**1. Introduction**

Being a generalization of graphs and yet having its own unique complexity and utility, hypergraph theory has emerged as a completely new dynamic research area. The fundamental concepts of path, tree, trail, cycle and their different well-known properties have already found plenty of applications in real-world problems in networking systems [1,2] of different types or in the field of bioinformatics [3–5]. The concept of the *hyperpath*, called also the path (both terms being used in a synonymous way), in a hypergraph represents the foundation of many research works. In the majority of these studies, the hypergraphs are considered to be directed, though there are papers related to paths in the case of undirected hypergraphs as well. Nguyen and Pallottino [6], in their work based on directed hypergraphs, have given some efficient algorithms in connection to some shortest path properties. In the same direction, we recall the work of Nielsen, Andersen and Pretolani [7], where the authors present the procedures for finding the *K*-shortest hyperpaths in a directed hypergraph. It is worth underlining that the area of research related to hyperpaths, shortest hyperpaths [6] and their links with vehicle navigation [1], network systems based on transit schedules [2], cellular networks [3], etc., is flourishing.

In this paper, we deal with two different problems related to hypergraphs. One concerns the behavior of hyperpaths under hyper-continuous mappings and pseudo-open mappings, while the other one is related to hyperpaths and hypertrees. Our study was motivated by the definition of the so-called *algebraic space* [8], introduced as a pair (*X*, *SX*), where *X* is a non-empty arbitrary set and *SX* ⊆ P(*X*) a non-empty family of subsets of *X*. An algebraic space can be seen as an extended version of a topological space but without having any closure property with respect to union or intersection, and it recalls the definition of the hypergraph to a great extent. As a result, the concept of pseudo-map or pseudo-continuity could be then defined between two hypergraphs. The key element of this parallel study is the new concept of the *knot*, which is a subset of hyperedge intersection

**Citation:** Rahman, S.; Chowdhury, M.; A., F.; Cristea, I. Knots and Knot-Hyperpaths in Hypergraphs. *Mathematics* **2022**, *10*, 424. https:// doi.org/10.3390/math10030424

Academic Editor: Mikhail Goubko

Received: 21 December 2021 Accepted: 27 January 2022 Published: 28 January 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

vertices. Since, in a hypergraph, the hyperedges appear as some subsets of the vertex set, it is trivial to note that the intersections of all possible adjacent hyperedges may contain more than one vertex. This fact leads to the intuitive notion of the knot that is the collection of explicit vertices. This notion further changes the dimension of perceiving the different concepts of hypergraphs such as walk, trail, path, tree, etc., where each of the adjacent hyperedge intersections gives rise to knots.

In graph theory, another important concept is that of the tree, which has been extensively used in networking, especially in theoretical computer science [9]. A graph *G* is a tree if there exists a unique path between any two vertices. Recall that the concept of the hypertree was introduced in hypergraph theory in terms of its host graph, as the hypergraph that admits a host graph that is a tree [10]. We emphasize that this fundamental characterization of trees is not generalized in hypergraph theory, in the sense that there is no characterization of hypertrees merely in terms of hyperpaths. This motivated us to present, in the second part of the paper, a characteristic of hypertrees in terms of hyperpaths, without using the concept of the host graph.

The structure of this work can be summarized as follows. First, in Section 2, we introduce the new concepts of point-hyperwalk, point-hypertrail and point-hyperpath, showing their differences in one illustrative example. Next, the key concepts of the knot and knot-hyperpath are defined. In Section 3, the notions of the hyper-continuous map, strictly hyper-continuous map and pseudo-open map between two hypergraphs are introduced and the behavior of point-hyperpaths and knot-hyperpaths under these notions is observed. In particular, we prove that the image of a point-hyperpath under an injective pseudo-open mapping is a point-hyperpath, while the image of a knot-hyperpath under a pseudoopen map is again a knot-hyperpath. Regarding the inverse image, we show that the inverse image of a knot-hyperpath under a surjective hyper-continuous map is a weak knot-hyperpath, or a knot-hyperpath if the map is surjective and strictly hyper-continuous. Section 4 is dedicated to the study of hypertrees. Based on the concept of equivalent entire knot-hyperpaths, we establish a sufficient condition under which a hypergraph becomes a hypertree. Moreover, we present an algorithm that extracts a host graph from a hypertree. A concluding section ends our study.

#### **2. Preliminaries**

Many definitions of hypergraphs exist; here, we will adopt the original one, given by Berge [11]. A *hypergraph* is a couple *H* = (*V*, *E*) defined by a finite set of *vertices* (called also *nodes*) *V* = {*v*1, ... , *vn*}, with *n* ∈ N, and the set *E* = {*Ei*}*i*∈<sup>N</sup> of non-empty subsets of *V*, called *hyperedges*. Two hyperedges *Ej*, *Ek* ∈ *E*, with *j* = *k*, such that *Ej* = *Ek* are called *repeated hyperedges* [12]. In this paper, all hypergraphs are considered to be with no repeated hyperedges.

**Definition 1** ([13])**.** *Let H* = (*V*, *E*) *be a hypergraph. By a hyperpath between two distinct vertices v*<sup>1</sup> *and vk in V, we mean a sequence v*1*E*1*v*2*E*<sup>2</sup> ... *vk*<sup>−</sup>1*Ek*<sup>−</sup>1*vk of vertices and hyperedges having the following properties:*


**Definition 2** ([13])**.** *A hypercycle in a hypergraph H* = (*V*, *E*) *on a vertex v*<sup>1</sup> *is a sequence v*1*E*1*v*2*E*<sup>2</sup> ... *vk*<sup>−</sup>1*Ek*<sup>−</sup>1*vkEkv*1*, having the following properties:*


It is important to note that a path in a graph does not contain repeated edges, while this property is not retained in the definition of a hyperpath in a hypergraph as it appears in Definition 1. Since, in some cases, it is necessary to distinguish this special case; we define the following types of hyperpaths.

**Definition 3.** *A point-hyperwalk in a hypergraph H* = (*V*, *E*) *is a hyperpath as defined in Definition 1, where the vertices may be repeated. A point-hyperwalk where no hyperedge is repeated (but vertices may be repeated) is called a point-hypertrail. A point-hyperpath is a point-hypertrail in which vertices are not repeated.*

In other words, a point-hyperpath is a point-hyperwalk where neither the edges nor the vertices are repeated.

The above definitions are illustrated in the following example.

**Example 1.** *Let H* = (*V*, *E*) *be a hypergraph with the vertex set V* = {*vi*|*i* = 1, 2, ... , 50} *and hyperedges E* = {*E*1, *E*2,..., *E*10} *such that*

*E*<sup>1</sup> = {*v*1, *v*2, *v*3, *v*4, *v*5, *v*47, *v*9, *v*10}, *E*<sup>2</sup> = {*v*12, *v*11, *v*15, *v*9, *v*10, *v*8, *v*6, *v*16, *v*7}, *E*<sup>3</sup> = {*v*11, *v*12, *v*15, *v*13, *v*46, *v*14, *v*30}, *E*<sup>4</sup> = {*v*14, *v*30, *v*31, *v*34, *v*33}, *E*<sup>5</sup> = {*v*14, *v*30, *v*20, *v*32, *v*22, *v*21}, *E*<sup>6</sup> = {*v*21, *v*22, *v*44, *v*48, *v*49, *v*50, *v*43}, *E*<sup>7</sup> = {*v*50, *v*43, *v*41, *v*42, *v*36, *v*45, *v*37}, *E*<sup>8</sup> = {*v*36, *v*45, *v*37, *v*46, *v*40, *v*27, *v*29}, *E*<sup>9</sup> = {*v*28, *v*23, *v*24, , *v*27, *v*29, *v*25, *v*26}, *E*<sup>10</sup> = {*v*16, *v*7, *v*18, *v*17, *v*28, *v*23, *v*24}*.*

*We represent this hypergraph in Figure 1.*

**Figure 1.** Hypergraph explaining point-hyperwalk, point-hypertrail and point-hyperpath notions.

*We notice that*


Suppose that *H* = (*V*, *E*) and *H* = (*V* , *E* ) are two hypergraphs. Let *f* : *V* → *V* be a mapping and let *P* ≡ *v*1*E*1*v*2*E*<sup>2</sup> ... *vk*<sup>−</sup>1*Ek*<sup>−</sup>1*vk* denote an alternating sequence of vertices and edges in the hypergraph *H*. Then, we denote the *f-image* of this sequence as *f*(*P*) ≡ *f*(*v*1)*f*(*E*1)*f*(*v*2)*f*(*E*2)... *f*(*vk*<sup>−</sup>1)*f*(*Ek*<sup>−</sup>1)*f*(*vk*), where *f*(*Ei*), *i* = 1, 2, ... , *k* is the *f*-image of *Ei*, *i* = 1, 2, . . . , *k*, respectively.

Generalizing the notions in Definition 1, we are ready to introduce the concepts of the knot and knot-hyperpath, where the vertices are replaced by a cluster of vertices, each of them behaving in a significant manner.

**Definition 4.** *A knot K in a hypergraph H* = (*V*, *E*) *is a non-empty subset of the intersections of some intersecting hyperedges. In other words, if H* = (*V*, *E*) *is a hypergraph and K is a knot, then K*( = ∅) ⊆ ∩*Ei for some intersecting hyperedges Ei*, *i* = 1, 2, ... , *k and k* ≥ 2*. In particular, if K* = ∩*Ei, then K is called an entire knot.*

**Definition 5.** *A knot-hyperpath in a hypergraph H* = (*V*, *E*) *between two vertices v*<sup>1</sup> *and vn is an alternating sequence of knots and hyperedges of the following type:*

$$\{\upsilon\_1\} E\_1 K\_1 E\_2 K\_2 E\_3 \dots E\_{n-1} K\_{n-1} E\_n \{\upsilon\_n\} \,. \tag{1}$$

*where Ki* <sup>⊆</sup> (*Ei* <sup>∩</sup> *Ei*+1)\(∪*i*−<sup>1</sup> *<sup>t</sup>*=1*Kt*)*, with <sup>i</sup>* <sup>=</sup> 1, ... , *<sup>n</sup>* <sup>−</sup> <sup>1</sup>*, <sup>v</sup>*<sup>1</sup> <sup>∈</sup> *<sup>E</sup>*1, *vn* <sup>∈</sup> *En and Eis are distinct hyperedges.*

*If Ki* = *Ei* ∩ *Ei*+<sup>1</sup> *for all i* = 1, 2, ... , *n* − 1*, then the knot-hyperpath is called the entire knot-hyperpath.*

Although the entire knot-hyperpath is a particular case of the knot-hyperpath, its significance can be seen in Section 4.

From the constructions of knots, it is clear that knots are mutually disjointed. Here, *n* is called the *length of the knot-hyperpath*.

**Example 2.** *By taking the hypergraph defined in Example 1, we can observe that*

{*v*4}*E*1{*v*9, *v*10}*E*2{*v*11, *v*12, *v*15}*E*3{*v*14, *v*30}*E*5{*v*22}*E*6{*v*50}

*is a knot-hyperpath of length 5.*

**Definition 6.** *Two knot-hyperpaths*

$$P\_1 \equiv \{v\_1\} E\_1 K\_1 E\_2 K\_2 E\_3 \dots E\_{n-1} K\_{n-1} E\_n \{v\_n\},$$

*and*

$$P\_2 \equiv \{v\_1\} E\_1' K\_1' E\_2' K\_2' E\_3' \dots E\_{n-1}' K\_{n-1}' E\_n' \{v\_n\}.$$

*of the same length of a hypergraph H* = (*V*, *E*) *are called* equivalent *or* isomorphic *if*


The above definition further can be generalized to a finite number of knot-hyperpaths (entire knot-hyperpaths) *P*1, *P*2, ... , *Pk*, where *k* ≥ 2 and the intersections in items (i) and (ii) are taken as follows:

(i) *<sup>k</sup> <sup>j</sup>*=<sup>1</sup> *<sup>E</sup><sup>j</sup> i* = ∅ (ii) *<sup>k</sup> <sup>j</sup>*=<sup>1</sup> *<sup>K</sup><sup>j</sup> i* = ∅ for all *i* = 1, 2, . . . , *n* − 1.

**Example 3.** *Consider the hypergraph H, with the vertex set*

*V* = {*v*1, *v*2, *v*3, *v*4, *v*5, *v*6, *v*7, *v*8, *v*9, *v*10, *v*11, *v*12, *v*13}

*and the hyperedges E*<sup>1</sup> = {*v*1, *v*2, *v*3, *v*4}, *E*<sup>2</sup> = {*v*3, *v*4, *v*5, *v*7, *v*9}, *E*<sup>3</sup> = {*v*2, *v*3, *v*4, *v*7, *v*8, *v*9}, *E*<sup>4</sup> = {*v*8, *v*9, *v*10, *v*11, *v*12}, *E*<sup>5</sup> = {*v*11, *v*12, *v*13}*. It can be easily verified that the following two knot-hyperpaths*

$$P\_1 \equiv \{v\_1\} E\_1\{v\_2, v\_3, v\_4\} E\_3\{v\_8, v\_9\} E\_4\{v\_{11}, v\_{12}\} E\_5\{v\_{13}\}$$

*and*

$$P\_2 \equiv \{v\_1\} E\_1 \{v\_{3\prime} v\_4\} E\_2 \{v\_9\} E\_4 \{v\_{11\prime} v\_{12}\} E\_5 \{v\_{13}\}$$

*are equivalent. We notice also that P*<sup>1</sup> *and P*<sup>2</sup> *are entire knot-hyperpaths, while*

$$P\_1' \equiv \{v\_1\} E\_1 \{v\_{2\prime}v\_{3\prime}v\_4\} E\_3 \{v\_{8\prime}v\_9\} E\_4 \{v\_{11}\} E\_5 \{v\_{13}\}$$

*and*

$$P\_2^{'} \equiv \{v\_1\} E\_1 \{v\_{3\prime} v\_4\} E\_2 \{v\_9\} E\_4 \{v\_{12}\} E\_5 \{v\_{13}\}$$

*are not equivalent because the last two knots of the knot-hyperpaths P* <sup>1</sup> *and P* <sup>2</sup> *have empty intersections.*

**Definition 7** ([8])**.** *A mapping f* : *V* → *V from the vertex set of a hypergraph H* = (*V*, *E*) *to the vertex set of another hypergraph K* = (*V* , *E* ) *is said to be pseudo-open (in short, ps-open) if, for each hyperedge Ei in E, the corresponding image f*(*Ei*) *is a hyperedge in E .*

**Example 4.** *Let H* = (*V*, *E*) *and K* = (*V* , *E* ) *be two hypergraphs with the vertex sets V* = {*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>v</sup>*4, *<sup>v</sup>*5, *<sup>v</sup>*6} *and <sup>V</sup>* = {*v* <sup>1</sup>, *v* <sup>2</sup>, *v* <sup>3</sup>, *v* <sup>4</sup>, *v* <sup>5</sup>} *and the hyperedge sets E* = {{*v*1, *v*2}, {*v*2, *v*3, *v*4}, {*v*3, *v*4, *v*5}}*, E* = {{*v* <sup>1</sup>}, {*v* <sup>2</sup>, *v* <sup>5</sup>}, {*v* <sup>1</sup>, *v* <sup>2</sup>, *v* <sup>5</sup>}}*, respectively. Define the map f* : *V* → *V such that f*(*v*1) = *v* <sup>1</sup> = *f*(*v*2)*, f*(*v*3) = *v* <sup>2</sup> = *f*(*v*5)*, f*(*v*4) = *v* <sup>5</sup>*, f*(*v*6) = *v* 3*. Then, f*({*v*1, *v*2}) = {*v* <sup>1</sup>}*, f*({*v*2, *v*3, *v*4}) = {*v* <sup>1</sup>, *v* <sup>2</sup>, *v* <sup>5</sup>}*, f*({*v*3, *v*4, *v*5}) = {*v* <sup>2</sup>, *v* <sup>5</sup>}*. Thus, for each Ei* ∈ *E, we have f*(*Ei*) ∈ *E . Hence, f is a ps-open mapping.*

**Definition 8.** *A hypergraph H* = (*V*, *E*) *is called connected if, for any two distinct vertices v*<sup>1</sup> *and v*2*, there exists a hyperpath joining v*<sup>1</sup> *and v*2*.*

**Definition 9.** *In a hypergraph H* = (*V*, *E*)*, a sequence*

{*v*1}*G*1*K*1*G*2*K*2*G*<sup>3</sup> ... *Gn*−1*Kn*−1*Gn*{*vn*}

*is called a weak knot-hypergraph if each Gi* <sup>⊃</sup> *Ei, (Ei* <sup>∈</sup> *E) with Ki* <sup>⊆</sup> (*Gi*−<sup>1</sup> <sup>∩</sup> *Gi*)\(∪*i*−<sup>1</sup> *<sup>t</sup>*=1*Kt*) *for all i* = 1, 2, . . . , *n* − 2*.*

## **3. Hyperpaths and Hypercontinuity**

In this section, we check whether the pseudo-open maps preserve the notion of the point-hyperpath and knot-hyperpath between two hypergraphs and under which conditions. Then, the notions of the hyper-continuous map and strictly hyper-continuous map between two hypergraphs are stated and various possible relationships between any two knot-hyperpaths under these notions are investigated.

**Definition 10.** *A mapping f* : *V* → *V between the vertex sets of two hypergraphs H* = (*V*, *E*) *and K* = (*V* , *E* ) *is called hyper-continuous if, for any E <sup>i</sup>* ∈ *E , there is some Ej* ∈ *E such that the corresponding inverse image satisfies f* <sup>−</sup>1(*E i* ) ⊇ *Ej.*

**Example 5.** *Suppose that H* = (*V*, *E*) *and K* = (*V* , *E* ) *are two hypergraphs, where <sup>V</sup>* <sup>=</sup> {*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>v</sup>*4, *<sup>v</sup>*5, *<sup>v</sup>*6} *and V* = {*v* <sup>1</sup>, *v* <sup>2</sup>, *v* <sup>3</sup>, *v* <sup>4</sup>, *v* <sup>5</sup>} *and E* = {{*v*1, *v*2}, {*v*3}, {*v*3, *v*4, *v*5}}*, E* = {{*v* <sup>1</sup>}, {*v* <sup>2</sup>, *v* <sup>3</sup>, *v* <sup>4</sup>}, {*v* <sup>1</sup>, *v* <sup>2</sup>, *v* <sup>3</sup>}}*. A map <sup>f</sup>* : *<sup>V</sup>* <sup>→</sup> *<sup>V</sup> is defined such that f*(*v*1) = *v* <sup>1</sup> = *f*(*v*2)*, f*(*v*3) = *v* <sup>2</sup> = *f*(*v*5)*, f*(*v*4) = *v* <sup>5</sup>*, f*(*v*6) = *v* <sup>3</sup>*. Now, we have* {*v* <sup>1</sup>} ∈ *E and*

*f* <sup>−</sup>1({*v* <sup>1</sup>}) = {*v*1, *v*2}⊇{*v*1, *v*2}(∈ *E*)*. Again,* {*v* <sup>2</sup>, *v* <sup>3</sup>, *v* <sup>4</sup>} ∈ *E and f* <sup>−</sup>1({*v* <sup>2</sup>, *v* <sup>3</sup>, *v* <sup>4</sup>}) = {*v*3, *v*5, *v*6}⊇{*v*3}(∈ *E*)*. Moreover,* {*v* <sup>1</sup>, *v* <sup>2</sup>, *v* <sup>3</sup>}∈*E and f* <sup>−</sup>1({*v* <sup>1</sup>, *v* <sup>2</sup>, *v* <sup>3</sup>})={*v*1, *v*2, *v*3, *v*5} ⊇ {*v*1, *v*2}, {*v*3}(∈ *E*)*.*

*Thus, for each E <sup>i</sup>* ∈ *E* , *i* = 1, 2, 3*, there is one Ej* ∈ *E*, *j* = 1, 2, 3*, such that f* <sup>−</sup>1(*E i* ) ⊇ *Ej*(∈ *E*)*. Thus, f is a hyper-continuous map from V to V .*

**Definition 11.** *A mapping f* : *V* → *V between the vertex sets of two hypergraphs H* = (*V*, *E*) *and K* = (*V* , *E* ) *is called strictly hyper-continuous if, for each E <sup>i</sup>* ∈ *E , there is an Ej* ∈ *E, such that f* <sup>−</sup>1(*E i* ) = *Ej.*

**Example 6.** *Suppose that H* = (*V*, *E*) *and K* = (*V* , *E* ) *are two hypergraphs, where V* = {*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>v</sup>*4, *<sup>v</sup>*5, *<sup>v</sup>*6} *and V* = {*v* <sup>1</sup>, *v* <sup>2</sup>, *v* <sup>3</sup>, *v* <sup>4</sup>, *v* <sup>5</sup>} *and E* = {{*v*1, *v*2},

{*v*2, *v*3, *v*4}, {*v*3, *v*4, *v*5}}*, E* = {{*v* <sup>1</sup>}, {*v* <sup>2</sup>, *v* <sup>5</sup>}, {*v* <sup>1</sup>, *v* <sup>4</sup>}}*. A map <sup>f</sup>* : *<sup>V</sup>* <sup>→</sup> *<sup>V</sup> is defined such that f*(*v*1) = *v* <sup>1</sup> = *f*(*v*2)*, f*(*v*3) = *v* <sup>2</sup> = *f*(*v*5)*, f*(*v*4) = *v* <sup>5</sup>*, f*(*v*6) = *v* <sup>3</sup>*. Now, we have* {*v* <sup>1</sup>} ∈ *E and f* <sup>−</sup>1({*v* <sup>1</sup>}) = {*v*1, *v*2} ∈ *E. Again,* {*v* <sup>2</sup>, *v* <sup>5</sup>} ∈ *E and f* <sup>−</sup>1({*v* <sup>2</sup>, *v* <sup>5</sup>}) = {*v*3, *v*4, *v*5} ∈ *E. Moreover, we have* {*v* <sup>1</sup>, *v* <sup>4</sup>} ∈ *E and f* <sup>−</sup>1({*v* <sup>1</sup>, *v* <sup>4</sup>}) = {*v*1, *v*2} ∈ *E.*

*Thus, for each E <sup>i</sup>* ∈ *E , i* = 1, 2, 3*, there exists an Ej* ∈ *E such that f* <sup>−</sup>1(*E i* ) = *Ej. Thus, f is strictly hyper-continuous.*

**Theorem 1.** *Suppose that H* = (*V*, *E*) *and K* = (*V* , *E* ) *are two hypergraphs and f is a mapping from V into V . If f is a ps-open mapping, then the f-image of a point-hyperwalk in H is a point-hyperwalk in K.*

**Proof.** Let

$$P \equiv \upsilon\_1 E\_1 \upsilon\_2 E\_2 \upsilon\_3 E\_3 \dots \upsilon\_{n-1} E\_n \upsilon\_n$$

be a point-hyperwalk in *H*. Then, we obtain its *f*-image

$$f(P) \equiv f(v\_1)f(E\_1)f(v\_2)f(E\_2)f(v\_3)f(E)\_3 \dots f(v\_{n-1})f(E\_n)f(v\_n).$$

Since *P* is a point-hyperwalk, it follows that *v*<sup>1</sup> ∈ *E*1, *v*<sup>2</sup> ∈ *E*<sup>1</sup> ∩ *E*2, ... , *vn*−<sup>1</sup> ∈ *En*−<sup>1</sup> ∩ *En* and *vn* ∈ *En*. Thus, *f*(*v*1) ∈ *f*(*E*1), *f*(*v*2) ∈ *f*(*E*<sup>1</sup> ∩ *E*2), ... , *f*(*vn*−1) ∈ *f*(*En*−<sup>1</sup> ∩ *En*), *f*(*vn*) ∈ *En*. Now, *E*<sup>1</sup> ∩ *E*<sup>2</sup> ⊆ *E*1, *E*<sup>2</sup> implies that *f*(*E*<sup>1</sup> ∩ *E*2) ⊆ *f*(*E*1), *f*(*E*2), whence *f*(*E*<sup>1</sup> ∩ *E*2) ⊆ *f*(*E*1) ∩ *f*(*E*2). Therefore, *f*(*v*2) ∈ *f*(*E*<sup>1</sup> ∩ *E*2) ⊆ *f*(*E*1) ∩ *f*(*E*2). Similarly, *f*(*v*3) ∈ *f*(*E*2) ∩ *f*(*E*3),..., *f*(*vn*) ∈ *f*(*En*). Hence, *f*(*P*) is a point-hyperwalk in *K*.

**Corollary 1.** *In Theorem 1, if f is an injective mapping, then the f-image of a point-hyperpath in H is a point-hyperpath in K, too.*

**Theorem 2.** *Suppose that H* = (*V*, *E*) *and K* = (*V* , *E* ) *are two hypergraphs and f is a ps-open mapping from V to V . Then, the f -image of a knot-hyperpath in H is a knot-hyperpath in K, too.*

**Proof.** Let *P* ≡ {*v*1}*E*1*K*1*E*2*K*2*E*3*K*<sup>3</sup> ... *Kn*−1*En*{*vn*} be a knot-hyperpath in *H* with *K*<sup>0</sup> = {*v*1} ⊆ *<sup>E</sup>*1, *Kn* <sup>=</sup> {*vn*} ⊆ *En* and *Ki* <sup>⊆</sup> (*Ei*+<sup>1</sup> <sup>∩</sup> *Ei*)\(∪*i*−<sup>1</sup> *<sup>t</sup>*=1*Kt*), *<sup>i</sup>* <sup>=</sup> 1, 2, ... , *<sup>n</sup>* <sup>−</sup> 1. Then, we have the *f*-image

$$f(P) \equiv f(\mathcal{K}\_0)f(E\_1)f(\mathcal{K}\_1)f(E\_2)f(\mathcal{K}\_2)f(E\_3)f(\mathcal{K}\_3)\dots f(\mathcal{K}\_{n-1})f(E\_n)f(\mathcal{K}\_n).$$

In order to prove that *f*(*P*) is a knot-hyperpath, we first show that *f*(*K*0) ⊆ *f*(*E*1) and *f*(*Kn*) ⊆ *f*(*En*). Since *K*<sup>0</sup> ⊆ *E*<sup>1</sup> and *Kn* ⊆ *En*, we have *f*(*K*0) ⊆ *f*(*E*1) and *f*(*Kn*) ⊆ *f*(*En*). Since *K*<sup>2</sup> ⊆ (*E*<sup>2</sup> ∩ *E*3)\*E*1, we have *K*<sup>2</sup> ⊆ (*E*<sup>2</sup> ∩ *E*3) ∩ *K<sup>c</sup>* <sup>1</sup>. It follows that *f*(*K*2) ⊆

*f*((*E*<sup>2</sup> ∩ *E*3) ∩ *K<sup>c</sup>* <sup>1</sup>) ⊆ *<sup>f</sup>*(*E*<sup>2</sup> ∩ *<sup>E</sup>*3) ∩ *<sup>f</sup>*(*K<sup>c</sup>* <sup>1</sup>) ⊆ *<sup>f</sup>*(*E*<sup>3</sup> ∩ *<sup>E</sup>*2) ∩ (*f*(*K*1))*<sup>c</sup>* ⊆ *<sup>f</sup>*(*E*<sup>3</sup> ∩ *<sup>E</sup>*2)\ *<sup>f</sup>*(*K*1). Hence, *f*(*K*2) ⊆ *f*(*E*<sup>3</sup> ∩ *E*2)\ *f*(*K*1).

Similarly, *K*<sup>3</sup> ⊆ (*E*<sup>4</sup> ∩ *E*3)\(*K*<sup>1</sup> ∪ *K*2) implies that *f*(*K*3) ⊆ *f*(*E*<sup>4</sup> ∩ *E*3)\ *f*(*K*1) ∪ *f*(*K*2) and so on. Thus, *Ki* <sup>⊆</sup> (*Ei* <sup>∩</sup> *Ei*+1)\(∪*i*−<sup>1</sup> *<sup>t</sup>*=1*Kt*) implies that

$$f(K\_i) \subseteq f(E\_i \cap E\_{i+1}) \backslash \cup\_{t=1}^{i-1} f(K\_t) \tag{2}$$

for any *i* = 1, 2, . . . , *n* − 1. Hence, we conclude that *f*(*P*) is a knot-hyperpath in *K*.

**Theorem 3.** *Suppose that H* = (*V*, *E*) *and K* = (*V* , *E* ) *are two hypergraphs. If f is a hypercontinuous map from V onto V , then the inverse image of a knot-hyperpath in K under f is a weak knot-hyperpath in H.*

**Proof.** Let *P* ≡ *K* 0*E* 1*K* 1*E* 2*K* <sup>2</sup> ... *<sup>K</sup> <sup>n</sup>*−1*E nK <sup>n</sup>* be a knot-hyperpath in *K*. As *f* is hypercontinuous, we have *f* <sup>−</sup>1(*E* <sup>1</sup>) <sup>⊇</sup> *<sup>E</sup>*1, *<sup>f</sup>* <sup>−</sup>1(*<sup>E</sup>* <sup>2</sup>) ⊇ *<sup>E</sup>*2, ... , *<sup>f</sup>* <sup>−</sup>1(*<sup>E</sup> <sup>n</sup>*) ⊇ *En*, for some hyperedges *E*1, *E*2, ... , *En* ∈ *E*. Moreover, the sets *f* <sup>−</sup>1(*Ki*), *i* = 0, 1, 2, ... , *n* are nonempty because *f* is an onto mapping. Now, the inverse image of the knot-hyperpath can be written as

$$f^{-1}(\boldsymbol{K\_0'})f^{-1}(\boldsymbol{E\_1'})f^{-1}(\boldsymbol{K\_1'})f^{-1}(\boldsymbol{E\_2'})\dots f^{-1}(\boldsymbol{K\_{n-1}'})f^{-1}(\boldsymbol{E\_n'})f^{-1}(\boldsymbol{K\_n'})\dots$$

where *f* <sup>−</sup>1(*E i* ) ⊇ *Ei*, for *i* = 1, 2, 3, ... , *n*. Since the inverse set function behaves well for union, intersection and complement, it follows that the conditions of a knot-hyperpath are easily satisfied. Hence, *f* <sup>−</sup>1(*P* ) is a weak knot-hyperpath.

**Corollary 2.** *The inverse image of an onto strictly hyper-continuous map of a knot-hyperpath is again a knot-hyperpath.*

**Proof.** Consider the knot-hyperpath

$$P' \equiv \mathcal{K}\_0' E\_1' \mathcal{K}\_1' E\_2' \mathcal{K}\_2' \dots \mathcal{K}\_{n-1}' E\_n' \mathcal{K}\_n'$$

as in the proof of Theorem 3. As *f* is strictly hyper-continuous, each *f* <sup>−</sup>1(*Ei*) belongs to *E* and, by using similar arguments, we can conclude that

$$f^{-1}(P^{'}) \equiv f^{-1}(\mathcal{K}\_0^{'})f^{-1}(E\_1^{'})f^{-1}(\mathcal{K}\_1^{'})\dots f^{-1}(\mathcal{K}\_{n-1}^{'})f^{-1}(E\_n^{'})f^{-1}(\mathcal{K}\_n^{'})$$

is a knot-hyperpath.

**Theorem 4.** *Let f* : *V* → *V be a ps-open mapping from a hypergraph H* = (*V*, *E*) *onto a hypergraph K* = (*V* , *E* )*. If H is connected, then K is connected, too.*

**Proof.** Let *v* <sup>1</sup> and *v* <sup>2</sup> be two any vertices in *K*. Since *f* is onto, there exists *v*1, *v*<sup>2</sup> ∈ *V* such that *f*(*v*1) = *v* <sup>1</sup> and *f*(*v*2) = *v* <sup>2</sup> ∈ *V*. Moreover, since *H* is connected and *v*1, *v*<sup>2</sup> ∈ *V*, there exists a knot-hyperpath *P* from *v*<sup>1</sup> to *v*2. Because the image of a knot-hyperpath under a ps-open mapping is again a knot-hyperpath in *K*, starting at *f*(*v*1) = *v*<sup>1</sup> and ending at *f*(*v*2) = *v*<sup>2</sup> , we immediately conclude that *K* is connected.

#### **4. Hyperpaths and Hypertrees**

In this section, we will present a sufficient condition, only involving hyperpaths, under which a hypergraph is a hypertree. Till now, the definition of a hypertree has been based on the concept of the host graph.

**Definition 12** ([14])**.** *Suppose that H* = (*V*, *E*) *is a hypergraph and G* = (*V*, *F*) *is a graph over the same vertex set V. We say that G is a host graph of H if each hyperedge Ei* ∈ *E induces a connected subgraph in G.*

**Lemma 1.** *There exists at least one host graph G of the hypergraph H in which the induced subgraph obtained from any two equivalent knot-hyperpaths never forms a cycle.*

**Proof.** Let *P*<sup>1</sup> and *P*<sup>2</sup> be any two equivalent knot-hyperpaths of the hypergraph *H*, which may be denoted as follows:

$$P\_1 \equiv K\_0 = \{v\_1\} E\_1 K\_1 E\_2 K\_2 E\_3 K\_3 \dots K\_{n-1} E\_n K\_n = \{v\_n\}$$

and

$$P\_2 \equiv \text{K}\_0^{'} = \{\upsilon\_1\} \\ E\_1^{'} \text{K}\_1^{'} E\_2^{'} \text{K}\_2^{'} E\_3^{'} \text{K}\_3^{'} \dots \\ \text{K}\_{n-1}^{'} E\_n^{'} \text{K}\_n^{'} = \{\upsilon\_n\}$$

and graphically represented in Figure 2.

Since they are equivalent knot-hyperpaths, it follows that *Ki* ∩ *Ki* = ∅, *Ei* ∩ *Ei* = ∅, *Ki* ∩ *Ki*+<sup>1</sup> = ∅, and *Ki* ∩ *Ki*+<sup>1</sup> = ∅.

**Figure 2.** A schematic diagram of two equivalent knot-hyperpaths *P*<sup>1</sup> and *P*2.

We note that *E*<sup>1</sup> ∪ *E* <sup>1</sup> can be expressed as the disjoint union of *E*1\*E*<sup>1</sup> , *E*<sup>1</sup> \*E*<sup>1</sup> and *E*<sup>1</sup> ∩ *E*<sup>1</sup> . As we know that, in any host graph of a hypergraph, all the vertices in a hyperedge are connected, and since *E*<sup>1</sup> ∩ *E*<sup>1</sup> is contained in *E*<sup>1</sup> and *E*<sup>1</sup> , it follows that all the vertices in *E*<sup>1</sup> ∩ *E*<sup>1</sup> can be connected to form a graph without cycles. Moreover, since *E*<sup>1</sup> ∩ *E*<sup>1</sup> and *E*1\*E*<sup>1</sup> are contained in *E*1, a graph can be drawn by connecting all the vertices in *E*1\*E*<sup>1</sup> without forming a cycle, which can be further connected with the cycle-free graph drawn in *E*<sup>1</sup> ∩ *E*<sup>1</sup> in the previous step. By connecting vertices in such a manner, the resultant graph will never form a cycle. Similarly, a graph can be drawn by connecting the cycle-free graph drawn in *E*<sup>1</sup> ∩ *E*<sup>1</sup> with a cycle-free graph in *E*<sup>1</sup> \*E*1. All these constructions are depicted in Figure 3.

**Figure 3.** Model of cycle-free connected induced subgraph of a host graph of the hypergraph H.

The model is constructed in such a way that the vertex *v*<sup>1</sup> is connected to *K*<sup>1</sup> ∩ *K*<sup>1</sup> and *K*<sup>1</sup> ∩ *K*<sup>1</sup> is connected to both *E*1\*E*<sup>1</sup> and *E*<sup>1</sup> \*E*<sup>1</sup> through *K*<sup>1</sup> ∪ *K*<sup>1</sup> without forming a cycle. Furthermore, it is to be noted that because *K*<sup>1</sup> ∩ *K*<sup>1</sup> is connected to *E*1\*E*<sup>1</sup> , in the next step, *K*<sup>2</sup> ∩ *K*<sup>2</sup> will connect to those vertices of *E*2\*E*<sup>2</sup> that are not in *E*1, in order to not create a cycle. Similarly, *K*<sup>2</sup> ∩ *K*<sup>2</sup> will connect to those vertices of *E*<sup>1</sup> \*E*<sup>1</sup> that are not in *E*<sup>1</sup> . This further continues till the last vertex *vn*, where *vn* is connected to *Kn*−<sup>1</sup> <sup>∩</sup> *<sup>K</sup> n*−1 and *Kn*−<sup>1</sup> <sup>∩</sup> *<sup>K</sup> <sup>n</sup>*−<sup>1</sup> is connected to *En*\*En* and *En* \*En* through *K*<sup>1</sup> ∪ *K*<sup>1</sup> , without forming a cycle. In this manner, a host graph can be drawn from the hypergraph *H*, where the induced subgraph obtained from the vertices in the edges of the two paths is cycle-free. We conclude that there exists at least one host graph G of H in which the induced subgraph obtained from the two equivalent knot-hyperpaths will never form a cycle.

**Remark 1.** *If the induced subgraph obtained from the vertex set of two knot-hyperpaths joining the same vertices of any host graph of a hypergraph always produces a cycle, then the knot-hyperpaths are not equivalent.*

**Theorem 5.** *Suppose that H is a connected hypergraph, which is a hypertree. Then, any entire knot-hyperpaths having the same length and connecting any two vertices are equivalent.*

**Proof.** Let *P*<sup>1</sup> and *P*<sup>2</sup> be any two entire knot-hyperpaths of the hypergraph *H*, which may be denoted as follows:

$$P\_1 \equiv K\_0 = \{v\_1\} E\_1 K\_1 E\_2 K\_2 E\_3 K\_3 \dots K\_{n-1} E\_n K\_n = \{v\_n\}$$

and

$$P\_2 \equiv \boldsymbol{K}\_0^{'} = \{\boldsymbol{v}\_1\} \boldsymbol{E}\_1^{'} \boldsymbol{K}\_1^{'} \boldsymbol{E}\_2^{'} \boldsymbol{K}\_2^{'} \boldsymbol{E}\_3^{'} \boldsymbol{K}\_3^{'} \dots \boldsymbol{K}\_{n-1}^{'} \boldsymbol{E}\_n^{'} \boldsymbol{K}\_n^{'} = \{\boldsymbol{v}\_n\}.$$

If *P*<sup>1</sup> and *P*<sup>2</sup> are equivalent knot-hyperpaths, then the result is proven.

On the contrary, if *P*<sup>1</sup> and *P*<sup>2</sup> are not equivalent, then there exists a pair of edges (*Ei*<sup>0</sup> , *E i*0 ), where *Ei*<sup>0</sup> is from *P*<sup>1</sup> and *E <sup>i</sup>*<sup>0</sup> is from *<sup>P</sup>*2, such that *Ei*<sup>0</sup> ∩ *<sup>E</sup> <sup>i</sup>*<sup>0</sup> = <sup>∅</sup>. Since *Ki*0−1, *Ki*<sup>0</sup> ⊆ *Ei*<sup>0</sup> and *<sup>K</sup> <sup>i</sup>*0−1, *<sup>K</sup> <sup>i</sup>*<sup>0</sup> ⊆ *<sup>E</sup> i*0 , we have *Ki*0−<sup>1</sup> <sup>∩</sup> *<sup>K</sup> <sup>i</sup>*0−<sup>1</sup> <sup>=</sup> <sup>∅</sup> <sup>=</sup> *Ki*<sup>0</sup> <sup>∩</sup> *<sup>K</sup> i*0 . Moreover, let *Ej*<sup>0</sup> , *E <sup>j</sup>*<sup>0</sup> be the edges such that *Ej*<sup>0</sup> ∩ *E j*0 = ∅, while *Ek* ∩ *E <sup>k</sup>* = ∅, for any *k* ∈ {*i*0, *i*<sup>0</sup> + 1, ... , *j*<sup>0</sup> − 1}. Then, the edges *Ei*0−<sup>1</sup> to *Ej*<sup>0</sup> and *E <sup>i</sup>*0−<sup>1</sup> to *E <sup>j</sup>*<sup>0</sup> will always form a cycle (see Figure 4) in any host graph of H, which is a contradiction. Therefore, *P*<sup>1</sup> and *P*<sup>2</sup> are equivalent. Thus, we can conclude that if *H* is a hypertree, then, between any two vertices, the entire knot-hyperpaths having the same length are unique up to isomorphism.

**Figure 4.** The cycle formed in a host graph of a hypergraph.

It is to be noted that two knot-hyperpaths joining two vertices in a hypertree may not always be equivalent. This can be observed in Example 7 by introducing an extra edge {*v*6, *v*7, *v*10} to the hypergraph, which subsequently produces two knot-hyperpaths joining *v*<sup>0</sup> and *v*1, but with different lengths.

**Theorem 6.** *Suppose that H is a hypergraph such that, between any two vertices, there exists a unique entire knot-hyperpath up to isomorphism. Then, H is a hypertree.*

**Proof.** By hypothesis, between any two vertices *v*<sup>1</sup> and *v*<sup>2</sup> of *H*, there exists an entire knothyperpath, which is unique up to isomorphism. It follows that *H* is connected. To show that H is a hypertree, it is enough to show that *H* admits a host graph that is a tree. Let

$$P \equiv K\_0 = \{v\_1\} E\_1 K\_1 E\_2 K\_2 E\_3 K\_3 \dots K\_{n-1} E\_n K\_n = \{v\_2\}$$

be an entire knot-hyperpath joining the vertices *v*<sup>1</sup> and *v*2. Then, the vertices contained in the edges of this knot-hyperpath can be joined without forming a cycle, in such a way that the constructed graph *G*<sup>1</sup> is an induced subgraph with vertex set *V*<sup>1</sup> = ∪*Ei* of some host graph *G* of the given hypergraph *H*. Now, if ∪*Ei* = *V*, then we can take *G* = *G*1, which is a tree. Hence, in this case, *H* is a hypertree and the theorem is proven.

If ∪*Ei* = *V*, then let *v*<sup>3</sup> ∈ *V* be such that *v*<sup>3</sup> ∈ ∪ / *Ei*. Let

$$P' \equiv \{v\_1\} E\_1' K\_1' E\_2' \dots K\_{k-1}' E\_k' \{v\_3\}.$$

be an entire knot-hyperpath joining the vertices *v*<sup>1</sup> and *v*3. We note that there may exist some hyperedges in *P* that coincide with the hyperedges of *P*. Now, excluding these common hyperedges, the rest of the hyperedges of *P* can be joined without forming a cycle. In this way, an induced subgraph *G*<sup>2</sup> can be formed with vertex set ∪*E <sup>j</sup>* and the edges set as the union of those edges common with *G*<sup>1</sup> and the edges newly formed from hyperedges of *P* , which are not in *P*. It is clear from the construction that both subgraphs *G*<sup>1</sup> and *G*<sup>2</sup> are not cyclic and the union *G*<sup>1</sup> ∪ *G*<sup>2</sup> is connected; otherwise, *H* would have two entire knot-hyperpaths joining the same vertices, but not equivalent (see proof of Theorem 5). Now, if (∪*Ei*) ∪ (∪*E j* ) = *V*, then *G* = *G*<sup>1</sup> ∪ *G*<sup>2</sup> is the host graph of *H* that is a tree and hence *H* is again a hypertree.

If (∪*Ei*) ∪ (∪*E j* ) = *V*, then there exists a vertex *v*<sup>4</sup> ∈ *V* that is not in (∪*Ei*) ∪ (∪*E j* ). Then, we will have an entire knot-hyperpath *P* joining *<sup>v</sup>*<sup>1</sup> and *<sup>v</sup>*<sup>4</sup> as follows:

$$P'' \equiv \{v\_1\} E\_1'' {k\_1''} E\_2'' \dots {k\_{l-1}''} E\_l'' \{v\_4\} . $$

Now, excluding those hyperedges of *P* that are common with *P* and *P* , the rest of the hyperedges of *P* can be joined without forming a cycle. In this way, an induced subgraph *G*<sup>3</sup> can be formed with vertex set ∪*E <sup>l</sup>* and the edges set as the union of those edges common with *G*<sup>1</sup> ∪ *G*<sup>2</sup> and the edges newly formed from hyperedges of *P* that are not in *P* and *P* . It is clear from the construction that all the subgraphs *G*1, *G*<sup>2</sup> and *G*<sup>3</sup> are not cyclic and the union *G*<sup>1</sup> ∪ *G*<sup>2</sup> ∪ *G*<sup>3</sup> is connected. Now, if (∪*Ei*) ∪ (∪*E j* ) ∪ (∪*E <sup>l</sup>* ) = *V*, then *G* = *G*<sup>1</sup> ∪ *G*<sup>2</sup> ∪ *G*<sup>3</sup> is the host graph of *H* that is a tree and hence *H* is a hypertree.

As the vertex set of the hypergraph is finite, the process has a finite number of steps. Thus, we can conclude that if *H* is a hypergraph such that, between any two vertices, there exists an entire knot-hyperpath unique up to isomorphism, then *H* is a hypertree.

**Remark 2.** *We can notice that the hypergraph considered in Example 3 is a hypertree, but the two knot-hyperpaths P* <sup>1</sup> *and P* <sup>2</sup> *joining the vertices v*<sup>1</sup> *and v*<sup>13</sup> *are not equivalent, even though they have the same length, while all the entire knot-hyperpaths (for example, P*<sup>1</sup> *and P*2*) are equivalent. Hence, the property of knots of being entire, in the above two theorems, is an important hypothesis to be considered.*

To illustrate the algorithm stated in the proof of Theorem 6, we present the following example, where the considered hypergraph is a hypertree and a host graph is drawn using the technique used in the proof of Theorem 6. This hypertree is represented in Figure 5.

**Example 7.** *Consider the hypergraph H* = (*V*, *E*)*, where V* = {*v*0, *v*1, *v*2, ... , *v*16} *and E* = {*E*<sup>1</sup> = {*v*0, *v*7, *v*6}, *E*<sup>2</sup> = {*v*6, *v*10, *v*11}, *E*<sup>3</sup> = {*v*11, *v*14, *v*15, *v*5, *v*16}, *E*<sup>4</sup> = {*v*3, *v*1, *v*13}, *E*<sup>5</sup> = {*v*6, *v*2}, *E*<sup>6</sup> = {*v*5, *v*16}, *E*<sup>7</sup> = {*v*4, *v*9, *v*12}, *E*<sup>8</sup> = {*v*5, *v*8, *v*13, *v*9}}*. One can easily verify that H is a hypertree and, between any two vertices, there exists an entire knot-hyperpath, unique up to isomorphism. Now, we will use the technique used in the proof of Theorem 6, in order to obtain a host graph that is a tree.*

**Figure 5.** A hypergraph that is a hypertree.

*Let us consider the vertices v*<sup>0</sup> *and v*<sup>1</sup> *and the knot-hyperpath*

*P* ≡ {*v*0}*E*1{*v*6}*E*2{*v*11}*E*3{*v*5}*E*4{*v*13}*E*8{*v*1}

*joining v*<sup>0</sup> *and v*1*. Now, the vertices in all hyperedges are connected and form a graph G*<sup>1</sup> *in such a way that it is not cyclic and it is an induced subgraph with vertex set V*<sup>1</sup> = *E*<sup>1</sup> ∪ *E*<sup>2</sup> ∪ *E*<sup>3</sup> ∪ *E*<sup>4</sup> ∪ *E*<sup>8</sup> *of some host graph G of H.*

*Clearly, V* = *V*1*, and so we consider the vertex v*<sup>2</sup> ∈ *V, which is not in V*1*. Now, a hyperpath P from v*<sup>0</sup> *to v*<sup>2</sup> *is constructed as follows:*

$$P' \equiv \{\upsilon\_0\} E\_1\{\upsilon\_6\} E\_2\{\upsilon\_{11}\} E\_3\{\upsilon\_5\} E\_6\{\upsilon\_2\}.$$

*Clearly, except E*6*, all other hyperedges of this knot-hyperpath appear in the previous knothyperpath, and so vertices of E*<sup>6</sup> *are joined in an acyclic way and represent a graph G*<sup>2</sup> *with vertex set V*<sup>2</sup> = *E*<sup>1</sup> ∪ *E*<sup>2</sup> ∪ *E*<sup>3</sup> ∪ *E*6*.*

*Here, we note that the union of the two graphs G*<sup>1</sup> *and G*<sup>2</sup> *is acyclic and connected. Moreover, V*<sup>1</sup> ∪ *V*<sup>2</sup> = *V. Therefore, we consider an arbitrary vertex from v*4, *v*12, *v*<sup>16</sup> *that is not in V*<sup>1</sup> ∪ *V*2*. Let us consider the vertex v*<sup>4</sup> *and the knot-hyperpath P constructed as follows:*

$$P'' \equiv \{\upsilon\_0\} E\_1\{\upsilon\_6\} E\_2\{\upsilon\_{11}\} E\_3\{\upsilon\_5\} E\_4\{\upsilon\_9\} E\_7\{\upsilon\_4\}.$$

*Clearly, except E*7*, all other hyperedges of this knot-hyperpath appear in the previous knothyperpaths, and so vertices of E*<sup>7</sup> *are joined in an acyclic way that represents a graph G*<sup>3</sup> *with vertex set V*<sup>3</sup> = *E*<sup>1</sup> ∪ *E*<sup>2</sup> ∪ *E*<sup>3</sup> ∪ *E*<sup>4</sup> ∪ *E*7*. Thus, G*<sup>1</sup> ∪ *G*<sup>2</sup> ∪ *G*<sup>3</sup> *is connected and acyclic. Since V*<sup>1</sup> ∪ *V*<sup>2</sup> ∪ *V*<sup>3</sup> = *V, we consider the vertex v*16*, the only one that is not in this union and the knot-hyperpath*

$$P^{\prime\prime} \equiv \{v\_0\} E\_1\{v\_6\} E\_5\{v\_{16}\}.$$

*Clearly, except E*5*, all other hyperedges of this knot-hyperpath appear in the previous knothyperpaths, and so vertices of E*<sup>5</sup> *are joined in an acyclic way that represents a graph G*<sup>4</sup> *with vertex set V*<sup>4</sup> = *E*<sup>1</sup> ∪ *E*5*. Now, G*<sup>1</sup> ∪ *G*<sup>2</sup> ∪ *G*<sup>3</sup> ∪ *G*<sup>4</sup> *is connected and acyclic, and V*<sup>1</sup> ∪ *V*<sup>2</sup> ∪ *V*<sup>3</sup> ∪ *V*<sup>4</sup> = *V. Therefore, G* = *G*<sup>1</sup> ∪ *G*<sup>2</sup> ∪ *G*<sup>3</sup> ∪ *G*<sup>4</sup> *is the required host graph, which is a tree.*

#### **5. Conclusions**

Based on the definition of a knot in a hypergraph *H*, which is a subset of the intersections of some intersecting hyperedges of *H*, we have introduced the notion of the knot-hyperpath, in order to better characterize the hyper-continuity and pseudo-continuity of functions between two hypergraphs. Moreover, in the second part of the paper, we have characterized the hypertrees without using the concept of a host graph. A sufficient condition is established to check whether or not a hypergraph is a hypertree. Furthermore, an algorithm is designed in order to extract from a hypertree a host graph that is a tree. This algorithm has the potential to determine whether a hypergraph is a hypertree or not. As we know, hypergraphs and hypertrees are extensively used in different branches of applied sciences, including networking and theoretical computer science, and therefore this investigation will give more future ideas towards the applicability of hypergraphs and hypertrees in these fields.

**Author Contributions:** Conceptualization, S.R. and M.C.; methodology, S.R., M.C. and I.C.; investigation, S.R., M.C., F.A. and I.C.; writing—original draft preparation, S.R., M.C. and F.A.; writing—review and editing, I.C.; funding acquisition, I.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** The third author acknowledges the financial support of the Slovenian Research Agency (research core funding No. P1-0285).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
