On MultiLine Graphs, MultiLine HyperGraphs, and MultiLine Super-HyperGraphs

1. Preliminaries

This section establishes the notation used throughout the paper and reviews the basic set-theoretic and combinatorial concepts needed in later sections.

1.1. SuperHyperGraphs

We begin with the set-theoretic operations underlying SuperHyperGraphs and then recall the relevant graph-theoretic definitions.

Definition 1 

(Base set). A base set S is the underlying universe of admissible objects in the setting under consideration, i.e.,

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

Accordingly, every element of $\mathcal{P}\left(S\right)$—and, more generally, of any iterated powerset—is ultimately formed from elements of S.

Definition 2 

(Powerset). (see [1]) For a set S, its powerset is the family of all subsets of S:

$$\mathcal{P}\left(S\right)=\{A\mid A\subseteq S\}.$$

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

Definition 3 

(Hypergraph). [2,3] A hypergraph is an ordered pair $H=(V,E)$ where

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

Thus a hyperedge may contain more than two vertices, allowing one to model genuinely multiway relations.

Definition 4 

(n-fold iterated powerset). [4,5] Let X be a set. Define ${\mathcal{P}}^1\left(X\right):=\mathcal{P}\left(X\right)$ and, for $n\ge 1$, set recursively

$${\mathcal{P}}^{n+1}\left(X\right):=\mathcal{P}\left({\mathcal{P}}^n\left(X\right)\right).$$

When excluding the empty set, we write

$${\mathcal{P}}_{*}^n\left(X\right):={\mathcal{P}}^n\left(X\right)\setminus \{\varnothing \}.$$

Definition 5 

(n-SuperHyperGraph). (see [6,7,8]) Let $V_0$ be a finite, nonempty base set, and define the iterated powersets by

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

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

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

satisfying

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

Elements of V are calledn-supervertices, and elements of E are calledsuperhyperedges; equivalently, each superhyperedge is a nonempty subset of the supervertex set V.

1.2. Line SuperHyperGraph

A line hypergraph replaces each hyperedge by a vertex; each original vertex yields a hyperedge of all incident hyperedges [9,10,11,12,13,14]. A line SuperHyperGraph makes superhyperedges vertices; each supervertex induces a hyperedge consisting of all recursively incident superhyperedges [15,16].

Definition 6 

(Line HyperGraph). (cf. [9,14]) Let $H=(V,E)$ be a (finite) hypergraph, i.e.,

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

For each vertex $v\in V$, define the star of v in H by

$${Star}_H\left(v\right):=\{e\in E\mid v\in e\}\subseteq E.$$

The line hypergraph of H is the hypergraph

$$L\left(H\right)=(V^L,E^L),$$

where

$$V^L:=E,E^L:=\left\{{Star}_H\left(v\right)\subseteq E\mid v\in V,{Star}_H\left(v\right)\ne \varnothing \right\}.$$

Thus the vertices of $L\left(H\right)$ are the hyperedges of H, and each original vertex $v\in V$ induces a hyperedge of $L\left(H\right)$ consisting of all hyperedges of H incident to v.

Definition 7 

(Line SuperHyperGraph). [15,16,17] Let $V_0$ be a finite nonempty base set and let $n\in \mathbb{N}\cup \left\{0\right\}$. Let

$$H^{\left(n\right)}=(V_n,E)$$

be a level-n SuperHyperGraph, i.e.,

$$V_n\subseteq {\mathcal{P}}^n\left(V_0\right)\mathrm{and}\varnothing \ne E\subseteq \mathcal{P}\left(V_n\right)\setminus \{\varnothing \}.$$

For each $v\in V_n$, define the star of v in $H^{\left(n\right)}$ by

$${Star}_H\left(v\right):=\{e\in E\mid v\in e\}\subseteq E.$$

The line SuperHyperGraph of $H^{\left(n\right)}$ is the pair

$$L\left(H^{\left(n\right)}\right)=(V_{n+1}^L,E_{n+1}^L),$$

defined by

$$V_{n+1}^L:=E,E_{n+1}^L:=\left\{{Star}_H\left(v\right)\subseteq E\mid v\in V_n,{Star}_H\left(v\right)\ne \varnothing \right\}.$$

Equivalently, $L\left(H^{\left(n\right)}\right)$ has one $(n+1)$-supervertex for each superhyperedge of $H^{\left(n\right)}$, and for every n-supervertex $v\in V_n$ it has a superhyperedge consisting of all superhyperedges of $H^{\left(n\right)}$ that contain v.

Remark 1 

(Level shift). Since $V_n\subseteq {\mathcal{P}}^n\left(V_0\right)$, we have

$$V_{n+1}^L=E\subseteq \mathcal{P}\left(V_n\right)\subseteq \mathcal{P}\left({\mathcal{P}}^n\left(V_0\right)\right)={\mathcal{P}}^{n+1}\left(V_0\right),$$

so the vertex set of $L\left(H^{\left(n\right)}\right)$ naturally lives at level $n+1$. Moreover, each element of $E_{n+1}^L$ is a nonempty subset of $V_{n+1}^L$ by construction.

2. Main Results

In this section, we present our main results.

2.1. MultiLine Graph

In an ordinary graph, adjacency is recorded only at the vertex level. By contrast, in a MultiLine Graph each vertex is equipped with several distinguishable local lines, and an edge is formed by pairing two such local lines. This provides a vertex-centered framework in which one can explicitly represent the fact that a vertex may carry several incident lines.

Definition 8 

(MultiLine Graph). Let V be a finite nonempty set. Let $\mathcal{L}$ be a finite set, and let

$$\pi :\mathcal{L}⟶V$$

be a surjective map. For each $v\in V$, define

$$\mathcal{L}\left(v\right):={\pi }^{-1}\left(v\right),$$

whose elements are called the local lines attached to v.

A MultiLine Graph is a triple

$$\mathcal{G}=(V,\mathcal{L},E)$$

satisfying the following conditions:

(i)

E is a set of 2-element subsets of $\mathcal{L}$, i.e.,

$$E\subseteq \left\{\{\ell ,{\ell }^{\prime }\}\subseteq \mathcal{L}|\left|\{\ell ,{\ell }^{\prime }\}\right|=2\right\};$$

(ii)

for every $\{\ell ,{\ell }^{\prime }\}\in E$, one has

$$\pi (\ell )\ne \pi \left({\ell }^{\prime }\right);$$

thus each edge joins local lines belonging to two distinct vertices;

(iii)

if $e_1,e_2\in E$ and $e_1\ne e_2$, then

$$e_1\cap e_2=\varnothing .$$

Hence each local line is used by at most one edge.

The elements of E are called multiline edges. For each vertex $v\in V$, the number

$$m\left(v\right):=\left|\mathcal{L}\right(v\left)\right|$$

is called the line multiplicity of v.

Remark 2. 

A MultiLine Graph is not the same as the classical line graph. Here the word “multi-line” means that each vertex carries several distinguishable local lines. In this sense, the structure is closer to a half-edge model with explicit vertex-wise line multiplicities.

Definition 9 

(Incident edge set and degree). Let $\mathcal{G}=(V,\mathcal{L},E)$ be a MultiLine Graph. For each $v\in V$, define the incident edge set of v by

$${Inc}_{\mathcal{G}}\left(v\right):=\{e\in E\mid e\cap \mathcal{L}\left(v\right)\ne \varnothing \}.$$

The degree of v in $\mathcal{G}$ is

$${deg}_{\mathcal{G}}\left(v\right):=\left|{Inc}_{\mathcal{G}}\left(v\right)\right|.$$

Proposition 1. 

Let $\mathcal{G}=(V,\mathcal{L},E)$ be a MultiLine Graph. Then, for every $v\in V$,

$${deg}_{\mathcal{G}}\left(v\right)\le m\left(v\right).$$

Proof. 

Fix $v\in V$. By Definition 8, distinct edges of E are pairwise disjoint. Therefore, if an edge $e\in E$ is incident with v, then it uses a unique local line from $\mathcal{L}\left(v\right)$. Since no local line can belong to two different edges, the number of edges incident with v cannot exceed the number of local lines attached to v. Hence

$${deg}_{\mathcal{G}}\left(v\right)\le \left|\mathcal{L}\left(v\right)\right|=m\left(v\right).$$

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Definition 10 

(Saturated and proper MultiLine Graphs). A MultiLine Graph $\mathcal{G}=(V,\mathcal{L},E)$ is calledsaturatedif

$$\bigcup _{e\in E}e=\mathcal{L},$$

that is, every local line is used by some multiline edge.

A saturated MultiLine Graph is called proper if

$$m\left(v\right)\ge 2\mathrm{for}\mathrm{all}v\in V.$$

Thus, in a proper MultiLine Graph, every vertex carries at least two local lines.

Proposition 2. 

If $\mathcal{G}=(V,\mathcal{L},E)$ is saturated, then for every $v\in V$,

$${deg}_{\mathcal{G}}\left(v\right)=m\left(v\right).$$

In particular, if $\mathcal{G}$ is proper, then every vertex is incident with at least two edges:

$${deg}_{\mathcal{G}}\left(v\right)\ge 2(v\in V).$$

Proof. 

Assume that $\mathcal{G}$ is saturated. Then every local line in $\mathcal{L}\left(v\right)$ belongs to some edge of E. Because distinct edges are disjoint, each local line of $\mathcal{L}\left(v\right)$ contributes to exactly one incident edge of v. Hence the number of incident edges at v is precisely the number of local lines attached to v, namely

$${deg}_{\mathcal{G}}\left(v\right)=\left|\mathcal{L}\left(v\right)\right|=m\left(v\right).$$

If, moreover, $\mathcal{G}$ is proper, then $m\left(v\right)\ge 2$ for all $v\in V$, and therefore

$${deg}_{\mathcal{G}}\left(v\right)=m\left(v\right)\ge 2.$$

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2.2. MultiLine HyperGraph

A MultiLine HyperGraph extends the MultiLine Graph framework by allowing one multiline edge to connect more than two local lines simultaneously. Thus, each vertex may carry several distinguishable local lines, and a multiline hyperedge is formed by selecting a finite set of such local lines attached to pairwise distinct vertices.

Definition 11 

(MultiLine HyperGraph). Let V be a finite nonempty set, let $\mathcal{L}$ be a finite set, and let

$$\pi :\mathcal{L}⟶V$$

be a surjective map. For each vertex $v\in V$, define

$$\mathcal{L}\left(v\right):={\pi }^{-1}\left(v\right),$$

whose elements are called the local lines attached to v.

A MultiLine HyperGraph is a triple

$$\mathcal{H}=(V,\mathcal{L},E)$$

such that the following conditions hold:

(i)

E is a finite set of subsets of $\mathcal{L}$ satisfying

$$E\subseteq \left\{F\subseteq \mathcal{L}\mid \left|F\right|\ge 2\right\};$$

(ii)

for every $F\in E$, the restriction

$${\pi |}_F:F⟶V$$

is injective; equivalently, if $\ell ,{\ell }^{\prime }\in F$ and $\ell \ne {\ell }^{\prime }$, then

$$\pi (\ell )\ne \pi \left({\ell }^{\prime }\right);$$

(iii)

distinct multiline hyperedges are pairwise disjoint, i.e., if $F_1,F_2\in E$ and $F_1\ne F_2$, then

$$F_1\cap F_2=\varnothing .$$

The elements of E are called multiline hyperedges. For each $v\in V$, the number

$$m\left(v\right):=\left|\mathcal{L}\right(v\left)\right|$$

is called the line multiplicity of v.

Remark 3. 

Condition (ii) means that a single multiline hyperedge cannot use two local lines attached to the same vertex. Hence each multiline hyperedge determines a genuine multiway relation among distinct vertices. Condition (iii) ensures that each local line participates in at most one multiline hyperedge.

Definition 12 

(Incident multiline hyperedges and degree). Let $\mathcal{H}=(V,\mathcal{L},E)$ be a MultiLine HyperGraph. For each $v\in V$, define

$${Inc}_{\mathcal{H}}\left(v\right):=\{F\in E\mid F\cap \mathcal{L}\left(v\right)\ne \varnothing \}.$$

The degree of v in $\mathcal{H}$ is

$${deg}_{\mathcal{H}}\left(v\right):=\left|{Inc}_{\mathcal{H}}\left(v\right)\right|.$$

Proposition 3. 

Let $\mathcal{H}=(V,\mathcal{L},E)$ be a MultiLine HyperGraph. Then, for every $v\in V$,

$${deg}_{\mathcal{H}}\left(v\right)\le m\left(v\right).$$

Proof. 

Fix $v\in V$. Since distinct multiline hyperedges are pairwise disjoint, each local line in $\mathcal{L}\left(v\right)$ can belong to at most one multiline hyperedge. Therefore the number of multiline hyperedges incident with v cannot exceed the number of local lines attached to v. Hence

$${deg}_{\mathcal{H}}\left(v\right)\le \left|\mathcal{L}\left(v\right)\right|=m\left(v\right).$$

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Definition 13 

(Saturated MultiLine HyperGraph). A MultiLine HyperGraph $\mathcal{H}=(V,\mathcal{L},E)$ is called saturated if

$$\bigcup _{F\in E}F=\mathcal{L}.$$

Equivalently, every local line is used by some multiline hyperedge.

Proposition 4. 

If $\mathcal{H}=(V,\mathcal{L},E)$ is saturated, then for every $v\in V$,

$${deg}_{\mathcal{H}}\left(v\right)=m\left(v\right).$$

Proof. 

Assume that $\mathcal{H}$ is saturated. Then every local line in $\mathcal{L}\left(v\right)$ belongs to some multiline hyperedge. Since distinct multiline hyperedges are disjoint, each local line of $\mathcal{L}\left(v\right)$ belongs to exactly one multiline hyperedge, and distinct local lines of $\mathcal{L}\left(v\right)$ cannot belong to the same multiline hyperedge by Definition 11(ii). Therefore the number of multiline hyperedges incident with v is exactly $\left|\mathcal{L}\right(v\left)\right|$. Hence

$${deg}_{\mathcal{H}}\left(v\right)=m\left(v\right).$$

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2.3. MultiLine SuperHyperGraph

A MultiLine SuperHyperGraph extends the previous construction from ordinary vertices to higher-level supervertices. In this setting, each n-supervertex carries several distinguishable local lines, and each multiline superhyperedge is a finite set of local lines attached to pairwise distinct n-supervertices.

Definition 14 

(MultiLine n-SuperHyperGraph). Let $V_0$ be a finite nonempty base set, let $n\in \mathbb{N}\cup \left\{0\right\}$, and let

$$V_n\subseteq {\mathcal{P}}^n\left(V_0\right).$$

Let $\mathcal{L}$ be a finite set, and let

$$\pi :\mathcal{L}⟶V_n$$

be a surjective map. For each $x\in V_n$, define

$$\mathcal{L}\left(x\right):={\pi }^{-1}\left(x\right),$$

whose elements are called the local lines attached to the n-supervertex x.

A MultiLine n-SuperHyperGraph is a triple

$${\mathcal{S}}^{\left(n\right)}=(V_n,\mathcal{L},E)$$

satisfying the following conditions:

(i)

E is a finite set of subsets of $\mathcal{L}$ satisfying

$$E\subseteq \left\{F\subseteq \mathcal{L}\mid \left|F\right|\ge 2\right\};$$

(ii)

for every $F\in E$, the restriction

$${\pi |}_F:F⟶V_n$$

is injective; equivalently, if $\ell ,{\ell }^{\prime }\in F$ and $\ell \ne {\ell }^{\prime }$, then

$$\pi (\ell )\ne \pi \left({\ell }^{\prime }\right);$$

(iii)

distinct multiline superhyperedges are pairwise disjoint, i.e., if $F_1,F_2\in E$ and $F_1\ne F_2$, then

$$F_1\cap F_2=\varnothing .$$

The elements of E are called multiline superhyperedges. For each $x\in V_n$, the number

$$m\left(x\right):=\left|\mathcal{L}\right(x\left)\right|$$

is called the line multiplicity of the n-supervertex x.

Remark 4. 

A MultiLine n-SuperHyperGraph is built over the level-n supervertex set $V_n\subseteq {\mathcal{P}}^n\left(V_0\right)$. Thus the local lines do not replace the n-supervertices; rather, they refine their incidence structure by allowing each n-supervertex to possess several distinguishable attachment slots.

Definition 15 

(Incident multiline superhyperedges and degree). Let ${\mathcal{S}}^{\left(n\right)}=(V_n,\mathcal{L},E)$ be a MultiLine n-SuperHyperGraph. For each $x\in V_n$, define

$${Inc}_{{\mathcal{S}}^{\left(n\right)}}\left(x\right):=\{F\in E\mid F\cap \mathcal{L}\left(x\right)\ne \varnothing \}.$$

The degree of x in ${\mathcal{S}}^{\left(n\right)}$ is

$${deg}_{{\mathcal{S}}^{\left(n\right)}}\left(x\right):=\left|{Inc}_{{\mathcal{S}}^{\left(n\right)}}\left(x\right)\right|.$$

Proposition 5. 

Let ${\mathcal{S}}^{\left(n\right)}=(V_n,\mathcal{L},E)$ be a MultiLine n-SuperHyperGraph. Then, for every $x\in V_n$,

$${deg}_{{\mathcal{S}}^{\left(n\right)}}\left(x\right)\le m\left(x\right).$$

Proof. 

The proof is identical to that of Proposition 3. Because distinct multiline superhyperedges are pairwise disjoint, each local line attached to x can belong to at most one multiline superhyperedge. Therefore the number of multiline superhyperedges incident with x cannot exceed the number of local lines attached to x. Hence

$${deg}_{{\mathcal{S}}^{\left(n\right)}}\left(x\right)\le \left|\mathcal{L}\left(x\right)\right|=m\left(x\right).$$

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Definition 16 

(Saturated MultiLine n-SuperHyperGraph). A MultiLine n-SuperHyperGraph ${\mathcal{S}}^{\left(n\right)}=(V_n,\mathcal{L},E)$ is calledsaturatedif

$$\bigcup _{F\in E}F=\mathcal{L}.$$

Proposition 6. 

If ${\mathcal{S}}^{\left(n\right)}=(V_n,\mathcal{L},E)$ is saturated, then for every $x\in V_n$,

$${deg}_{{\mathcal{S}}^{\left(n\right)}}\left(x\right)=m\left(x\right).$$

Proof. 

The proof is the same as in the hypergraph case. Under saturation, every local line attached to x is used by some multiline superhyperedge; by pairwise disjointness of multiline superhyperedges and injectivity of ${\pi |}_F$ on each $F\in E$, this correspondence is one-to-one. Therefore

$${deg}_{{\mathcal{S}}^{\left(n\right)}}\left(x\right)=\left|\mathcal{L}\left(x\right)\right|=m\left(x\right).$$

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