Introducing soft directed hypergraphs: a fusion of soft set theory and directed hypergraphs

1. Introduction

Directed hypergraphs [7,8,10] serve as a powerful modelling tool in both Operations Research and Computer Science, offering a broader representation than traditional directed graphs. They allow for the representation of relationships among multiple entities, making them particularly useful in scenarios where complex interactions need to be captured.

The concept of soft sets, pioneered by Molodtsov [27] in 1999, revolutionized mathematical approaches to dealing with uncertainties that evade conventional methods. Soft sets provide a flexible framework for handling imprecise or uncertain information, enabling more robust decision-making processes. Subsequent research by Maji, Roy, and Biswas [25,26] has delved deeper into the theory of soft sets, exploring their applications in various decision-making contexts.

Thumbakara and George [31] introduced soft graphs and some of their properties [33,34], which have since been refined and extended by researchers like Akram and Nawas [2,3]. Their work has led to the development of fuzzy soft graphs and fuzzy soft trees, further expanding the applicability of soft graph theory [4,5,6]. Contributions from Thenge, Jain, and Reddy [30,28,29] have also significantly advanced the field of soft graphs.

George, Thumbakara, and Jose have made substantial contributions to the domain by introducing concepts such as soft hypergraphs [11] and soft directed graphs [23,22]. These extensions have broadened the scope of soft graph theory, enabling the modelling of more complex systems and phenomena. Moreover, they have investigated various product operations on soft graphs, including modular products and homomorphic products, which have implications for graph analysis and manipulation [12,14,15,19,21,20,24].

Baghernejad and Borzooei [9] have demonstrated the practical utility of soft graphs and soft multigraphs in managing urban traffic flows, showcasing the real-world applicability of these theoretical constructs. Additionally, innovations like soft semigraphs [13,18,17] and soft disemigraphs [16] have further enriched the field, providing new avenues for exploration and application.

In this paper, the authors introduce the concept of soft directed hypergraphs, which represent a further extension of soft graph theory. They also investigate various operations on soft directed hypergraphs, aiming to elucidate their properties.

2. Preliminaries

For basic concepts of a directed hypergraph, we refer [1,4,8]. “A directed hypergraph ${\mathrm{\Delta }}^{\ast }=(\mathrm{\Gamma },\mathrm{\Xi })$ consists of a vertex set $\mathrm{\Gamma }$ and a set of directed hyperedges or hyperarcs $\mathrm{\Xi }=\{e=(T(e),H(e))|T(e)\subseteq \mathrm{\Gamma }$ and $H(e)\subseteq \mathrm{\Gamma }\}$, where $T(e)\ne \varphi$ and $H(e)\ne \varphi$. The sets $T(e)$ and $H(e)$ are called tail and head of the hyperarc $e$, respectively. A directed hypergraph is called k-uniform if $|T(e)|=|H(e)|=k$ for all $e\in \mathrm{\Xi }$. Two hyperarcs $e$ and $e^{\prime }$ are said to be parallel if $T(e)=T(e^{\prime })$ and $H(e)=H(e^{\prime })$. A hyperarc $e$ is said to be a loop if $T(e)=H(e)$. A directed hypergraph ${\mathrm{\Delta }}^{\ast }=(\mathrm{\Gamma },\mathrm{\Xi })$ is called simple if it has no parallel hyperarcs and loops. A directed hypergraph is called trivial if $|\mathrm{\Gamma }|=1$ and $\mathrm{\Xi }=\varphi$. If the two vertices $u$ and $v$ of ${\mathrm{\Delta }}^{\ast }$ are such that $u\in T(e)$ and $v\in H(e)$ then we say that $v$ is adjacent from $u$ or $u$ is adjacent to $v$. The indegree of a vertex $v$ in ${\mathrm{\Delta }}^{\ast }$, denoted by $d^{-}(v)$ is the number of hyperarcs that contain $v$ in their head. The outdegree of a vertex $v$ in ${\mathrm{\Delta }}^{\ast }$, denoted by $d^{+}(v)$ is the number of hyperarcs that contain $v$ in their tail. A directed hypergraph ${\mathrm{\Delta }}^{\prime }=({\mathrm{\Gamma }}^{\prime },{\mathrm{\Xi }}^{\prime })$ is a weak subhypergraph of the directed hypergraph ${\mathrm{\Delta }}^{\ast }=(\mathrm{\Gamma },\mathrm{\Xi })$ if ${\mathrm{\Gamma }}^{\prime }\subseteq \mathrm{\Gamma }$ and ${\mathrm{\Xi }}^{\prime }$ consists of hyperarcs $e^{\prime }$ with $T(e^{\prime })=\{v|v\in T(e)\cap {\mathrm{\Gamma }}^{\prime }\}$ and $H(e^{\prime })=\{v|v\in H(e)\cap {\mathrm{\Gamma }}^{\prime }\}$ for some $e\in \mathrm{\Xi }$. A directed hypergraph ${\mathrm{\Delta }}^{\prime }=({\mathrm{\Gamma }}^{\prime },{\mathrm{\Xi }}^{\prime })$ is a weak induced subhypergraph of the directed hypergraph ${\mathrm{\Delta }}^{\ast }=(\mathrm{\Gamma },\mathrm{\Xi })$ if ${\mathrm{\Gamma }}^{\prime }\subseteq \mathrm{\Gamma }$ and hyperarc set ${\mathrm{\Xi }}^{\prime }=\{(T(e)\cap {\mathrm{\Gamma }}^{\prime },H(e)\cap {\mathrm{\Gamma }}^{\prime })|e\in \mathrm{\Xi }$and$T(e)\cap {\mathrm{\Gamma }}^{\prime }\ne \varphi$and$H(e)\cap {\mathrm{\Gamma }}^{\prime }\ne \varphi \}$.”

In 1999 Molodtsov [27] initiated the concept of soft sets. Let $U$ be an initial universe set and let $\mathrm{\Pi }$ be a set of parameters. A pair $(F,\mathrm{\Pi })$ is called a soft set (over $U$) if and only $F$ is a mapping of $\mathrm{\Pi }$ into the set of all subsets of the set $U$. That is, $F:\mathrm{\Pi }\to \mathcal{P}(U)$.

3. Soft directed hypergraphs

Example 3.3 Consider a directed hypergraph ${\mathrm{\Delta }}^{\ast }=(\mathrm{\Gamma },\mathrm{\Xi })$ given in Figure 1.

Let $\mathrm{\Pi }=\{v_2,v_{10}\}\subseteq \mathrm{\Gamma }$ be a parameter set. Define a function $X:\mathrm{\Pi }\to \mathcal{P}(\mathrm{\Gamma })$ defined by $\mathrm{\Omega }(\pi )=\{v\in \mathrm{\Gamma }:\pi Rv\iff v=\pi$ or $v$ is adjacent from $\pi$ or $v$ is adjacent to $\pi$} for all $\pi \in P$. That is, $\mathrm{\Omega }(v_2)=\{v_1,v_2,v_3,v_7,v_9\}$ and $\mathrm{\Omega }(v_{10})=\{v_5,v_{10},v_{11}\}$. Then $(\mathrm{\Omega },\mathrm{\Pi })$ is a soft set over $\mathrm{\Gamma }$. Define another function $\mathrm{\Psi }:\mathrm{\Pi }\to \mathcal{P}({\mathrm{\Xi }}_s)$ defined by $\mathrm{\Psi }(\pi )=\{(T(e)\cap \mathrm{\Omega }(\pi ),H(e)\cap \mathrm{\Omega }(\pi ))|e\in \mathrm{\Xi }$and$T(e)\cap \mathrm{\Omega }(\pi )\ne \varphi$and$H(e)\cap \mathrm{\Omega }(\pi )\ne \varphi \}$. That is, $\mathrm{\Psi }(v_2)=\{(\{v_1,v_3\},\{v_2\}),(\{v_2\},\{v_7,v_9\})\}$ and $v_{10})=\{(\{v_5\},\{v_{10}\}),(\{v_{10}\},\{v_{11}\})\}$. Then $(\mathrm{\Psi },\mathrm{\Pi })$ is a soft set over ${\mathrm{\Xi }}_s$. Also $Z(v_2)=(\mathrm{\Omega }(v_2),\mathrm{\Psi }(v_2))$ and $Z(v_{10})=(\mathrm{\Omega }(v_{10}),\mathrm{\Psi }(v_{10}))$ are weak induced subhypergraphs of ${\mathrm{\Delta }}^{\ast }$ as shown in Figure 2. Hence $\mathrm{\Delta }=\{Z(v_2),Z(v_{10})\}$ is a soft directed hypergraph of ${\mathrm{\Delta }}^{\ast }$.

Example 3.5 Consider a directed hypergraph ${\mathrm{\Delta }}^{\ast }=(\mathrm{\Gamma },\mathrm{\Xi })$ given in Figure 3.

Let ${\mathrm{\Pi }}_1=\{v_3,v_7\}\subseteq \mathrm{\Gamma }$ be a parameter set. Define a function ${\mathrm{\Omega }}_1:{\mathrm{\Pi }}_1\to \mathcal{P}(\mathrm{\Gamma })$ defined by ${\mathrm{\Omega }}_1(\pi )=\{v\in \mathrm{\Gamma }:\pi Rv\iff v=\pi$ or $v$ is adjacent from $\pi$ or $v$ is adjacent to $\pi$} for all $\pi \in {\mathrm{\Pi }}_1$. That is, ${\mathrm{\Omega }}_1(v_3)=\{v_3,v_4,v_5,v_9,v_{11}\}$ and ${\mathrm{\Omega }}_1(v_7)=\{v_6,v_7,v_9,v_{10},v_{12}\}$.

Then $({\mathrm{\Omega }}_1,{\mathrm{\Pi }}_1)$ is a soft set over $\mathrm{\Gamma }$. Define another function ${\mathrm{\Psi }}_1:{\mathrm{\Pi }}_1\to \mathcal{P}({\mathrm{\Xi }}_s)$ defined by ${\mathrm{\Psi }}_1(\pi )=\{(T(e)\cap {\mathrm{\Omega }}_1(\pi ),H(e)\cap {\mathrm{\Omega }}_1(\pi ))|e\in \mathrm{\Xi }$and$T(e)\cap {\mathrm{\Omega }}_1(\pi )\ne \varphi$and$H(e)\cap {\mathrm{\Omega }}_1(\pi )\ne \varphi \}$. That is, ${\mathrm{\Psi }}_1(v_3)=\{(\{v_{11}\},\{v_3\}),(\{v_3\},\{v_4,v_5,v_9\})\}$ and ${\mathrm{\Psi }}_1(v_7)=\{(\{v_6,v_9\},\{v_7\}),(\{v_{10}\},\{v_7\}),(\{v_7\},\{v_{12}\})\}$. Then $({\mathrm{\Psi }}_1,{\mathrm{\Pi }}_1)$ is a soft set over ${\mathrm{\Xi }}_s$. Also $Z_1(v_3)=({\mathrm{\Omega }}_1(v_3),{\mathrm{\Psi }}_1(v_3))$ and $Z_1(v_7)=({\mathrm{\Omega }}_1(v_7),{\mathrm{\Psi }}_1(v_7))$ are weak induced subhypergraphs of ${\mathrm{\Delta }}^{\ast }$ as shown in Figure 4. Hence ${\mathrm{\Delta }}_1=\{Z_1(v_3),Z_1(v_7)\}$ is a soft directed hypergraph of ${\mathrm{\Delta }}^{\ast }$.

Let ${\mathrm{\Pi }}_2=\{v_7\}\subseteq \mathrm{\Gamma }$ be another parameter set. Define a function ${\mathrm{\Omega }}_2:{\mathrm{\Pi }}_2\to \mathcal{P}(\mathrm{\Gamma })$ defined by ${\mathrm{\Omega }}_2(\pi )=\{v\in \mathrm{\Gamma }:\pi Rv\iff v=\pi$ or $v$ is adjacent to $\pi$} for all $\pi \in {\mathrm{\Pi }}_2$. That is, ${\mathrm{\Omega }}_2(v_7)=\{v_6,v_7,v_9,v_{10}\}$. Then $({\mathrm{\Omega }}_2,{\mathrm{\Pi }}_2)$ is a soft set over $\mathrm{\Gamma }$. Define another function ${\mathrm{\Psi }}_2:{\mathrm{\Pi }}_2\to \mathcal{P}({\mathrm{\Xi }}_s)$ defined by ${\mathrm{\Psi }}_2(\pi )=\{(T(e)\cap {\mathrm{\Omega }}_2(\pi ),H(e)\cap {\mathrm{\Omega }}_2(\pi ))|e\in \mathrm{\Xi }$and$T(e)\cap {\mathrm{\Omega }}_2(\pi )\ne \varphi$and$H(e)\cap {\mathrm{\Omega }}_2(\pi )\ne \varphi \}$. That is, ${\mathrm{\Psi }}_2(v_7)=\{(\{v_6,v_9\},\{v_7\}),(\{v_{10}\},\{v_7\})\}$. Then $({\mathrm{\Psi }}_2,{\mathrm{\Pi }}_2)$ is a soft set over ${\mathrm{\Xi }}_s$. Also $Z_2(v_7)=({\mathrm{\Omega }}_2(v_7),{\mathrm{\Psi }}_2(v_7))$ is a weak induced subhypergraph of ${\mathrm{\Delta }}^{\ast }$ as shown in Figure 5. Hence ${\mathrm{\Delta }}_2=\{Z_2(v_7)\}$ is a soft directed hypergraph of ${\mathrm{\Delta }}^{\ast }$.

Here ${\mathrm{\Delta }}_2$ is a soft weak induced subhypergraph of ${\mathrm{\Delta }}_1$ since

1. ${\mathrm{\Pi }}_2\subseteq {\mathrm{\Pi }}_1$,

2. $Z_2(v_7)=({\mathrm{\Omega }}_2(v_7),{\mathrm{\Psi }}_2(v_7))$ is a weak induced subhypergraph of $Z_1(v_7)=({\mathrm{\Omega }}_1(v_7),{\mathrm{\Psi }}_1(v_7))$.

4. Extended union of two soft directed hypergraphs

Example 4.2. Consider a simple directed hypergraph ${\mathrm{\Delta }}^{\ast }=(\mathrm{\Gamma },\mathrm{\Xi })$ given in Figure 6.

Let ${\mathrm{\Pi }}_1=\{v_7,v_9\}\subseteq \mathrm{\Gamma }$ be a parameter set. Define a function ${\mathrm{\Omega }}_1:{\mathrm{\Pi }}_1\to \mathcal{P}(\mathrm{\Gamma })$ defined by ${\mathrm{\Omega }}_1(\pi )=\{v\in \mathrm{\Gamma }:\pi Rv\iff v=\pi$ or $v$ is adjacent from $\pi$ or $v$ is adjacent to $\pi$}, for all $\pi \in {\mathrm{\Pi }}_1$. That is, ${\mathrm{\Omega }}_1(v_7)=\{v_1,v_2,v_7,v_9,v_{12}\}$ and ${\mathrm{\Omega }}_1(v_9)=\{v_3,v_6,v_7,v_9,v_{14}\}$. Then $({\mathrm{\Omega }}_1,{\mathrm{\Pi }}_1)$ is a soft set over $\mathrm{\Gamma }$. Define another function ${\mathrm{\Psi }}_1:{\mathrm{\Pi }}_1\to \mathcal{P}({\mathrm{\Xi }}_s)$ defined by ${\mathrm{\Psi }}_1(\pi )=\{(T(e)\cap {\mathrm{\Omega }}_1(\pi ),H(e)\cap {\mathrm{\Omega }}_1(\pi ))|e\in \mathrm{\Xi }$and$T(e)\cap {\mathrm{\Omega }}_1(\pi )\ne \varphi$and$H(e)\cap {\mathrm{\Omega }}_1(\pi )\ne \varphi \}$. That is, ${\mathrm{\Psi }}_1(v_7)=\{(\{v_1,v_{12}\},\{v_7\}),(\{v_2,v_9\},\{v_7\})\}$ and ${\mathrm{\Psi }}_1(v_9)=\{(\{v_9\},\{v_6,v_7\}),(\{v_3\},\{v_9\}),(\{v_9\},\{v_{14}\})\}$. Then $({\mathrm{\Psi }}_1,{\mathrm{\Pi }}_1)$ is a soft set over ${\mathrm{\Xi }}_s$. Also $Z_1(v_7)=({\mathrm{\Omega }}_1(v_7),{\mathrm{\Psi }}_1(v_7))$ and $Z_1(v_9)=({\mathrm{\Omega }}_1(v_9),{\mathrm{\Psi }}_1(v_9))$ are weak induced subhypergraphs of ${\mathrm{\Delta }}^{\ast }$ as shown in Figure 7.

Hence ${\mathrm{\Delta }}_1=\{Z_1(v_7),Z_1(v_9)\}$ is a soft directed hypergraph of ${\mathrm{\Delta }}^{\ast }$.

Let ${\mathrm{\Pi }}_2=\{v_9,v_{15}\}\subseteq \mathrm{\Gamma }$ be another parameter set. Define a function ${\mathrm{\Omega }}_2:{\mathrm{\Pi }}_2\to \mathcal{P}(\mathrm{\Gamma })$ defined by ${\mathrm{\Omega }}_2(\pi )=\{v\in \mathrm{\Gamma }:\pi Rv\iff v=\pi$ or $v$ is adjacent from $\pi$}, for all $\pi \in {\mathrm{\Pi }}_2$. That is, ${\mathrm{\Omega }}_2(v_9)=\{v_6,v_7,v_9,v_{14}\}$ and ${\mathrm{\Omega }}_2(v_{15})=\{v_3,v_{10},v_{13},v_{14},v_{15}\}$. Then $({\mathrm{\Omega }}_2,{\mathrm{\Pi }}_2)$ is a soft set over $\mathrm{\Gamma }$. Define another function ${\mathrm{\Psi }}_2:{\mathrm{\Pi }}_2\to \mathcal{P}({\mathrm{\Xi }}_s)$ defined by ${\mathrm{\Psi }}_2(\pi )=\{(T(e)\cap {\mathrm{\Omega }}_2(\pi ),H(e)\cap {\mathrm{\Omega }}_2(\pi ))|e\in \mathrm{\Xi }$and$T(e)\cap {\mathrm{\Omega }}_2(\pi )\ne \varphi$and$H(e)\cap {\mathrm{\Omega }}_2(\pi )\ne \varphi \}$. That is, ${\mathrm{\Psi }}_2(v_9)=\{(\{v_9\},\{v_6,v_7\}),(\{v_9\},\{v_{14}\})\}$ and ${\mathrm{\Psi }}_2(v_{15})=\{(\{v_{15}\},\{v_3,v_{10},v_{14}\}),(\{v_{15}\},\{v_{13}\})\}$. Then $({\mathrm{\Psi }}_2,{\mathrm{\Pi }}_2)$ is a soft set over ${\mathrm{\Xi }}_s$. Also $Z_2(v_9)=({\mathrm{\Omega }}_2(v_9),{\mathrm{\Psi }}_2(v_9))$ and $Z_2(v_{15})=({\mathrm{\Omega }}_2(v_{15}),{\mathrm{\Psi }}_2(v_{15}))$ are weak induced subhypergraphs of ${\mathrm{\Delta }}^{\ast }$ as shown in Figure 8.

Hence ${\mathrm{\Delta }}_2=\{Z_2(v_9),Z_2(v_{15})\}$ is a soft directed hypergraph of ${\mathrm{\Delta }}^{\ast }$.

The extended union of two soft directed hypergraphs ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ is $\mathrm{\Delta }={\mathrm{\Delta }}_1{\cup }_E{\mathrm{\Delta }}_2=({\mathrm{\Delta }}^{\ast },\mathrm{\Omega },\mathrm{\Psi },\mathrm{\Pi })$ where $\mathrm{\Pi }={\mathrm{\Pi }}_1\cup {\mathrm{\Pi }}_2=\{v_7,v_9,v_{15}\}$. Also $\mathrm{\Omega }(v_7)={\mathrm{\Omega }}_1(v_7)=\{v_1,v_2,v_7,v_9,v_{12}\}$, $\mathrm{\Psi }(v_7)={\mathrm{\Psi }}_1(v_7)=\{(\{v_1,v_{12}\},\{v_7\}),(\{v_2,v_9\},\{v_7\})\}$, $\mathrm{\Omega }(v_9)={\mathrm{\Omega }}_1(v_9)\cup {\mathrm{\Omega }}_2(v_9)={\mathrm{\Omega }}_1(v_9)=\{v_3,v_6,v_7,v_9,v_{14}\}$, $\mathrm{\Psi }(v_9)=\{(T(e)\cap \mathrm{\Omega }(v_9),H(e)\cap \mathrm{\Omega }(v_9))|e\in \mathrm{\Xi }$and$T(e)\cap \mathrm{\Omega }(v_9)\ne \varphi$and$$\begin{gathered} H(e)\cap \mathrm{\Omega }(v_9)\ne \varphi \}=\{(\{v_9\},\{v_6,v_7\}), \\ (\{v_3\},\{v_9\}),(\{v_9\},\{v_{14}\})\} \end{gathered}$$, $\mathrm{\Omega }(v_{15})={\mathrm{\Omega }}_2(v_{15})=\{v_3,v_{10},v_{13},v_{14},v_{15}\}$ and $$\begin{gathered} \mathrm{\Psi }(v_{15})={\mathrm{\Psi }}_2(v_{15})=\{(\{v_{15}\}, \\ \{v_3,v_{10},v_{14}\}),(\{v_{15}\},\{v_{13}\})\} \end{gathered}$$. Here $(\mathrm{\Omega },\mathrm{\Pi })$ is a soft set over $\mathrm{\Gamma }$ and $(\mathrm{\Psi },\mathrm{\Pi })$ is a soft set over ${\mathrm{\Xi }}_s$. Also $Z(v_7)=(\mathrm{\Omega }(v_7),\mathrm{\Psi }(v_7)),Z(v_9)=(\mathrm{\Omega }(v_9),\mathrm{\Psi }(v_9))$ and $Z(v_{15})=(\mathrm{\Omega }(v_{15}),\mathrm{\Psi }(v_{15}))$ are weak induced subhypergraphs of ${\mathrm{\Delta }}^{\ast }$. Hence ${\mathrm{\Delta }}_1{\cup }_E{\mathrm{\Delta }}_2=\{Z(v_7),Z(v_9),Z(v_{15})\}$ is a soft directed hypergrah of ${\mathrm{\Delta }}^{\ast }$ and is given in Figure 9.

5. Extended intersection of two soft directed hypergraphs

Example 5.2. Examine a simple directed hypergraph ${\mathrm{\Delta }}^{\ast }=(\mathrm{\Gamma },\mathrm{\Xi })$ depicted in Figure 6, along with its corresponding soft directed hypergraphs ${\mathrm{\Delta }}_1$ illustrated in Figure 7 and ${\mathrm{\Delta }}_2$ depicted in Figure 8. The extended intersection of these two soft directed hypergraphs ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ is $\mathrm{\Delta }={\mathrm{\Delta }}_1{\cap }_E{\mathrm{\Delta }}_2=({\mathrm{\Delta }}^{\ast },\mathrm{\Omega },\mathrm{\Psi },\mathrm{\Pi })$ where $\mathrm{\Pi }={\mathrm{\Pi }}_1\cup {\mathrm{\Pi }}_2=\{v_7,v_9,v_{15}\}$. Also $\mathrm{\Omega }(v_7)={\mathrm{\Omega }}_1(v_7)=\{v_1,v_2,v_7,v_9,v_{12}\}$, $\mathrm{\Psi }(v_7)={\mathrm{\Psi }}_1(v_7)=\{(\{v_1,v_{12}\},\{v_7\}),(\{v_2,v_9\},\{v_7\})\}$, $\mathrm{\Omega }(v_9)={\mathrm{\Omega }}_1(v_9)\cap {\mathrm{\Omega }}_2(v_9)={\mathrm{\Omega }}_2(v_9)=\{v_6,v_7,v_9,v_{14}\}$, $\mathrm{\Psi }(v_9)=\{(T(e)\cap \mathrm{\Omega }(v_9),H(e)\cap \mathrm{\Omega }(v_9))|e\in \mathrm{\Xi }$and$T(e)\cap \mathrm{\Omega }(v_9)\ne \varphi$and$H(e)\cap \mathrm{\Omega }(v_9)\ne \varphi \}=\{(\{v_9\},\{v_6,v_7\}),(\{v_9\},\{v_{14}\})\}$, $\mathrm{\Omega }(v_{15})={\mathrm{\Omega }}_2(v_{15})=\{v_3,v_{10},v_{13},v_{14},v_{15}\}$ and $\mathrm{\Psi }(v_{15})={\mathrm{\Psi }}_2(v_{15})=\{(\{v_{15}\},\{v_3,v_{10},v_{14}\}),(\{v_{15}\},\{v_{13}\})\}.$ Here $(\mathrm{\Omega },\mathrm{\Pi })$ is a soft set over $\mathrm{\Gamma }$ and $(\mathrm{\Psi },\mathrm{\Pi })$ is a soft set over ${\mathrm{\Xi }}_s$. Also $Z(v_7)=(\mathrm{\Omega }(v_7),\mathrm{\Psi }(v_7)),Z(v_9)=(\mathrm{\Omega }(v_9),\mathrm{\Psi }(v_9))$ and $Z(v_{15})=(\mathrm{\Omega }(v_{15}),\mathrm{\Psi }(v_{15}))$ are weak induced subhypergraphs of ${\mathrm{\Delta }}^{\ast }$. Hence ${\mathrm{\Delta }}_1{\cap }_E{\mathrm{\Delta }}_2=\{Z(v_7),Z(v_9),Z(v_{15})\}$ is a soft directed hypergrah of ${\mathrm{\Delta }}^{\ast }$ and is given in Figure 10.