Tree-Vertex Graph: New Hierarcal Graph Class

1. Preliminaries

This section introduces the notation and basic terminology.

1.1. SuperHyperGraphs

Graph theory studies vertices and edges, analyzing connectivity, structure, and algorithms for modeling relationships in mathematics, computer science, and applications [1]. A hypergraph generalizes an ordinary graph by allowing an edge to connect any nonempty subset of the vertex set. Hence, hypergraphs provide a direct language for modeling multiway interactions [2,3]. In particular, such structures have played an important role in recent years in methods such as neural networks [3,4,5,6,7].

By iterating the powerset operation one step further, one obtains nested (higher-order) vertex objects and, consequently, a finite SuperHyperGraph whose vertices and edges may themselves be set-valued at multiple levels [8,9]. Such hierarchical representations are useful, for instance, in molecular design, complex-network analysis, neural networks, and related applications [10,11,12,13,14,15,16]. Moreover, as related concepts, research has also been progressing on notions such as Directed SuperHyperGraphs [17,18] and MetaSuperHyperGraphs [19].

Throughout, the index n in ${\mathrm{PS}}_n(·)$ and in an n-SuperHyperGraph is always taken to be a nonnegative integer.

Definition 1

(Base set). A base set S is the underlying universe of discourse:

$$S=\{x\mid x\mathit{is}\mathit{an}\mathit{admissible}\mathit{object}\mathit{in}\mathit{the}\mathit{context}\mathit{under}\mathit{consideration}\}.$$

All sets appearing in $\mathrm{PS}\left(S\right)$ and in the iterated powersets ${\mathrm{PS}}_n\left(S\right)$ are ultimately built from elements of S.

Definition 2

(Powerset). (see [20,21,22]) For a set S, the powerset of S is

$$\mathrm{PS}\left(S\right)=\{A\mid A\subseteq S\}.$$

In particular, $\varnothing \in \mathrm{PS}\left(S\right)$ and $S\in \mathrm{PS}\left(S\right)$.

Definition 3

(Hypergraph [23,24]). A hypergraph is a pair $H=(V,E)$ such that:

• V is a finite set (the vertices), and
• E is a finite family of nonempty subsets of V (the hyperedges).

Thus, a hyperedge may involve more than two vertices, capturing genuinely multiway relations.

Definition 4

(Iterated powerset and flattening). Let $V_0$ be a finite nonempty set. Define ${\mathrm{PS}}^0\left(V_0\right):=V_0$ and ${\mathrm{PS}}^{k+1}\left(V_0\right):=\mathrm{PS}\left({\mathrm{PS}}^k\left(V_0\right)\right)$ for $k\ge 0$. For each $k\ge 0$, define the flattening map

$${Flat}_k:{\mathrm{PS}}^k\left(V_0\right)\setminus \{\varnothing \}⟶\mathrm{PS}\left(V_0\right)\setminus \{\varnothing \}$$

recursively by

$${Flat}_0\left(x\right):=\left\{x\right\}(x\in V_0),{Flat}_{k+1}\left(X\right):=\bigcup _{Y\in X}{Flat}_k\left(Y\right)(X\in {\mathrm{PS}}^{k+1}\left(V_0\right)\setminus \{\varnothing \}).$$

Definition 5

(n-SuperHyperGraph). (see [25]) Let $V_0$ be a finite, nonempty base set. Define

$${\mathrm{PS}}^0\left(V_0\right):=V_0,{\mathrm{PS}}^{k+1}\left(V_0\right):=\mathrm{PS}\left({\mathrm{PS}}^k\left(V_0\right)\right)(k\in \mathbb{N}).$$

For $n\ge 0$, an n-SuperHyperGraph on $V_0$ is a pair

$${\mathrm{SHG}}^{\left(n\right)}=(V,E)$$

satisfying

$$V\subseteq {\mathrm{PS}}^n\left(V_0\right)\mathit{and}E\subseteq \mathrm{PS}\left(V\right)\setminus \{\varnothing \}.$$

Elements of V are called n-supervertices, and elements of E are called n-superedges (i.e., each n-superedge is a nonempty subset of V).

As a reference, a high-level comparison of Graphs, Hypergraphs, and SuperHyperGraphs is presented in Table 1.

1.2. Rooted Tree

A rooted tree is a connected acyclic graph with a distinguished root, where each non-root node has exactly one parent.

Definition 6

(Rooted tree). A rooted tree is a triple $T=(\mathcal{N},\mathcal{F},r)$ where $\mathcal{N}$ is a finite node set, $r\in \mathcal{N}$ is the root, and $\mathcal{F}\subseteq \mathcal{N}\times \mathcal{N}$ is a set of parent–child arcs such that every node $u\in \mathcal{N}\setminus \left\{r\right\}$ has a unique parent and the underlying undirected graph is connected and acyclic. For $u\in \mathcal{N}$, let $\mathrm{Ch}\left(u\right)$ denote its children and let $\mathrm{Leaf}\left(T\right)\subseteq \mathcal{N}$ denote the set of leaves.

2. Main Results

This section presents the main results of this paper.

2.1. Depth-n Tree-Vertex Graph

A depth-n Tree-Vertex Graph is a rooted tree of depth n whose nodes represent hierarchical units via nested iterated-powerset labels, with edges only within each level.

Definition 7

(Depth-n Tree-Vertex Graph with level edges). Let $V_0$ be a finite, nonempty set of base vertices and let $n\in \mathbb{N}$. A depth-n Tree-Vertex Graph (TVG) on $V_0$ is a quadruple

$${\mathrm{TVG}}^{\left(n\right)}=\left(V_0,T,\eta ,{\left\{E^{\left(k\right)}\right\}}_{k=0}^n\right),$$

where:

(i)

$T=(\mathcal{N},\mathcal{F},r)$ is a rooted tree whose leaves have depth 0 and whose root has depth n. For each $k\in \{0,\dots ,n\}$ define the level set

$${\mathcal{N}}_k:=\{u\in \mathcal{N}\mid \mathrm{depth}\left(u\right)=k\}.$$

(ii)

η is a nested labeling (level-typed label map)

$$\eta :\mathcal{N}⟶\bigcup _{k=0}^n{\mathrm{PS}}^k\left(V_0\right)\setminus \{\varnothing \}$$

satisfying:

(a)

(Level-typing) For every $u\in {\mathcal{N}}_k$, one has

$$\eta \left(u\right)\in {\mathrm{PS}}^k\left(V_0\right)\setminus \{\varnothing \}.$$

(b)

(Leaf grounding) The restriction ${\eta |}_{{\mathcal{N}}_0}:{\mathcal{N}}_0\to V_0$ is a bijection (so each leaf represents exactly one base vertex).

(c)

(Recursive nesting along the tree) For every internal node $u\in {\mathcal{N}}_k$ with $k\ge 1$,

$$\eta \left(u\right)=\left\{\eta \right(v)\mid v\in \mathrm{Ch}(u\left)\right\},$$

so the label of u is literally the set of its children’s labels (hence lives in the next iterated powerset).

(iii)

The support (flattened base-vertex set) of a tree-vertex $u\in {\mathcal{N}}_k$ is defined by

$$\lambda \left(u\right):={Flat}_k\left(\eta \left(u\right)\right)\subseteq V_0,$$

where ${Flat}_k$ is from Definition 4.

(iv)

For each level $k\in \{0,\dots ,n\}$, $E^{\left(k\right)}$ is a set of level-k graph edges :

$$E^{\left(k\right)}\subseteq \left\{\{u,v\}\subseteq {\mathcal{N}}_k\mid u\ne v\right\}.$$

An edge $\{u,v\}\in E^{\left(k\right)}$ encodes a binary relation within the same abstraction depth k between the hierarchical units represented by u and v.

Remark 1

(Why this matches the n-th powerset depth). In Definition 7, the constraint

$$\eta \left(u\right)\in {\mathrm{PS}}^{\mathrm{depth}\left(u\right)}\left(V_0\right)$$

is enforced by construction: leaves satisfy $\eta (\ell )\in V_0={\mathrm{PS}}^0\left(V_0\right)$, and each upward step applies one powerset operation via $\eta \left(u\right)=\left\{\eta \right(v)\mid v\in \mathrm{Ch}(u\left)\right\}$. Hence the tree depth is literally the iterated-powerset depth. The flattened support $\lambda \left(u\right)={Flat}_{\mathrm{depth}\left(u\right)}\left(\eta \left(u\right)\right)$ recovers the base vertices contained in u.

Remark 2

(Option: level-wise hyperedges instead of edges). If one wants multiway relations at each level, replace each $E^{\left(k\right)}$ by a hyperedge family

$${\mathcal{E}}^{\left(k\right)}\subseteq \mathrm{PS}\left({\mathcal{N}}_k\right)\setminus \{\varnothing \},$$

so that each hyperedge $e\in {\mathcal{E}}^{\left(k\right)}$ relates an arbitrary finite subset of nodes at level k. This yields a level-hypergraph TVG while retaining the same nested labeling η.

A comparison of an n-SuperHyperGraph and a depth-n Tree-Vertex Graph is presented in Table 2.

Specific examples are given below.

Example 1

(A depth-2 TVG on four base vertices). Let

$$V_0=\{a,b,c,d\},n=2.$$

We define ${\mathrm{TVG}}^{\left(2\right)}=(V_0,T,\eta ,{\left\{E^{\left(k\right)}\right\}}_{k=0}^2)$ as follows.

(i)

Rooted tree. Let the node set be

$$\mathcal{N}=\{{\ell }_a,{\ell }_b,{\ell }_c,{\ell }_d,u_1,u_2,r\}.$$

Declare r to be the root. Define the parent–child arcs

$$\mathcal{F}=\{(r,u_1),(r,u_2),(u_1,{\ell }_a),(u_1,{\ell }_b),(u_2,{\ell }_c),(u_2,{\ell }_d)\}.$$

Then the leaves are ${\mathcal{N}}_0=\{{\ell }_a,{\ell }_b,{\ell }_c,{\ell }_d\}$, the level-1 nodes are ${\mathcal{N}}_1=\{u_1,u_2\}$, and the unique level-2 node is ${\mathcal{N}}_2=\left\{r\right\}$.

(ii)

Nested labeling $\eta$. We specify $\eta :\mathcal{N}\to {\bigcup }_{k=0}^2{\mathrm{PS}}^k\left(V_0\right)\setminus \{\varnothing \}$ by

$$\eta \left({\ell }_a\right)=a,\eta \left({\ell }_b\right)=b,\eta \left({\ell }_c\right)=c,\eta \left({\ell }_d\right)=d,$$

$$\eta \left(u_1\right)=\{\eta \left({\ell }_a\right),\eta \left({\ell }_b\right)\}=\{a,b\},\eta \left(u_2\right)=\{\eta \left({\ell }_c\right),\eta \left({\ell }_d\right)\}=\{c,d\},$$

$$\eta \left(r\right)=\{\eta \left(u_1\right),\eta \left(u_2\right)\}=\left\{\{a,b\},\{c,d\}\right\}.$$

By construction:

• $\eta \left({\ell }_x\right)\in {\mathrm{PS}}^0\left(V_0\right)=V_0$ for each ${\ell }_x\in {\mathcal{N}}_0$,
• $\eta \left(u_i\right)\in {\mathrm{PS}}^1\left(V_0\right)$ for $u_i\in {\mathcal{N}}_1$,
• $\eta \left(r\right)\in {\mathrm{PS}}^2\left(V_0\right)$ for $r\in {\mathcal{N}}_2$,

and the recursive nesting condition $\eta \left(u\right)=\left\{\eta \right(v)\mid v\in \mathrm{Ch}(u\left)\right\}$ holds for the internal nodes.

(iii)

Supports (flattened base-vertex sets). Using $\lambda \left(u\right)={Flat}_{\mathrm{depth}\left(u\right)}\left(\eta \left(u\right)\right)$, we obtain

$$\lambda \left({\ell }_a\right)=\left\{a\right\},\lambda \left({\ell }_b\right)=\left\{b\right\},\lambda \left({\ell }_c\right)=\left\{c\right\},\lambda \left({\ell }_d\right)=\left\{d\right\},$$

$$\lambda \left(u_1\right)=\{a,b\},\lambda \left(u_2\right)=\{c,d\},\lambda \left(r\right)=\{a,b,c,d\}.$$

(iv)

Level edges ${\left\{E^{\left(k\right)}\right\}}_{k=0}^2$. Define

$$E^{\left(0\right)}=\left\{\{{\ell }_a,{\ell }_c\},\{{\ell }_b,{\ell }_d\}\right\},E^{\left(1\right)}=\left\{\{u_1,u_2\}\right\},E^{\left(2\right)}=\varnothing .$$

Example 2

(A depth-3 TVG on six base vertices with nontrivial middle levels). Let

$$V_0=\{p_1,p_2,p_3,p_4,p_5,p_6\},n=3.$$

We define ${\mathrm{TVG}}^{\left(3\right)}=(V_0,T,\eta ,{\left\{E^{\left(k\right)}\right\}}_{k=0}^3)$ as follows.

(i)

Rooted tree. Let the leaves (depth 0) be

$${\mathcal{N}}_0=\{{\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6\}.$$

Let the level-1 nodes be

$${\mathcal{N}}_1=\{a_1,a_2,a_3\},$$

the level-2 nodes be

$${\mathcal{N}}_2=\{b_1,b_2\},$$

and the root (depth 3) be

$${\mathcal{N}}_3=\left\{r\right\}.$$

Set $\mathcal{N}={\mathcal{N}}_0\cup {\mathcal{N}}_1\cup {\mathcal{N}}_2\cup {\mathcal{N}}_3$ and define arcs

$$\mathcal{F}=\{(r,b_1),(r,b_2),(b_1,a_1),(b_1,a_2),(b_2,a_3),(a_1,{\ell }_1),(a_1,{\ell }_2),(a_2,{\ell }_3),(a_2,{\ell }_4),(a_3,{\ell }_5),(a_3,{\ell }_6)\}.$$

(ii)

Nested labeling $\eta$. Define η on leaves by the bijection

$$\eta \left({\ell }_i\right)=p_i(i=1,\dots ,6).$$

Then define η on internal nodes by the recursive rule:

$$\eta \left(a_1\right)=\{\eta \left({\ell }_1\right),\eta \left({\ell }_2\right)\}=\{p_1,p_2\},\eta \left(a_2\right)=\{\eta \left({\ell }_3\right),\eta \left({\ell }_4\right)\}=\{p_3,p_4\},\eta \left(a_3\right)=\{\eta \left({\ell }_5\right),\eta \left({\ell }_6\right)\}=\{p_5,p_6\},$$

$$\eta \left(b_1\right)=\{\eta \left(a_1\right),\eta \left(a_2\right)\}=\left\{\{p_1,p_2\},\{p_3,p_4\}\right\},\eta \left(b_2\right)=\left\{\eta \left(a_3\right)\right\}=\left\{\{p_5,p_6\}\right\},$$

$$\eta \left(r\right)=\{\eta \left(b_1\right),\eta \left(b_2\right)\}=\left\{\left\{\{p_1,p_2\},\{p_3,p_4\}\right\},\left\{\{p_5,p_6\}\right\}\right\}.$$

(iii)

Supports (flattened base-vertex sets). Using $\lambda \left(u\right)={Flat}_{\mathrm{depth}\left(u\right)}\left(\eta \left(u\right)\right)$, we obtain

$$\lambda \left({\ell }_i\right)=\left\{p_i\right\}(i=1,\dots ,6),$$

$$\lambda \left(a_1\right)=\{p_1,p_2\},\lambda \left(a_2\right)=\{p_3,p_4\},\lambda \left(a_3\right)=\{p_5,p_6\},$$

$$\lambda \left(b_1\right)=\{p_1,p_2,p_3,p_4\},\lambda \left(b_2\right)=\{p_5,p_6\},\lambda \left(r\right)=V_0.$$

(iv)

Level edges ${\left\{E^{\left(k\right)}\right\}}_{k=0}^3$. Define

$$E^{\left(0\right)}=\left\{\{{\ell }_2,{\ell }_3\},\{{\ell }_4,{\ell }_6\}\right\},E^{\left(1\right)}=\left\{\{a_1,a_2\}\right\},E^{\left(2\right)}=\left\{\{b_1,b_2\}\right\},E^{\left(3\right)}=\varnothing .$$

Notation 1

(Descendants, subtree leaves, and level-k container). Let $T=(\mathcal{N},\mathcal{F},r)$ be a rooted tree whose leaves have depth 0 and root has depth n. Write $v⪯u$ if v is a descendant of u (possibly $v=u$), i.e. there is a directed path from u to v. For $u\in \mathcal{N}$ define the subtree leaf set

$$\mathcal{L}\left(u\right):=\{\ell \in {\mathcal{N}}_0\mid \ell ⪯u\}.$$

For each level $k\in \{0,\dots ,n\}$ and each base vertex $x\in V_0$, let ${\ell }_x\in {\mathcal{N}}_0$ denote the unique leaf with $\eta \left({\ell }_x\right)=x$ (existence and uniqueness by leaf grounding). Define the level-k container of x by

$${\pi }_k\left(x\right):=\mathit{the}\mathit{unique}\mathit{node}u\in {\mathcal{N}}_k\mathit{such}\mathit{that}{\ell }_x\in \mathcal{L}\left(u\right).$$

(The uniqueness of ${\pi }_k\left(x\right)$ will be proved in Theorem 2).

Theorem 1

(Support equals the set of leaves in the subtree). Let ${\mathrm{TVG}}^{\left(n\right)}=(V_0,T,\eta ,{\left\{E^{\left(k\right)}\right\}}_{k=0}^n)$ be a depth-n TVG and let $\lambda \left(u\right)={Flat}_{\mathrm{depth}\left(u\right)}\left(\eta \left(u\right)\right)\subseteq V_0$. Then for every $u\in \mathcal{N}$,

$$\lambda \left(u\right)=\{\eta (\ell )\mid \ell \in \mathcal{L}\left(u\right)\}\subseteq V_0.$$

In particular, $\left|\lambda \right(u\left)\right|=\left|\mathcal{L}\right(u\left)\right|$.

Proof. 

We argue by induction on $k=\mathrm{depth}\left(u\right)$.

Base case $k=0$. Then $u\in {\mathcal{N}}_0$ is a leaf, so $\mathcal{L}\left(u\right)=\left\{u\right\}$. Moreover $\eta \left(u\right)\in {\mathrm{PS}}^0\left(V_0\right)=V_0$ and by definition $\lambda \left(u\right)={Flat}_0\left(\eta \left(u\right)\right)=\left\{\eta \left(u\right)\right\}$. Hence

$$\lambda \left(u\right)=\left\{\eta \right(u\left)\right\}=\left\{\eta \right(\ell )\mid \ell \in \mathcal{L}(u\left)\right\}.$$

Inductive step. Assume the claim holds for all nodes of depth $<k$, and let $u\in {\mathcal{N}}_k$ with $k\ge 1$. By recursive nesting,

$$\eta \left(u\right)=\left\{\eta \right(v)\mid v\in \mathrm{Ch}(u\left)\right\}.$$

By the recursive definition of ${Flat}_k$ (as in Definition 4),

$$\lambda \left(u\right)={Flat}_k\left(\eta \left(u\right)\right)=\bigcup _{v\in \mathrm{Ch}\left(u\right)}{Flat}_{k-1}\left(\eta \left(v\right)\right)=\bigcup _{v\in \mathrm{Ch}\left(u\right)}\lambda \left(v\right).$$

By the induction hypothesis, for each child $v\in \mathrm{Ch}\left(u\right)$ we have $\lambda \left(v\right)=\left\{\eta \right(\ell )\mid \ell \in \mathcal{L}(v\left)\right\}$, hence

$$\lambda \left(u\right)=\bigcup _{v\in \mathrm{Ch}\left(u\right)}\{\eta (\ell )\mid \ell \in \mathcal{L}\left(v\right)\}.$$

Finally, the leaves under u are exactly the disjoint union of the leaves under its children:

$$\mathcal{L}\left(u\right)=\underset{v\in \mathrm{Ch}\left(u\right)}{⨆}\mathcal{L}\left(v\right),$$

so the right-hand side equals $\left\{\eta \right(\ell )\mid \ell \in \mathcal{L}(u\left)\right\}$. This proves the desired identity for depth k. The cardinality statement $\left|\lambda \right(u\left)\right|=\left|\mathcal{L}\right(u\left)\right|$ follows because the map ${\eta |}_{{\mathcal{N}}_0}:{\mathcal{N}}_0\to V_0$ is a bijection, so distinct leaves have distinct base labels. □

Theorem 2

(Level supports form a partition of $V_0$). Fix $k\in \{0,\dots ,n\}$. Then the family of supports at level k,

$${\mathcal{L}}_k:=\{\lambda \left(u\right)\mid u\in {\mathcal{N}}_k\},$$

is a partition of $V_0$, i.e.

$$V_0=\bigcup _{u\in {\mathcal{N}}_k}\lambda \left(u\right)\mathit{and}u\ne v\Rightarrow \lambda \left(u\right)\cap \lambda \left(v\right)=\varnothing (u,v\in {\mathcal{N}}_k).$$

Equivalently, for every $x\in V_0$ there exists a unique $u\in {\mathcal{N}}_k$ with $x\in \lambda \left(u\right)$.

Proof. 

Covering. Let $x\in V_0$, and let ${\ell }_x\in {\mathcal{N}}_0$ be the unique leaf with $\eta \left({\ell }_x\right)=x$. Consider the unique root-to-leaf path from r to ${\ell }_x$. Since depths decrease by 1 along each parent–child arc and the root has depth n while ${\ell }_x$ has depth 0, this path contains exactly one node u at depth k. Then ${\ell }_x\in \mathcal{L}\left(u\right)$. By Theorem 1,

$$x=\eta \left({\ell }_x\right)\in \{\eta (\ell )\mid \ell \in \mathcal{L}\left(u\right)\}=\lambda \left(u\right),$$

so x lies in the union of the level-k supports.

Disjointness and uniqueness. Let $u,v\in {\mathcal{N}}_k$ with $u\ne v$. If $\lambda \left(u\right)\cap \lambda \left(v\right)\ne \varnothing$, choose $x\in \lambda \left(u\right)\cap \lambda \left(v\right)$. By Theorem 1, there exist leaves $\ell ,{\ell }^{\prime }\in {\mathcal{N}}_0$ such that $\ell \in \mathcal{L}\left(u\right)$, ${\ell }^{\prime }\in \mathcal{L}\left(v\right)$, and $\eta (\ell )=x=\eta \left({\ell }^{\prime }\right)$. Since ${\eta |}_{{\mathcal{N}}_0}$ is injective, $\ell ={\ell }^{\prime }$. Thus there is a leaf ℓ that lies in both subtrees rooted at u and v. In a tree, the path from the root to ℓ is unique, hence it cannot pass through two distinct nodes at the same depth. Therefore $u=v$, a contradiction. Hence $\lambda \left(u\right)\cap \lambda \left(v\right)=\varnothing$.

The equivalent uniqueness statement follows: if x were contained in $\lambda \left(u\right)$ and $\lambda \left(v\right)$ for two level-k nodes, then $\lambda \left(u\right)\cap \lambda \left(v\right)\ne \varnothing$, forcing $u=v$. □

Corollary 1

(Support recursion and additivity across children). Let $u\in {\mathcal{N}}_k$ with $k\ge 1$. Then

$$\lambda \left(u\right)=\bigcup _{v\in \mathrm{Ch}\left(u\right)}\lambda \left(v\right),\mathit{and}\mathit{the}\mathit{union}\mathit{is}\mathit{disjoint},\mathit{hence}\left|\lambda \left(u\right)\right|=\sum _{v\in \mathrm{Ch}\left(u\right)}\left|\lambda \left(v\right)\right|.$$

Proof. 

The identity $\lambda \left(u\right)={\bigcup }_{v\in \mathrm{Ch}\left(u\right)}\lambda \left(v\right)$ was shown in the inductive step of Theorem 1. Children of u all lie at the same depth $k-1$, so by Theorem 2 (applied at level $k-1$), their supports are pairwise disjoint. Additivity of cardinalities follows. □

Theorem 3

(Laminarity of supports across all tree-vertices). For any $u,v\in \mathcal{N}$, exactly one of the following holds:

$$\lambda \left(u\right)\cap \lambda \left(v\right)=\varnothing ,\lambda \left(u\right)\subseteq \lambda \left(v\right),\lambda \left(v\right)\subseteq \lambda \left(u\right).$$

Moreover,

$$\lambda \left(u\right)\cap \lambda \left(v\right)\ne \varnothing ⟺(u⪯v)\mathrm{or}(v⪯u).$$

Proof. 

If $u⪯v$, then every leaf under u is also a leaf under v, i.e. $\mathcal{L}\left(u\right)\subseteq \mathcal{L}\left(v\right)$. Applying Theorem 1 gives

$$\lambda \left(u\right)=\left\{\eta \right(\ell )\mid \ell \in \mathcal{L}(u\left)\right\}\subseteq \left\{\eta \right(\ell )\mid \ell \in \mathcal{L}(v\left)\right\}=\lambda \left(v\right).$$

The same reasoning applies if $v⪯u$.

Now assume neither $u⪯v$ nor $v⪯u$. Then the subtrees rooted at u and v are disjoint: if there were a leaf ℓ belonging to both $\mathcal{L}\left(u\right)$ and $\mathcal{L}\left(v\right)$, the unique root-to-ℓ path would contain both u and v, forcing one to be an ancestor of the other, contradicting the assumption. Hence $\mathcal{L}\left(u\right)\cap \mathcal{L}\left(v\right)=\varnothing$, and by Theorem 1,

$$\lambda \left(u\right)\cap \lambda \left(v\right)=\left\{\eta \right(\ell )\mid \ell \in \mathcal{L}(u\left)\right\}\cap \left\{\eta \right(\ell )\mid \ell \in \mathcal{L}(v\left)\right\}=\varnothing ,$$

because ${\eta |}_{{\mathcal{N}}_0}$ is injective and the leaf sets are disjoint. This establishes the trichotomy and the equivalence. □

2.2. n-SuperHyperGraph with Intra-Level and Inter-Level Edges

An n-SuperHyperGraph with intra-level and inter-level edges extends an n-SuperHyperGraph by adding edges among same-level supervertices and cross-level edges linking different nesting depths.

Definition 8

(n-SuperHyperGraph with intra-level and inter-level edges). Let $V_0$ be a finite, nonempty set and let $n\in \mathbb{N}$. An n-SuperHyperGraph with intra-level and inter-level edges is a tuple

$${\mathrm{SHG}}_{\times }^{(\le n)}=\left(V_0,V,\mathcal{E},{\left\{E^{\left(k\right)}\right\}}_{k=0}^n,{\left\{E^{(k\to \ell )}\right\}}_{0\le k<\ell \le n}\right),$$

where:

(i)

(Graded supervertex set)

$$V\subseteq \bigcup _{k=0}^n{\mathrm{PS}}^k\left(V_0\right),V_k:=V\cap {\mathrm{PS}}^k\left(V_0\right)(k=0,1,\dots ,n).$$

Optionally (and typically), one assumes grounding $V_0\subseteq V$ (equivalently $V_0=V\cap {\mathrm{PS}}^0\left(V_0\right)$), and $V_k\subseteq {\mathrm{PS}}^k\left(V_0\right)\setminus \{\varnothing \}$ for $k\ge 1$.

(ii)

(Superhyperedges)

$$\mathcal{E}\subseteq \mathrm{PS}\left(V\right)\setminus \{\varnothing \}.$$

Each $e\in \mathcal{E}$ is a nonempty set of supervertices, possibly mixing levels.

(iii)

(Support / flattening) For $u\in V_k$ define its support

$$\lambda \left(u\right):={Flat}_k\left(u\right)\subseteq V_0,$$

where ${Flat}_k$ is the flattening map from Definition 4. (Thus $\lambda \left(u\right)$ is the set of base vertices “contained in” the nested object u.)

(iv)

(Intra-level (graph) edges) For each $k\in \{0,\dots ,n\}$,

$$E^{\left(k\right)}\subseteq \left\{\{u,v\}\subseteq V_k\mid u\ne v\right\}.$$

An edge $\{u,v\}\in E^{\left(k\right)}$ encodes a binary relation between two supervertices at the same level k.

(v)

(Inter-level (directed) edges) For each pair $0\le k<\ell \le n$,

$$E^{(k\to \ell )}\subseteq V_k\times V_{\ell }.$$

An ordered pair $(u,v)\in E^{(k\to \ell )}$ is an inter-level edge from level k to level ℓ.

(vi)

(Optional support-compatibility constraints) Depending on semantics, one may require every $(u,v)\in E^{(k\to \ell )}$ to satisfy, for example,

$$\lambda \left(u\right)\subseteq \lambda \left(v\right)(\mathit{upward}/\mathit{abstraction}),\mathit{or}\lambda \left(u\right)\cap \lambda \left(v\right)\ne \varnothing \left(\mathit{overlap}\right).$$

Remark 3

(Relation to the classical n-SuperHyperGraph). If one restricts V to a single level $V=V_n\subseteq {\mathrm{PS}}^n\left(V_0\right)$ and discards all ${\left\{E^{\left(k\right)}\right\}}_{k<n}$ and ${\left\{E^{(k\to \ell )}\right\}}_{k<\ell }$, then $(V,\mathcal{E})$ is precisely an n-SuperHyperGraph in the sense of Definition 5. The present definition enriches the model by allowing graded supervertices (levels 0 through n) and by equipping them with both intra-level and inter-level graph-type edges in addition to hyperedges.

As a reference, a comparison between an n-SuperHyperGraph and an n-SuperHyperGraph with intra-level and inter-level edges is presented in Table 3.

Example 3

(A depth-2 instance with both hyperedges and cross-level edges). Let $V_0=\{a,b,c,d\}$ and $n=2$. Define the graded supervertex sets

$$V_0=\{a,b,c,d\},V_1=\{A,B,C\},V_2=\{U,W\},$$

where

$$A=\{a,b\},B=\{c,d\},C=\{b,c\}\in {\mathrm{PS}}^1\left(V_0\right)=\mathrm{PS}\left(V_0\right),$$

and

$$U=\{A,B\},W=\{A,C\}\in {\mathrm{PS}}^2\left(V_0\right)=\mathrm{PS}\left(\mathrm{PS}\left(V_0\right)\right).$$

Set $V:=V_0\cup V_1\cup V_2\subseteq {\mathrm{PS}}^0\left(V_0\right)\cup {\mathrm{PS}}^1\left(V_0\right)\cup {\mathrm{PS}}^2\left(V_0\right)$.

Supports (via flattening). Using $\lambda \left(u\right)={Flat}_{\mathrm{level}\left(u\right)}\left(u\right)$,

$$\lambda \left(a\right)=\left\{a\right\},\lambda \left(b\right)=\left\{b\right\},\lambda \left(c\right)=\left\{c\right\},\lambda \left(d\right)=\left\{d\right\},$$

$$\lambda \left(A\right)=\{a,b\},\lambda \left(B\right)=\{c,d\},\lambda \left(C\right)=\{b,c\},$$

$$\lambda \left(U\right)=\{a,b,c,d\},\lambda \left(W\right)=\{a,b,c\}.$$

Superhyperedges. Let the hyperedge family be

$$\mathcal{E}=\left\{\{a,A\},\{b,A,C\},\{c,C,B\},\{A,C,W\},\{U,W,B\}\right\}\subseteq \mathrm{PS}\left(V\right)\setminus \{\varnothing \}.$$

These hyperedges allow genuinely multiway relations, and they may mix levels (e.g., $\{U,W,B\}$ lives across levels 2 and 1).

Intra-level edges. Define

$$E^{\left(0\right)}=\left\{\{a,b\},\{c,d\}\right\},E^{\left(1\right)}=\left\{\{A,C\}\right\},E^{\left(2\right)}=\left\{\{U,W\}\right\}.$$

For instance, $\{A,C\}\in E^{\left(1\right)}$ encodes a binary relation between two level-1 group-vertices (perhaps “overlapping communities” or “related clusters”).

Inter-level edges. Let

$$E^{(0\to 1)}=\{(a,A),(b,A),(b,C),(c,C),(c,B),(d,B)\},$$

$$E^{(1\to 2)}=\{(A,U),(B,U),(A,W),(C,W)\},E^{(0\to 2)}=\{(a,U),(d,U)\}.$$

If one enforces the optional upward constraint $\lambda \left(u\right)\subseteq \lambda \left(v\right)$, then it holds here: for example, $(b,C)\in E^{(0\to 1)}$ satisfies $\lambda \left(b\right)=\left\{b\right\}\subseteq \lambda \left(C\right)=\{b,c\}$, and $(C,W)\in E^{(1\to 2)}$ satisfies $\lambda \left(C\right)=\{b,c\}\subseteq \lambda \left(W\right)=\{a,b,c\}$. Thus the inter-level edges can be read as membership/abstraction links across levels.

Consequently,

$${\mathrm{SHG}}_{\times }^{(\le 2)}=\left(V_0,V,\mathcal{E},\{E^{\left(0\right)},E^{\left(1\right)},E^{\left(2\right)}\},\{E^{(0\to 1)},E^{(0\to 2)},E^{(1\to 2)}\}\right)$$

is a concrete 2-SuperHyperGraph with both intra-level and inter-level edges in the sense of Definition 8.

2.3. Depth-n TVG with Intra-Level and Inter-Level Edges

A depth-n TVG with intra-level and inter-level edges is a depth-n rooted tree whose nodes carry iterated-powerset labels; edges relate nodes within the same level and across levels.

Definition 9

(Depth-n TVG with intra-level and inter-level edges). Let $V_0$ be a finite, nonempty set and let $n\in \mathbb{N}$. A depth-n Tree-Vertex Graph with cross-level edges is a tuple

$${\mathrm{TVG}}_{\times }^{\left(n\right)}=\left(V_0,T,\eta ,{\left\{E^{\left(k\right)}\right\}}_{k=0}^n,{\left\{E^{(k\to \ell )}\right\}}_{0\le k<\ell \le n}\right),$$

where:

(i)

$T=(\mathcal{N},\mathcal{F},r)$ is a rooted tree whose leaves have depth 0 and whose root has depth n. Let ${\mathcal{N}}_k=\{u\in \mathcal{N}\mid \mathrm{depth}\left(u\right)=k\}$.

(ii)

$\eta :\mathcal{N}\to {\bigcup }_{k=0}^n{\mathrm{PS}}^k\left(V_0\right)\setminus \{\varnothing \}$ is a nested labeling such that:

(a)

(Level-typing) If $u\in {\mathcal{N}}_k$, then $\eta \left(u\right)\in {\mathrm{PS}}^k\left(V_0\right)\setminus \{\varnothing \}$.

(b)

(Leaf grounding) ${\eta |}_{{\mathcal{N}}_0}:{\mathcal{N}}_0\to V_0$ is a bijection.

(c)

(Recursive nesting) For every $u\in {\mathcal{N}}_k$ with $k\ge 1$,

$$\eta \left(u\right)=\left\{\eta \right(v)\mid v\in \mathrm{Ch}(u\left)\right\}.$$

(iii)

Define the flattened support $\lambda \left(u\right)\subseteq V_0$ by $\lambda \left(u\right)={Flat}_{\mathrm{depth}\left(u\right)}\left(\eta \left(u\right)\right)$.

(iv)

(Intra-level edges) For each $k\in \{0,\dots ,n\}$,

$$E^{\left(k\right)}\subseteq \left\{\{u,v\}\subseteq {\mathcal{N}}_k\mid u\ne v\right\}.$$

(v)

(Inter-level edges / cross edges) For each pair $0\le k<\ell \le n$,

$$E^{(k\to \ell )}\subseteq {\mathcal{N}}_k\times {\mathcal{N}}_{\ell }.$$

An ordered pair $(u,v)\in E^{(k\to \ell )}$ is an inter-level edge from level k to level ℓ.

(vi)

(Support-compatibility constraint, optional) One may additionally require that every inter-level edge $(u,v)\in E^{(k\to \ell )}$ satisfies a support constraint such as either:

$$\lambda \left(u\right)\subseteq \lambda \left(v\right)(\mathit{upward}/\mathit{abstraction}\mathit{edge}),\mathit{or}\lambda \left(v\right)\subseteq \lambda \left(u\right)(\mathit{downward}/\mathit{refinement}\mathit{edge}),$$

or the weaker overlap condition $\lambda \left(u\right)\cap \lambda \left(v\right)\ne \varnothing$, depending on the intended semantics.

Remark 4

(Undirected cross-edges). If one prefers undirected inter-level edges, replace $E^{(k\to \ell )}\subseteq {\mathcal{N}}_k\times {\mathcal{N}}_{\ell }$ by a symmetric set

$$E^{(k,\ell )}\subseteq \left\{\{u,v\}\mid u\in {\mathcal{N}}_k,v\in {\mathcal{N}}_{\ell }\right\},$$

and interpret $\{u,v\}$ as an undirected relation between levels k and ℓ.

For reference, the comparison between a depth-n TVG and a depth-n TVG with intra-level and inter-level edges is presented in Table 4, and the comparison between a depth-n TVG with intra-level and inter-level edges and an n-SuperHyperGraph with intra-level and inter-level edges is presented in Table 5.

Specific examples are given below.

Example 4

(A depth-2 TVG with both intra-level and inter-level edges (overlap-compatible cross edges)). Let

$$V_0=\{a,b,c,d\},n=2.$$

We define

$${\mathrm{TVG}}_{\times }^{\left(2\right)}=\left(V_0,T,\eta ,{\left\{E^{\left(k\right)}\right\}}_{k=0}^2,{\left\{E^{(k\to \ell )}\right\}}_{0\le k<\ell \le 2}\right)$$

by specifying each component.

(i)

Rooted tree T. Let the node set be

$$\mathcal{N}=\{{\ell }_a,{\ell }_b,{\ell }_c,{\ell }_d,u_1,u_2,r\}.$$

Take r as the root and define the parent–child arcs

$$\mathcal{F}=\{(r,u_1),(r,u_2),(u_1,{\ell }_a),(u_1,{\ell }_b),(u_2,{\ell }_c),(u_2,{\ell }_d)\}.$$

Thus

$${\mathcal{N}}_0=\{{\ell }_a,{\ell }_b,{\ell }_c,{\ell }_d\},{\mathcal{N}}_1=\{u_1,u_2\},{\mathcal{N}}_2=\left\{r\right\}.$$

(ii)

Nested labeling $\eta$. Define η on the leaves by

$$\eta \left({\ell }_a\right)=a,\eta \left({\ell }_b\right)=b,\eta \left({\ell }_c\right)=c,\eta \left({\ell }_d\right)=d,$$

and on internal nodes by the recursive nesting rule:

$$\eta \left(u_1\right)=\{\eta \left({\ell }_a\right),\eta \left({\ell }_b\right)\}=\{a,b\},\eta \left(u_2\right)=\{\eta \left({\ell }_c\right),\eta \left({\ell }_d\right)\}=\{c,d\},$$

$$\eta \left(r\right)=\{\eta \left(u_1\right),\eta \left(u_2\right)\}=\left\{\{a,b\},\{c,d\}\right\}.$$

(iii)

Flattened supports $\lambda$. Using $\lambda \left(u\right)={Flat}_{\mathrm{depth}\left(u\right)}\left(\eta \left(u\right)\right)$, we obtain

$$\lambda \left({\ell }_a\right)=\left\{a\right\},\lambda \left({\ell }_b\right)=\left\{b\right\},\lambda \left({\ell }_c\right)=\left\{c\right\},\lambda \left({\ell }_d\right)=\left\{d\right\},$$

$$\lambda \left(u_1\right)=\{a,b\},\lambda \left(u_2\right)=\{c,d\},\lambda \left(r\right)=\{a,b,c,d\}.$$

(iv)

Intra-level edges. Set

$$E^{\left(0\right)}=\left\{\{{\ell }_a,{\ell }_c\},\{{\ell }_b,{\ell }_d\}\right\},E^{\left(1\right)}=\left\{\{u_1,u_2\}\right\},E^{\left(2\right)}=\varnothing .$$

(v)

Inter-level edges. We give cross edges at both pairs of levels:

$$E^{(0\to 1)}=\{({\ell }_a,u_1),({\ell }_c,u_2)\},E^{(0\to 2)}=\left\{({\ell }_b,r)\right\},E^{(1\to 2)}=\{(u_1,r),(u_2,r)\}.$$

(vi)

Compatibility check (overlap condition). Every listed inter-level edge $(u,v)$ satisfies $\lambda \left(u\right)\cap \lambda \left(v\right)\ne \varnothing$: for instance, $({\ell }_b,r)\in E^{(0\to 2)}$ has $\lambda \left({\ell }_b\right)=\left\{b\right\}$ and $\lambda \left(r\right)=\{a,b,c,d\}$, so the intersection is $\left\{b\right\}$; similarly $({\ell }_a,u_1)$ and $({\ell }_c,u_2)$ overlap by construction.

Example 5

(A depth-3 TVG with both intra-level and inter-level edges (mix of upward and downward semantics)). Let

$$V_0=\{p_1,p_2,p_3,p_4,p_5,p_6\},n=3.$$

We define

$${\mathrm{TVG}}_{\times }^{\left(3\right)}=\left(V_0,T,\eta ,{\left\{E^{\left(k\right)}\right\}}_{k=0}^3,{\left\{E^{(k\to \ell )}\right\}}_{0\le k<\ell \le 3}\right).$$

(i)

Rooted tree T. Let the levels be

$${\mathcal{N}}_0=\{{\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6\},{\mathcal{N}}_1=\{a_1,a_2,a_3\},{\mathcal{N}}_2=\{b_1,b_2\},{\mathcal{N}}_3=\left\{r\right\}.$$

Let the arc set be

$$\mathcal{F}=\{(r,b_1),(r,b_2),(b_1,a_1),(b_1,a_2),(b_2,a_3),(a_1,{\ell }_1),(a_1,{\ell }_2),(a_2,{\ell }_3),(a_2,{\ell }_4),(a_3,{\ell }_5),(a_3,{\ell }_6)\}.$$

(ii)

Nested labeling $\eta$. Define $\eta \left({\ell }_i\right)=p_i$ for $i=1,\dots ,6$. Then recursively:

$$\eta \left(a_1\right)=\{\eta \left({\ell }_1\right),\eta \left({\ell }_2\right)\}=\{p_1,p_2\},\eta \left(a_2\right)=\{\eta \left({\ell }_3\right),\eta \left({\ell }_4\right)\}=\{p_3,p_4\},\eta \left(a_3\right)=\{\eta \left({\ell }_5\right),\eta \left({\ell }_6\right)\}=\{p_5,p_6\},$$

$$\eta \left(b_1\right)=\{\eta \left(a_1\right),\eta \left(a_2\right)\}=\left\{\{p_1,p_2\},\{p_3,p_4\}\right\},$$

$$\eta \left(b_2\right)=\left\{\eta \left(a_3\right)\right\}=\left\{\{p_5,p_6\}\right\},$$

$$\eta \left(r\right)=\{\eta \left(b_1\right),\eta \left(b_2\right)\}=\left\{\left\{\{p_1,p_2\},\{p_3,p_4\}\right\},\left\{\{p_5,p_6\}\right\}\right\}.$$

(iii)

Flattened supports $\lambda$.

$$\lambda \left({\ell }_i\right)=\left\{p_i\right\}(i=1,\dots ,6),\lambda \left(a_1\right)=\{p_1,p_2\},\lambda \left(a_2\right)=\{p_3,p_4\},\lambda \left(a_3\right)=\{p_5,p_6\},$$

$$\lambda \left(b_1\right)=\{p_1,p_2,p_3,p_4\},\lambda \left(b_2\right)=\{p_5,p_6\},\lambda \left(r\right)=V_0.$$

(iv)

Intra-level edges. We specify relations within each abstraction depth:

$$E^{\left(0\right)}=\left\{\{{\ell }_2,{\ell }_3\},\{{\ell }_4,{\ell }_6\}\right\},E^{\left(1\right)}=\left\{\{a_1,a_2\}\right\},E^{\left(2\right)}=\left\{\{b_1,b_2\}\right\},E^{\left(3\right)}=\varnothing .$$

(v)

Inter-level edges (cross edges).We include a mix of upward cross edges (support inclusion $\lambda \left(u\right)\subseteq \lambda \left(v\right)$) and downward cross edges (support inclusion $\lambda \left(v\right)\subseteq \lambda \left(u\right)$) by choosing:

$$E^{(0\to 1)}=\{({\ell }_1,a_1),({\ell }_4,a_2),({\ell }_6,a_3)\},E^{(1\to 2)}=\{(a_2,b_1),(a_3,b_2)\},E^{(2\to 3)}=\{(b_1,r),(b_2,r)\},$$

together with two downward edges from the top:

$$E^{(3\to 1)}=\left\{(r,a_1)\right\},E^{(3\to 0)}=\left\{(r,{\ell }_5)\right\}.$$

(Equivalently, one may encode these as $E^{(1\to 3)}$ and $E^{(0\to 3)}$ with reversed orientation, depending on whether the intended semantics is “refines” or “abstracts.”)

(vi)

Compatibility check (inclusion/overlap). For each upward edge $(u,v)\in E^{(0\to 1)}\cup E^{(1\to 2)}\cup E^{(2\to 3)}$, we have $\lambda \left(u\right)\subseteq \lambda \left(v\right)$, e.g. $\lambda \left({\ell }_4\right)=\left\{p_4\right\}\subseteq \lambda \left(a_2\right)=\{p_3,p_4\}\subseteq \lambda \left(b_1\right)=\{p_1,p_2,p_3,p_4\}\subseteq \lambda \left(r\right)$. For each downward edge $(r,a_1)$ and $(r,{\ell }_5)$ we have $\lambda \left(a_1\right)\subseteq \lambda \left(r\right)$ and $\lambda \left({\ell }_5\right)\subseteq \lambda \left(r\right)$, so in particular $\lambda \left(r\right)\cap \lambda \left(a_1\right)\ne \varnothing$ and $\lambda \left(r\right)\cap \lambda \left({\ell }_5\right)\ne \varnothing$.

Notation 2

(Descendants, subtree leaves, and level-k container). Let $T=(\mathcal{N},\mathcal{F},r)$ be a rooted tree whose leaves have depth 0 and whose root has depth n. Write $v⪯u$ if v is a descendant of u (possibly $v=u$). For $u\in \mathcal{N}$ define the subtree leaf set

$$\mathcal{L}\left(u\right):=\{\ell \in {\mathcal{N}}_0\mid \ell ⪯u\}.$$

For each $x\in V_0$, let ${\ell }_x\in {\mathcal{N}}_0$ denote the unique leaf with $\eta \left({\ell }_x\right)=x$. For each $k\in \{0,\dots ,n\}$ define the level-k container map

$${\pi }_k:V_0\to {\mathcal{N}}_k,{\pi }_k\left(x\right):=\mathit{the}\mathit{unique}\mathit{node}u\in {\mathcal{N}}_k\mathit{such}\mathit{that}{\ell }_x\in \mathcal{L}\left(u\right).$$

(The existence and uniqueness of ${\pi }_k$ is established in Theorem 5.)

Theorem 4

(Support equals the base labels of subtree leaves). Let ${\mathrm{TVG}}_{\times }^{\left(n\right)}$ be as in Definition 9. Then for every $u\in \mathcal{N}$,

$$\lambda \left(u\right)=\{\eta (\ell )\mid \ell \in \mathcal{L}\left(u\right)\}\subseteq V_0.$$

In particular, $\left|\lambda \right(u\left)\right|=\left|\mathcal{L}\right(u\left)\right|$.

Proof. 

Identical to the proof of the corresponding statement for level-only TVGs: induct on $k=\mathrm{depth}\left(u\right)$ using recursive nesting $\eta \left(u\right)=\left\{\eta \right(v)\mid v\in \mathrm{Ch}(u\left)\right\}$ and the recursive definition of ${Flat}_k$. The presence of additional edge sets $\left\{E^{(k\to \ell )}\right\}$ does not affect the labeling argument. □

Theorem 5

(Level supports partition $V_0$). Fix $k\in \{0,\dots ,n\}$. Then the family

$${\mathcal{L}}_k:=\{\lambda \left(u\right)\mid u\in {\mathcal{N}}_k\}$$

is a partition of $V_0$:

$$V_0=\bigcup _{u\in {\mathcal{N}}_k}\lambda \left(u\right),u\ne v\Rightarrow \lambda \left(u\right)\cap \lambda \left(v\right)=\varnothing (u,v\in {\mathcal{N}}_k).$$

Equivalently, for every $x\in V_0$ there exists a unique $u\in {\mathcal{N}}_k$ with $x\in \lambda \left(u\right)$, and thus ${\pi }_k$ in Notation 2 is well-defined.

Proof. 

Let $x\in V_0$ and let ${\ell }_x$ be its unique leaf. The unique root-to-${\ell }_x$ path contains exactly one node u at depth k. Then ${\ell }_x\in \mathcal{L}\left(u\right)$, so by Theorem 4 we have $x=\eta \left({\ell }_x\right)\in \lambda \left(u\right)$, proving the covering.

For disjointness, let $u,v\in {\mathcal{N}}_k$ with $u\ne v$ and suppose $\lambda \left(u\right)\cap \lambda \left(v\right)\ne \varnothing$. Choose x in the intersection. By Theorem 4 there exist leaves $\ell \in \mathcal{L}\left(u\right)$ and ${\ell }^{\prime }\in \mathcal{L}\left(v\right)$ with $\eta (\ell )=x=\eta \left({\ell }^{\prime }\right)$. Injectivity of ${\eta |}_{{\mathcal{N}}_0}$ implies $\ell ={\ell }^{\prime }$. But a root-to-leaf path cannot pass through two distinct nodes at the same depth, contradiction. Hence the supports are disjoint. □

Theorem 6

(Laminarity of supports across all tree-vertices). For any $u,v\in \mathcal{N}$, exactly one of the following holds:

$$\lambda \left(u\right)\cap \lambda \left(v\right)=\varnothing ,\lambda \left(u\right)\subseteq \lambda \left(v\right),\lambda \left(v\right)\subseteq \lambda \left(u\right).$$

Moreover,

$$\lambda \left(u\right)\cap \lambda \left(v\right)\ne \varnothing ⟺(u⪯v)\mathrm{or}(v⪯u).$$

Proof. 

If $u⪯v$, then $\mathcal{L}\left(u\right)\subseteq \mathcal{L}\left(v\right)$, hence

$$\lambda \left(u\right)=\left\{\eta \right(\ell )\mid \ell \in \mathcal{L}(u\left)\right\}\subseteq \left\{\eta \right(\ell )\mid \ell \in \mathcal{L}(v\left)\right\}=\lambda \left(v\right)$$

by Theorem 4. Symmetrically for $v⪯u$.

If neither is an ancestor of the other, the subtrees are disjoint: $\mathcal{L}\left(u\right)\cap \mathcal{L}\left(v\right)=\varnothing$. Then, using injectivity of ${\eta |}_{{\mathcal{N}}_0}$,

$$\lambda \left(u\right)\cap \lambda \left(v\right)=\left\{\eta \right(\ell )\mid \ell \in \mathcal{L}(u\left)\right\}\cap \left\{\eta \right(\ell )\mid \ell \in \mathcal{L}(v\left)\right\}=\varnothing .$$

This yields the trichotomy and the equivalence. □

Definition 10

(Cross-edge expansion to a base relation). Fix $0\le k<\ell \le n$. For $(u,v)\in {\mathcal{N}}_k\times {\mathcal{N}}_{\ell }$ define its base expansion as the set

$$\mathrm{Exp}(u,v):=\lambda \left(u\right)\times \lambda \left(v\right)\subseteq V_0\times V_0.$$

Given a cross-edge set $E^{(k\to \ell )}\subseteq {\mathcal{N}}_k\times {\mathcal{N}}_{\ell }$, define the induced base relation

$$R_0^{(k\to \ell )}:=\bigcup _{(u,v)\in E^{(k\to \ell )}}\mathrm{Exp}(u,v)\subseteq V_0\times V_0.$$

Theorem 7

(Uniqueness of lifted endpoints for induced base pairs). Fix $0\le k<\ell \le n$ and define $R_0^{(k\to \ell )}$ as in Definition 10. If $(x,y)\in R_0^{(k\to \ell )}$, then the nodes

$$u={\pi }_k\left(x\right)\in {\mathcal{N}}_k,v={\pi }_{\ell }\left(y\right)\in {\mathcal{N}}_{\ell }$$

are uniquely determined and satisfy $(u,v)\in E^{(k\to \ell )}$. Conversely, if $(u,v)\in E^{(k\to \ell )}$ and $x\in \lambda \left(u\right)$, $y\in \lambda \left(v\right)$, then $(x,y)\in R_0^{(k\to \ell )}$.

Proof. 

Assume $(x,y)\in R_0^{(k\to \ell )}$. Then by definition there exists $(u^{\prime },v^{\prime })\in E^{(k\to \ell )}$ such that $x\in \lambda \left(u^{\prime }\right)$ and $y\in \lambda \left(v^{\prime }\right)$. By Theorem 5 applied at level k and at level ℓ, the level containers are unique:

$$u^{\prime }={\pi }_k\left(x\right),v^{\prime }={\pi }_{\ell }\left(y\right).$$

Hence $({\pi }_k\left(x\right),{\pi }_{\ell }\left(y\right))\in E^{(k\to \ell )}$, and uniqueness is immediate from the partition property.

Conversely, if $(u,v)\in E^{(k\to \ell )}$ and $x\in \lambda \left(u\right)$, $y\in \lambda \left(v\right)$, then $(x,y)\in \lambda \left(u\right)\times \lambda \left(v\right)=\mathrm{Exp}(u,v)\subseteq R_0^{(k\to \ell )}$. □

Theorem 8

(Support-compatibility implies monotone containment on base pairs). Fix $0\le k<\ell \le n$ and suppose the following upward support-compatibility holds:

$$\forall (u,v)\in E^{(k\to \ell )}:\lambda \left(u\right)\subseteq \lambda \left(v\right).$$

Then the induced base relation $R_0^{(k\to \ell )}$ satisfies:

$$(x,y)\in R_0^{(k\to \ell )}⟹x\in \lambda \left({\pi }_{\ell }\left(y\right)\right).$$

Equivalently, every base pair $(x,y)$ induced by a cross-edge points from a base vertex x that lies inside the (unique) level-ℓ cluster containing y.

Proof. 

Take $(x,y)\in R_0^{(k\to \ell )}$. By Theorem 7, $({\pi }_k\left(x\right),{\pi }_{\ell }\left(y\right))\in E^{(k\to \ell )}$. By the assumed compatibility,

$$\lambda \left({\pi }_k\left(x\right)\right)\subseteq \lambda \left({\pi }_{\ell }\left(y\right)\right).$$

Since $x\in \lambda \left({\pi }_k\left(x\right)\right)$ by definition of ${\pi }_k$, we conclude $x\in \lambda \left({\pi }_{\ell }\left(y\right)\right)$. □

Theorem 9

(No cross-edges between disjoint supports under overlap-compatibility). Fix $0\le k<\ell \le n$ and assume the overlap-compatibility constraint:

$$\forall (u,v)\in E^{(k\to \ell )}:\lambda \left(u\right)\cap \lambda \left(v\right)\ne \varnothing .$$

Then every cross-edge connects comparable nodes in the tree:

$$(u,v)\in E^{(k\to \ell )}⟹u⪯v.$$

In particular, under overlap-compatibility, cross-edges cannot jump between disjoint branches.

Proof. 

Let $(u,v)\in E^{(k\to \ell )}$. By assumption $\lambda \left(u\right)\cap \lambda \left(v\right)\ne \varnothing$. By Theorem 6, this implies $u⪯v$ or $v⪯u$. But $\mathrm{depth}\left(u\right)=k<\ell =\mathrm{depth}\left(v\right)$, so $v⪯u$ is impossible (a descendant must have smaller depth). Hence $u⪯v$. □

Corollary 2

(Under overlap-compatibility, cross-edges are determined by ancestor constraints). Assume overlap-compatibility as in Theorem 9. Then for each $0\le k<\ell \le n$,

$$E^{(k\to \ell )}\subseteq \{(u,v)\in {\mathcal{N}}_k\times {\mathcal{N}}_{\ell }\mid u⪯v\}.$$

Moreover, for any $(u,v)\in E^{(k\to \ell )}$ one has $\lambda \left(u\right)\subseteq \lambda \left(v\right)$.

Proof. 

The containment is exactly Theorem 9. If $u⪯v$, then $\mathcal{L}\left(u\right)\subseteq \mathcal{L}\left(v\right)$ and thus $\lambda \left(u\right)\subseteq \lambda \left(v\right)$ by Theorem 4. □

3. Conclusions

In this paper, we defined a new class of graphs called Tree-Vertex Graphs. We expect that future work will explore extensions based on Fuzzy Sets [26], Neutrosophic Sets [27,28], and Plithogenic Sets [29,30], as well as applications to methods such as neural networks.