomorphism from $(G,\lambda )$ to $(G', \lambda ')$ , written $\sigma \colon (G,\lambda ) \longrightarrow (G', \lambda ')$ , is a map $\sigma \colon N_{G} \longrightarrow N_{G'}$ such that for all $u, v \in N_G$ , $\{u, v\} \in E_{G}$ implies $\{\sigma (u), \sigma (v)\} \in E_{G'}$ and $ \lambda (u, v) = \lambda '(\sigma (u), \sigma (v))$ .

We call $\sigma $ an isomorphism if $\sigma $ is bijective and its inverse is a label-preserving homomorphism. The edge-labeled graphs $(G,\lambda )$ and $(G', \lambda ')$ are said to be isomorphic, written $(G,\lambda ) \simeq (G', \lambda ')$ if there exists an isomorphism $\sigma \colon (G,\lambda ) \longrightarrow (G', \lambda ')$ . If $(G,\lambda ) \simeq (G, \lambda ')$ , we say that two labelings $\lambda $ and $ \lambda '$ are the same (or isomorphic).

If $(G,\lambda ) \simeq (G', \lambda ')$ and $(G,\lambda )$ is an MAT-labeled graph, then $(G', \lambda ')$ is also an MAT-labeled graph.

Definition 6.2 (Category of MAT-labeled (complete) graphs).

The category $\mathbf {\mathsf {MG}}$ of MAT-labeled graphs is the category whose objects are the MAT-labeled graphs and whose morphisms are the label-preserving homomorphisms. The category $\mathbf {\mathsf {MCG}}$ of MAT-labeled complete graphs is a full subcategory of $\mathsf {MG}$ whose objects are the MAT-labeled complete graphs.

Recall the definition of rank and join-preserving homomorphisms of graded posets from Definition 3.5.

Definition 6.3 (Category of (L)R-vines).

The category $\mathbf {\mathsf {LRV}}$ of LR-vines is the category whose objects are the LR-vines and whose morphisms are the homomorphisms preserving rank and join. The category $\mathbf {\mathsf {RV}}$ of R-vines is a full subcategory of $\mathsf {LRV}$ whose objects are the R-vines.

First we need some lemmas.

Lemma 6.4. Let $\varphi \colon \mathcal {P} \longrightarrow \mathcal {P}'$ be a rank-preserving homomorphism of LR-vines. Suppose $\varphi $ preserves join of minimal pairs (i.e., if $x, y \in \min (\mathcal {P})$ such that the join $ x\vee y$ exists, then $ \varphi (x) \vee \varphi (y)$ exists and $\varphi ( x\vee y) = \varphi (x) \vee \varphi (y)$ ). Then $\varphi $ induces an isomorphism $ \mathcal {P}_{\le v} \simeq \mathcal {P}^{\prime }_{\le \varphi (v)}$ for every $v\in \mathcal {P}$ .

Proof. Clearly, $\varphi $ induces a homomorphism $\varphi |_{ \mathcal {P}_{\le v}} \colon \mathcal {P}_{\le v} \longrightarrow \mathcal {P}^{\prime }_{\le \varphi (v)}$ . In particular, $\varphi (U_v) \subseteq U_{\varphi (v)}$ . Note that for any distinct $i, j \in U_v$ , the join $i \vee j$ exists in the R-vine $\mathcal {P}_{\le v}$ by Remark 5.4. Since $\varphi $ preserves rank, $\operatorname {rk}'( \varphi ( i \vee j) ) = \operatorname {rk}( i \vee j)> \operatorname {rk}( i)$ . Since $\varphi $ preserves join of minimal pairs, $ \varphi (i) \vee \varphi (j)$ exists and $\varphi ( i\vee j) = \varphi (i) \vee \varphi (j)$ . In particular, $\varphi (i) \ne \varphi (j)$ . Hence, the elements in $\varphi (U_v)$ are pairwise distinct. Thus, $\varphi (U_v) = U_{\varphi (v)}$ since $|\varphi (U_v) | = |U_v |= \operatorname {rk}(v) = \operatorname {rk}'(\varphi (v)) = |U_{\varphi (v)} |$ .

Let $a,a' \in \mathcal {P}_{\le v} $ be such that $\varphi (a) = \varphi (a')$ . We may write $a = i \vee j$ and $a' = i' \vee j'$ for minimal nodes $i \ne j$ , $i' \ne j'$ . Thus, $ \varphi (i) \vee \varphi (j) = \varphi (i') \vee \varphi (j')$ . Again by Remark 5.4, $\{\varphi (i), \varphi (j)\} =\{\varphi (i'), \varphi (j')\} $ . Since the elements in $\varphi (U_v)$ are pairwise distinct, $\{ i , j\} = \{i' ,j'\}$ . Hence, $a=a'$ . This implies that $\varphi |_{ \mathcal {P}_{\le v}}$ is injective. Moreover, $\left | \mathcal {P}_{\le v} \right |=\left | \mathcal {P}^{\prime }_{\le \varphi (v)} \right |$ by Remark 3.15. Hence, $\varphi |_{ \mathcal {P}_{\le v}}$ is bijective.

Now let $a,b \in \mathcal {P}_{\le v} $ be such that $\varphi (a) \le \varphi (b)$ . Therefore, $U_{\varphi (a)} \subseteq U_{\varphi (b)}$ ; hence, $\varphi (U_a) \subseteq \varphi (U_b)$ . It follows that $U_a \subseteq U_b$ . By Corollary 4.6, $a \le b$ . Thus, the inverse of $\varphi |_{ \mathcal {P}_{\le v}}$ is a homomorphism. We conclude that $\varphi |_{ \mathcal {P}_{\le v}}$ is an isomorphism.

Lemma 6.5. Let $\varphi \colon \mathcal {P} \longrightarrow \mathcal {P}'$ be a rank-preserving homomorphism of LR-vines such that $\varphi $ preserves join of minimal pairs (Lemma 6.4). Then $\varphi $ is join-preserving.

Proof. Let $a,b \in \mathcal {P} $ and suppose the join $a \vee b$ exists in $\mathcal {P}$ . Write $v =a \vee b$ . Note that $\varphi (a), \varphi (b) \in \mathcal {P}^{\prime }_{\le \varphi (v)}$ . By Lemma 6.4, $\varphi |_{ \mathcal {P}_{\le v}} \colon \mathcal {P}_{\le v} \longrightarrow \mathcal {P}^{\prime }_{\le \varphi (v)}$ is a poset isomorphism. It follows that $\varphi |_{ \mathcal {P}_{\le v}}$ is join-preserving. Therefore, $\varphi (a) \vee \varphi (b)$ exists in $ \mathcal {P}^{\prime }_{\le \varphi (v)}$ and hence in $\mathcal {P}'$ and $\varphi ( a\vee b) = \varphi (a) \vee \varphi (b)$ . Thus, $\varphi $ is join-preserving.

Lemma 6.6. Let $\mathcal {P} $ and $ \mathcal {P}'$ be LR-vines and suppose there is a homomorphism $\varphi \colon \mathcal {P} \longrightarrow \mathcal {P}'$ preserving rank and join. Let $(G,\lambda )$ (resp. $(G', \lambda ')$ ) denote the MAT-labeled graph defined by $\mathcal {P} $ (resp. $\mathcal {P}'$ ) from Definition 5.16 and Theorem 5.17. Define a map $\Omega (\varphi )\colon N_{G} \longrightarrow N_{G'}$ by sending each node i in G to a node $\varphi (i)$ in $G'$ for $1 \le i \le \ell $ . Then $\Omega (\varphi )$ is a label-preserving homomorphism from $(G,\lambda )$ to $(G', \lambda ')$ .

Proof. Let $\{i, j\} \in E_{G}$ be an edge in G for distinct nodes $i, j \in N_G$ . Then the join $ i \vee j$ exists in $\mathcal {P}$ . Since $\varphi $ is join-preserving, $ \varphi (i) \vee \varphi (j)$ exists and $\varphi ( i\vee j) = \varphi (i) \vee \varphi (j)$ . Since $\varphi $ is rank-preserving, $\varphi (i) \ne \varphi (j)$ . Therefore, $\{\varphi (i), \varphi (j)\} \in E_{G'}$ . Furthermore,

$$ \begin{align*}\lambda'(\varphi (i), \varphi(j)) =\operatorname{rk}'( \varphi (i) \vee \varphi(j) ) -1= \operatorname{rk}( i \vee j)-1= \lambda(i,j).\end{align*} $$

Thus, $\Omega (\varphi )$ is a label-preserving homomorphism.

Lemma 6.7. Let $(G,\lambda )$ and $(G', \lambda ')$ be MAT-labeled graphs and suppose there is a label-preserving homomorphism $\sigma \colon (G,\lambda ) \longrightarrow (G', \lambda ')$ . Let $\mathcal {P} $ (resp. $\mathcal {P}'$ ) denote the LR-vine defined by $(G,\lambda ) $ (resp. $(G', \lambda ')$ ) from Definition 4.11 and Theorem 4.13. Define a map $\Psi (\sigma ) \colon \mathcal {P} \longrightarrow \mathcal {P}'$ by sending $v\in \mathcal {P}$ to $\{\sigma (a) \mid a \in v\}\in \mathcal {P}'$ . Then $\Psi (\sigma ) $ is a homomorphism preserving rank and join between $\mathcal {P} $ and $ \mathcal {P}'$ .

Proof. Let $\varphi =\Psi (\sigma ) $ . Note that since $\varphi $ is label-preserving, if $v = K_e \in \mathcal {P}$ where $K_e$ is a principal clique in $(G,\lambda )$ for $e \in E_G$ , then $\sigma (K_e)=\{\sigma (a) \mid a \in K_e\}=K_{\sigma (e) }$ is a principal clique in $(G', \lambda ')$ . Hence, $\varphi $ is indeed well-defined. It is also easily seen that $\varphi $ is a rank-preserving homomorphism since $\lambda (e) = |K_e|-1$ . Let $\{i\} , \{j\} \in \mathcal {P}$ be distinct minimal nodes in $\mathcal {P}$ such that $\{i\} \vee \{j\}$ exists in $\mathcal {P}$ . By Corollary 4.14, we may write $K_e=\{i\} \vee \{j\} \in \mathcal {P}$ for $e=\{i,j\} \in E_G$ . Also by Corollary 4.14, $ K_{\sigma (e) }=\{ \sigma (i) \} \vee \{ \sigma (j) \} \in \mathcal {P}'$ since $\sigma (e)=\{\sigma (i) ,\sigma (j)\} \in E_{G'}$ . Therefore, $\varphi ( \{i\} \vee \{j\} ) = \varphi (\{i\}) \vee \varphi (\{j\})$ . Thus, $\varphi $ preserves join of minimal pairs. By Lemma 6.5, $\varphi $ is join-preserving.

The following two lemmas give a construction of functors between $\mathsf {MG}$ and $\mathsf {LRV} $ . The proofs are routine.

Lemma 6.8. Define a mapping $\Omega \colon \mathsf {LRV}\longrightarrow \mathsf {MG}$ by associating

1. (1) each object $ \mathcal {P}$ in $\mathsf {LRV} $ to an object $\Omega (\mathcal {P}) = (G,\lambda )$ in $\mathsf {MG}$ , where $ (G,\lambda )$ is the MAT-labeled graph defined by $\mathcal {P} $ from Definition 5.16 and Theorem 5.17, and

2. (2) each morphism $\varphi \colon \mathcal {P} \longrightarrow \mathcal {P}'$ in $\mathsf {LRV} $ to a morphism $\Omega (\varphi ) \colon \Omega (\mathcal {P}) \longrightarrow \Omega (\mathcal {P}')$ in $\mathsf {MG}$ , where $\Omega (\varphi ) $ is the label-preserving homomorphism from Lemma 6.6.

Then $\Omega $ is a functor from $\mathsf {LRV} $ to $\mathsf {MG}$ .

Lemma 6.9. Define a mapping $\Psi \colon \mathsf {MG} \longrightarrow \mathsf {LRV} $ by associating

1. (1) each object $(G,\lambda )$ in $\mathsf {MG}$ to an object $\Psi (G,\lambda ) =\mathcal {P}$ in $\mathsf {LRV} $ , where $\mathcal {P} $ is the LR-vine defined by $(G,\lambda ) $ from Definition 4.11 and Theorem 4.13, and

2. (2) each morphism $\sigma \colon (G,\lambda ) \longrightarrow (G', \lambda ')$ in $\mathsf {MG}$ to a morphism $\Psi (\sigma ) \colon \Psi (G,\lambda ) \longrightarrow \Psi (G', \lambda ')$ in $\mathsf {LRV} $ , where $\Psi (\sigma ) $ is the homomorphism preserving rank and join from Lemma 6.7.

Then $\Psi $ is a functor from $\mathsf {MG}$ to $\mathsf {LRV} $ .

We are now ready to prove the main result of the paper.

Theorem 6.10. The composite functor $\Psi \Omega $ (resp. $\Omega \Psi $ ) is naturally isomorphic to the identity functor $ 1_{\mathsf {LRV}}$ (resp. $1_{\mathsf {MG}}$ ). As a result, the categories $\mathsf {MG}$ and $\mathsf {LRV} $ are equivalent.

Proof. For every LR-vine $\mathcal {P}$ , recall from Proposition 4.8 the LR-vine $ \widehat {\mathcal {P}} =\{ U_v \mid v \in \mathcal {P}\}$ and the isomorphism

$$ \begin{align*}\eta_{\mathcal{P}}\colon \mathcal{P} \longrightarrow \widehat{\mathcal{P}} \quad \text{via} \quad v \mapsto U_v.\end{align*} $$

By Lemmas 6.9 and 6.8, the functor $\Psi \Omega \colon \mathsf {LRV}\longrightarrow \mathsf {LRV}$ assigns

1. (1) each LR-vine $ \mathcal {P}$ to the LR-vine $\Psi \Omega (\mathcal {P})= \widehat {\mathcal {P}}$ , and

2. (2) each morphism $\varphi \colon \mathcal {P} \longrightarrow \mathcal {P}'$ in $\mathsf {LRV} $ to a morphism $\Psi \Omega (\varphi ) \colon \widehat {\mathcal {P}} \longrightarrow \widehat {\mathcal {P}'}$ in $\mathsf {LRV} $ defined by sending $U_v \in \widehat {\mathcal {P}}$ to $\varphi (U_v)=U_{\varphi (v)} \in \widehat {\mathcal {P}'}$ for every $ v\in \mathcal {P}$ . The equality holds by Lemma 6.4.

Similarly, the functor $ \Omega \Psi \colon \mathsf {MG}\longrightarrow \mathsf {MG}$ assigns

1. (1’) each MAT-labeled graph $ (G,\lambda )$ to an MAT-labeled graph $ \Omega \Psi (G,\lambda )= (\widehat G,\widehat \lambda )$ , where $N_{\widehat G} = \{ \{i\} \mid i \in N_G\}$ , $E_{\widehat G} = \{ \{\{i\},\{j\}\} \mid \{i,j\} \in E_G\}$ and $\widehat \lambda (\{i\},\{j\}) =\lambda (i,j)$ (see also Corollary 4.14), and

2. (2’) each morphism $\sigma \colon (G,\lambda ) \longrightarrow (G', \lambda ')$ in $\mathsf {MG}$ to a morphism $ \Omega \Psi (\sigma ) \colon (\widehat G,\widehat \lambda ) \longrightarrow (\widehat {G'},\widehat {\lambda '})$ in $\mathsf {MG}$ defined by sending $\{i\} \in N_{\widehat G}$ to $\{ \sigma (i) \} \in N_{\widehat {G'}}$ for every $ i \in N_G$ .

First we prove $\Psi \Omega \simeq 1_{\mathsf {LRV}}$ (i.e., $\Psi \Omega $ is naturally isomorphic to $ 1_{\mathsf {LRV}}$ ). For every morphism $\varphi \colon \mathcal {P} \longrightarrow \mathcal {P}'$ in $\mathsf {LRV} $ , we have a commutative diagram in Figure 3.

Figure 3 Commutative diagram that shows $\Psi \Omega \simeq 1_{\mathsf {LRV}}$ .

This follows that $\eta \colon 1_{\mathsf {LRV}} \longrightarrow \Psi \Omega $ with component $\eta _{\mathcal {P}}$ at $\mathcal {P}$ is a natural isomorphism. Thus, $\Psi \Omega $ is naturally isomorphic to $ 1_{\mathsf {LRV}}$ .

The proof for $\Omega \Psi \simeq 1_{\mathsf {MG}}$ is similar and easier. For every object $(G,\lambda ) $ in $\mathsf {MG}$ , the following map is an isomorphism:

$$ \begin{align*}\epsilon_{(G,\lambda) }\colon (G,\lambda) \longrightarrow (\widehat G,\widehat\lambda) \quad \text{via} \quad N_G \ni i \mapsto \{i\} \in N_{\widehat G}.\end{align*} $$

Furthermore, $ \Omega \Psi $ is naturally isomorphic to $ 1_{\mathsf {MG}}$ via the natural isomorphism $\epsilon \colon 1_{\mathsf {MG}} \longrightarrow \Omega \Psi $ with component $\epsilon _{(G,\lambda ) }$ at $(G,\lambda )$ .

The following corollary is straightforward from Theorem 6.10.

Corollary 6.11. The restriction $\Psi |_{\mathsf {MCG}}$ (resp. $\Omega |_{\mathsf {RV}}$ ) is a functor from $\mathsf {MCG}$ (resp. $ \mathsf {RV}$ ) to $ \mathsf {RV}$ (resp. $\mathsf {MCG}$ ). Furthermore, the composite functor $\Psi |_{\mathsf {MCG}} \Omega |_{\mathsf {RV}}$ (resp. $\Omega |_{\mathsf {RV}} \Psi |_{\mathsf {MCG}}$ ) is naturally isomorphic to the identity functor $ 1_{\mathsf {RV}}$ (resp. $1_{\mathsf {MCG}}$ ). As a result, the categories $\mathsf {MCG}$ and $\mathsf {RV} $ are equivalent.

Another interesting consequence is the existence of a pushout in the category $\mathsf {MG}$ and hence in $\mathsf {LRV}$ owing to the gluing method in Lemma 2.21.

Corollary 6.12. Suppose we are in the situation of Lemma 2.21. Denote by , , and the embeddings (in particular, morphisms in $\mathsf {MG}$ ) of the MAT-labeled graphs. Then $((G, \lambda ), \sigma _1, \sigma _2)$ is a pushout of $ \mu _1$ and $\mu _2$ .

Proof. It is easily seen that $\sigma _1 \mu _1 = \sigma _2 \mu _2$ . Hence, the square diagram commutes. Now given another triple $((G_3, \lambda _3), \sigma ^{\prime }_1, \sigma ^{\prime }_2)$ with $\sigma ^{\prime }_1 \mu _1 = \sigma ^{\prime }_2 \mu _2$ , let $\theta \colon (G, \lambda )\longrightarrow (G_3, \lambda _3)$ be defined by $\theta (u) = \sigma ^{\prime }_1(u)$ for $u \in N_{G_1}$ , $\theta (u) = \sigma ^{\prime }_2(u)$ for $u \in N_{G_2}$ . Thus, $\theta \sigma _1 = \sigma ^{\prime }_1$ and $\theta \sigma _2 = \sigma ^{\prime }_2$ . It is also easy to see that $\theta $ is the unique morphism making the diagram commute.

We conclude that $((G, \lambda ), \sigma _1, \sigma _2)$ is a pushout of $ \mu _1$ and $\mu _2$ .

6.2. Equivalence of LR-vines and m-vines

Theorem 6.13. Let $\mathcal {P}$ be a vine. The following are equivalent:

1. (1) $\mathcal {P}$ is an m-vine (i.e., by definition, $\mathcal {P}$ is an ideal of an R-vine).

2. (2) $\mathcal {P}$ satisfies the proximity condition.

3. (3) $\mathcal {P}$ is an LR-vine.

Proof. $(2) \Leftrightarrow (3)$ is shown in Proposition 4.4. $(1) \Rightarrow (3)$ is straightforward from Remark 3.17. It remains to show $(1) \Leftarrow (3)$ . The proof is based on the following diagram:

First, given an LR-vine $\mathcal {P}$ , let $ (G,\lambda ) = \Omega (\mathcal {P}) $ be the MAT-labeled graph associated to $\mathcal {P}$ via the functor $\Omega $ from Lemma 6.8. Next, let $(K,\widetilde \lambda )$ be the MAT-labeled complete graph such that $ (G, \lambda ) \le (K,\widetilde \lambda ) $ from Proposition 2.20. In particular, there exists an embedding . Then let $\Psi |_{\mathsf {MCG}} (K,\widetilde \lambda )= \widehat {\mathcal {R}} $ be the R-vine associated to $(K,\widetilde \lambda )$ via the functor $\Psi |_{\mathsf {MCG}}$ from Corollary 6.11. Finally, let $\mathcal {R}$ be the R-vine isomorphic to $ \widehat {\mathcal {R}} $ via the (inverse of) poset isomorphism $\eta _{\mathcal {R}}$ from Proposition 4.8.

$(1) \Leftarrow (3)$ is proved once we show that $\mathcal {P}$ is an ideal of $\mathcal {R}$ . Indeed, by the construction, $\mathcal {P}$ is an induced subposet of $\mathcal {R}$ . (One may see this via the sequence in the diagram.) Note also that $\mathcal {P}$ and $\mathcal {R}$ have the same set of minimal nodes. Let $w,v \in \mathcal {R}$ be two nodes with $w \le v$ and $v \in \mathcal {P}$ . We need to show that $w \in \mathcal {P}$ . The assertion follows easily if either w or v is a minimal node. We may assume that both w and v are not minimal. Since $C_w \subseteq U_w \subseteq U_v$ in $\mathcal {R}$ , and $U_v$ in $\mathcal {P}$ is the same as $U_v$ in $\mathcal {R}$ , we have $C_w \subseteq U_v$ in $\mathcal {P}$ . By Remark 5.4, there exists a non-minimal node $w'$ in the R-vine $\mathcal {P}_{\le v}$ such that $C_w =C_{w'}$ . Applying Corollary 5.7 for two non-minimal nodes $w,w'$ in $\mathcal {R}$ , we have $w=w'$ . Hence, $w $ is an element in $\mathcal {P}$ , as desired.

7. Applications

7.1. From LR-vines to MAT-labeled graphs

7.1.1. A poset characterization of MAT-free graphic arrangements

The most important application of our main result (Theorem 6.10) is an affirmative answer for the question of Cuntz-Mücksch (Question 1.3) in the case of graphic arrangements: MAT-free graphic arrangements have a poset characterization by LR-vines. (Note that LR-vines generalize the root poset of type A by Remark 4.16.) We give below two examples to illustrate the correspondence. First we need a definition.

Definition 7.1 (C-vine).

An R-vine is called a C-Vine if each associated tree has the largest possible number of vertices of degree $1$ . Equivalently, each associated tree is a star graph.

D-vine and C-vine can be regarded as the ‘extreme’ cases of R-vines.

Example 7.2. In dimension $4$ , there are exactly two non-isomorphic R-vine structures: D-vine and C-vine. Likewise, there are exactly two non-isomorphic MAT-labeled complete graphs on $4$ vertices. Figure 4 depicts a C-vine on $[4]$ (top right), the associated forests (top left), the associated MAT-labeled complete graph (bottom right) via the functor $\Omega $ from Lemma 6.8, and the MAT-partition (bottom left) of the corresponding graphic arrangement. The C-vine in dimension $\ge 4$ is not an ideal of any D-vine; hence, the corresponding MAT-partition is not obtained from an ideal of the type A root poset.

Figure 4 C-vine on $4$ elements, the associated trees, MAT-labeled complete graph and MAT-partition from Example 7.2.

Figure 5 An MAT-labeled graph on $5$ vertices (right) and the LR-vine $\mathcal {P} = \Psi (G,\lambda ) $ (left) from Lemma 6.9 with its $3$ -lower truncation.

Example 7.3. Figure 5 depicts on the right an MAT-labeled graph $ (G, \lambda )$ on $5$ vertices and an MAT-labeled complete graph $(K,\widetilde \lambda )$ such that $ (G, \lambda ) \le (K,\widetilde \lambda ) $ . The complementary edges are shown in dashed lines. The graphs $ (G, \lambda )$ and $(K,\widetilde \lambda )$ correspond (via the functor $ \Psi $ from Lemma 6.9) to the LR-vine $\mathcal {P}$ and R-vine $\mathcal {R}$ on the left, respectively. In this case, $\mathcal {P}$ is the $3$ -lower truncation of $\mathcal {R}$ .

7.1.2. Number of non-isomorphic MAT-labelings of complete graphs

The number of equivalence classes of R-vines in dimension $\ell $ is given in [17, §10.3]. By our Corollary 6.11, this number is equal to the number of non-isomorphic MAT-labelings of the complete graph on $\ell $ vertices. We immediately have the following:

Theorem 7.4. The number $E_\ell $ of non-isomorphic MAT-labelings of the complete graph $K_\ell $ for $\ell \ge 1$ is given by $E_1 =E_2=E_3=1$ and $E_\ell =(A_\ell +B_\ell )/2$ for $\ell \ge 4$ , where

$$ \begin{align*}A_\ell = 2^{(\ell-2)(\ell-3)/2}, \quad B_\ell = \sum_{k=1}^{\lfloor \ell/2 \rfloor -1}A_\ell c_k 2^{-k + \sum_{i=0}^{k-1} (\ell-4-2i)},\end{align*} $$

and $c_k = 1$ for all k except $c_{\lfloor \ell /2 \rfloor -1} = 2$ .

The first $8$ elements of the sequence $(E_\ell )$ are $1,1 ,1 , 2 , 6,40, 560, 17024$ mentioned in §1.1. In particular, $E_4=2$ and these MAT-labelings are given in Figures 2 and 4.

7.1.3. Upper truncation of MAT-labeled graphs

In Remark 4.10, we discussed two ways to obtain a new LR-vine from a given LR-vine by upper or lower truncation. From our main result 6.10, the lower truncation simply corresponds to deleting the edges of high label in the associated MAT-labeled graph (Figure 5). The upper truncation, however, gives rise to a nontrivial graph operation which produces an MAT-labeled graph from a given one. We shall not give an explicit formulation of the operation but instead illustrate it by an example in Figure 6.

Figure 6 An MAT-labeled graph on $7$ vertices (top left), the LR-vine $\mathcal {P} = \Psi (G,\lambda ) $ (bottom) from Lemma 6.9 and its $1$ -upper truncation $\overline {\mathcal {P}}_{\ge 1}$ , and the MAT-labeled graph $\Omega (\overline {\mathcal {P}}_{\ge 1})$ (top right) from Lemma 6.8.

In terms of hyperplane arrangement, any upper truncation of an MAT-free graphic arrangement is again MAT-free. This fact is not true in general. For example, the Weyl subarrangement defined by the $1$ -upper truncation of the root poset of type $B_3$ is not free, and hence not MAT-free. It would be interesting to find for which MAT-free arrangement or for which upper truncation of a given MAT-free arrangement this property holds true.

7.2. From MAT-labeled graphs to LR-vines

7.2.1. Strongly chordal graphs and m-vines

Given a strongly chordal graph G, Zhu-Kurowicka [28, §3.4] showed that there exists an m-vine, equivalently, an LR-vine $\mathcal {P}$ (by our Theorem 6.13) such that the principal ideals generated by the maximal elements of $\mathcal {P}$ are in one-to-one correspondence with the maximal cliques of G. Their method is based on the existence of a strong clique tree of G. We give below a different construction of such an LR-vine thanks to the equivalence between the LR-vines and MAT-labeled graphs from Theorem 6.10.

Theorem 7.5. Given a strongly chordal graph G, there exists an LR-vine $\mathcal {P}$ such that the principal ideals generated by the maximal elements of $\mathcal {P}$ are in one-to-one correspondence with the maximal cliques of G.

Proof. By Theorem 1.5, there exists an MAT-labeling $\lambda $ of G. The construction of such a $\lambda $ can be found in [27, Theorem 5.12] based on the notion of clique intersection poset of G first appeared in [21]. Let $ \mathcal {P}= \Psi (G,\lambda ) $ (Lemma 6.9). Theorem 7.5 is proved once we prove that the set $\max ( \mathcal {P})$ of maximal elements of $ \mathcal {P}$ coincides with the set $\mathcal {K}(G)$ of maximal cliques of G. By Lemma 2.23, any maximal clique C in G is principal and hence an element in $ \mathcal {P}$ . Moreover, $C \in \max ( \mathcal {P})$ . Otherwise, there exists a clique $C' \in \max ( \mathcal {P})$ such that $C \subsetneq C'$ . This contradicts the maximality of C. Hence, $\mathcal {K}(G) \subseteq \max ( \mathcal {P})$ . The reserve inclusion is proved similarly. Thus, $ \mathcal {P}$ is a desired LR-vine.

7.2.2. Marginalization and sampling order

The notions of marginalization and sampling order of an R-vine were introduced in [8, §3]. We give below an extension of these notions to an LR-vine.

Definition 7.6 (Marginalization).

Let $(\mathcal {P}, \le _{\mathcal {P}}, \operatorname {rk}_{\mathcal {P}})$ be an LR-vine. Let $v \in \min (\mathcal {P})$ be a minimal node. The marginalization $(\mathcal {P}\,\|\, v, \le _{\mathcal {P}})$ of $\mathcal {P}$ by v is the induced subposet of $\mathcal {P}$ obtained by removing v and the nodes whose conditioned sets contain v; that is,

$$ \begin{align*}\mathcal{P}\,\|\, v:=\mathcal{P} \setminus (\{v\} \cup \{ x \in \mathcal{P} \mid v \in C_x\} ).\end{align*} $$

Let $\mathcal {Q}$ be an induced subposet of a finite graded poset $(\mathcal {P}, \le _{\mathcal {P}}, \operatorname {rk}_{\mathcal {P}})$ . We say that $\mathcal {Q}$ is graded by $\mathbf {\operatorname {rk}_{\mathcal {P}}}$ if the restriction $\operatorname {rk}_{\mathcal {P}} |_{\mathcal {Q}}$ is a rank function on $\mathcal {Q}$ . The marginalization of $\mathcal {P}$ by a minimal node is not necessarily graded by $\operatorname {rk}_{\mathcal {P}}$ in general. For example, let $\mathcal {P}$ be the C-vine with $4$ minimal elements $1,2,3,4$ in Figure 4. Then the marginalization $\mathcal {Q}:=(\mathcal {P}\,\|\, 2, \le _{\mathcal {P}})$ is not graded by $\operatorname {rk}_{\mathcal {P}}$ since while cannot have rank $4$ in $\mathcal {Q}$ .

Definition 7.7 (Sampling order).

Let $(\mathcal {P}, \le _{\mathcal {P}}, \operatorname {rk}_{\mathcal {P}})$ be an $\ell $ -dimensional (L)R-vine. An ordering $ (v_{1}, \dots , v_{\ell }) $ of minimal nodes in $ \mathcal {P}$ is a sampling order if the marginalization $\mathcal {P}_{i}:= \mathcal {P}_{i+1} \,\|\, v_{i+1}$ is an (L)R-vine graded by $\operatorname {rk}_{\mathcal {P}}$ for each $1 \le i \le \ell -1$ . Here, we let $\mathcal {P}=\mathcal {P}_\ell $ .

It is shown in [8, Theorem 5.1] that an R-vine always has a sampling order. Now we generalize this result to LR-vines.

Theorem 7.8. Let $\mathcal {P} $ be an $\ell $ -dimensional LR-vine and let $ (G,\lambda ) = \Omega ( \mathcal {P})$ (Lemma 6.8). If $ (v_{1}, \dots , v_{\ell }) $ is an MAT-PEO of $ (G,\lambda )$ , then $ (v_{1}, \dots , v_{\ell }) $ is a sampling order of $\mathcal {P}$ . As a consequence, an LR-vine always has a sampling order.

Proof. The assertion is clearly true when $\ell \le 1$ . Suppose that $\ell \ge 2$ and $ (v_{1}, \dots , v_{\ell }) $ is an MAT-PEO of $ (G,\lambda )$ . By Lemma 2.15, $ (G_{i}, \lambda _{i}) $ is an MAT-labeled graph for each $1\le i \le \ell -1 $ , where $ G_{i}:= G[\{v_{1}, \dots , v_{i}\}] $ and $ \lambda _{i}:= \lambda |_{E_{G_{i}}} $ . By the proof of Theorem 4.13, for each i, $\Psi (G_{i}, \lambda _{i})=\widehat {\mathcal {P}_{i}}$ where $ \mathcal {P}_{i}:= \mathcal {P}_{i+1} \,\|\, v_{i+1}$ , $\mathcal {P}_\ell := \mathcal {P}$ and $ \widehat {\mathcal {P}_{i}}$ is the poset isomorphic to $ \mathcal {P}_{i}$ in Proposition 4.8. Thus, each $ {\mathcal {P}_{i}}$ is an LR-vine graded by $\operatorname {rk}_{\mathcal {P}}$ . Hence, $ (v_{1}, \dots , v_{\ell }) $ is a sampling order of $\mathcal {P}$ . The consequence is straightforward since an MAT-labeled graph always has an MAT-PEO by Theorem 2.17.

Remark 7.9. The converse of the main assertion in Theorem 7.8 is not true in general. Namely, a sampling order of $\mathcal {P}$ is not necessarily an MAT-PEO of $ (G,\lambda )$ . The reason is that even if the marginalization $\mathcal {P}\,\|\, v$ for some $v \in \min (\mathcal {P})$ is an LR-vine graded by $\operatorname {rk}_{\mathcal {P}}$ , the node v is not necessarily an MAT-simplicial vertex of $ (G,\lambda )$ . For example, let $\mathcal {P}$ be the $3$ -lower truncation of the C-vine in Figure 4 (see Figure 7 below). The associated MAT-labeled graph $ (G,\lambda )= \Omega ( \mathcal {P})$ is the graph obtained from the corresponding complete graph by removing the edge $\{v_3,v_4\}$ . Then the marginalization $\mathcal {P}\,\|\, 2 $ is an LR-vine graded by $\operatorname {rk}_{\mathcal {P}}$ , but the vertex $v_2$ is not (MAT-)simplicial in $ (G,\lambda )$ since its neighborhood does not form a clique. In addition, $(1,3,4,2)$ is a sampling order of $\mathcal {P}$ , but $(v_1,v_3,v_4,v_2)$ is not an MAT-PEO of $ (G,\lambda )$ .

Figure 7 An LR-vine and its marginalization by the node $4$ visualized as ‘vertical’ truncation.

While upper or lower truncation of an LR-vine can be visualized as ‘horizontal’ truncation, the marginalization is a ‘vertical’ truncation. Figure 7 depicts the $3$ -lower truncation $\mathcal {P}$ of the C-vine in Figure 4 and the marginalization $\mathcal {P}\,\|\, 4 $ . We may continue marginalizing and get the sampling order $(1,2,3,4)$ of $\mathcal {P}$ . Furthermore, $(v_1,v_2,v_3,v_4)$ is an MAT-PEO of the MAT-labeled graph associated to $\mathcal {P}$ .

7.2.3. Number of ideals in a D-vine

Given an arbitrary R-vine $\mathcal {P}$ , a natural question is to find the number of m-vines or ideals in $\mathcal {P}$ . This question is still open in general (see, for example, [28, §4]). However, in the particular case of D-vine, we have an explicit answer owing to a classical result of Shi [23] in the theory of hyperplane arrangements.

Theorem 7.10. The number of ideals in the D-vine of dimension $\ell -1$ for $\ell \ge 2$ is given by the $\ell $ -th Catalan number $\mathrm {Cat}_\ell ={\frac {1}{\ell +1}}{2\ell \choose \ell } =\prod \limits _{k=2}^{\ell }{\frac {\ell +k}{k}}.$

Proof. By Remark 4.16, the D-vine of dimension $\ell -1$ is isomorphic to the type $A_{\ell -1}$ root poset. It is known that the number of ideals in the latter is given by the $\ell $ -th Catalan number $\mathrm {Cat}_\ell $ [23, Theorem 1.4].

Some calculation on C-vines suggests us the following conjecture:

Conjecture 7.11. The number of ideals in the C-vine of rank $\ell \ge 1$ is given by $\sum _{i=0}^{\ell +1}( (i+1)^{\ell +1-i} - i^{\ell +1-i})$ , the $(\ell +1)$ -th number in the OEIS sequence A047970.