New concepts of domination in cubic graphs with application

Abstract

The cubic set, introduced as a combination of a fuzzy set and an interval-valued fuzzy set, provided researchers with more flexibility than the previous two sets in dealing with complex and uncertain problems. Fuzzy graphs, based on this type of set, are among the emerging fuzzy graphs that have a great potential to model the surrounding phenomena. Consistent with the special role that cubic graphs play in decision-making and selecting superior options, dominating these graphs is of great importance and value. In this paper, we introduce the domination of the cubic graphs in terms of strong edges and examine their properties. In addition, we examine domination in terms of independent sets and since many of the phenomena surrounding us are hybrid, we also discuss the domination concept on its fuzzy operations. Finally, we present an application of this graph on the subject of domination.

Keywords

Cubic graph dominating set independent cubic set cubic operations

1 Introduction

The research that Zadeh [71] conducted which focused on advancing the uncertainty theory led to the introduction of a new concept called fuzzy sets, which led to the many problems solution in areas such as topology, abstract algebra, geometry, graph theory, and analysis. This development is still ongoing in various branches. He later introduced a branch of fuzzy sets that showed the degree of membership, instead of assigning a number between 0 and 1 to each member of the set with an interval of numbers between 0 and 1. Zadeh’s ideas established many applications in various sciences such as decision analysis, information sciences, system sciences, control engineering, expert systems, pattern recognition, management sciences, etc. Kauffmann’s [23] ideas directed Rosenfeld [57] to introduce the concept of the fuzzy graph (FG). In FGs, fuzzy relations are introduced on fuzzy sets, which means that in addition to vertices, the edges are also in varying degrees between 0 and 1. Bhattacharya [6] presented some remarks on FGs and some operations on FGs were introduced by Mordeson and Peng [33]. Akram and Dudek [2] gave the idea of an interval-valued fuzzy graph (IVFG) in 2011. Atanassov [5] adopted the notion of membership and non-membership of an element in a set and proposed the idea of intuitionistic fuzzy sets (IFS). Talebi et al. [64 –68] analyzed several concepts on FG and interval-valued intuitionistic fuzzy graph (IVIFG). Rashmanlou et al. [50 –55] investigated novel concepts in fuzzy graphs. Other researchers have studied topics related to fuzzy graphs [4 , 58].

Jun et al. [17] introduced the idea of the cubic set (CS) and it was characterized by interval-valued fuzzy set and fuzzy set, which is a more general tool to capture uncertainty and vagueness. Through applying this concept, we can solve various problems instigated by uncertainties and have the best choice by using CSs in decision-making. Jun et al. [20] combined neutrosophic set with CSs and proposed the Neutrosophic-CS idea, and defined different operations. He being aided by his colleagues also applied this concept to algebraic disciplines [15 , 60]. Kang and Kim [22] investigated CSs mappings. Muhiuddin et al. [32] presented the stable-CSs idea. Rashid et al. [38] introduced the notion of a cubic graph (CG) where they introduced many new types of graphs and provided their application. Muhiuddin et al. [31] provided a modified definition of a CG. Kishore Kumar et al. [26] examined the regularity concept in CG. Other studies have been performed by some researchers in this regard [1 , 70].

The concept of dominating set (DS) in graphs was inspired by the chess moves introduced by O.Ore [43]. The initial foundation of domination in graphs was laid by T.W. Haynes et al. [12]. The domination in FG was introduced by Somasundaram and Somasundaram [61, 62]. Gani et al. [35 –42] conducted many studies on this concept. Gani and Chandrasekaran [35] introduced domination in FGs using strong edges. This concept was also studied by Mohideen and Ismayil [30]. Different definitions of domination in FGs were proposed and its different types were studied in FGs by researchers. Borzooei et al. [7 , 69] described several concepts on vague graphs. Parvathi et al. [44] introduced the domination in an intuitionistic fuzzy graph (IFG).

In this paper, we introduce the domination and some of its properties in a CG by strong edges. Next, we introduce an independent cubic set (ICS) and examined its relationship with the DS. We also studied the domination of some of the most important CG operations. Finally, we present an application of domination in the emergency stairs of buildings.

2 Preliminaries

In this section, some basic concepts are reviewed to enter the main discussion.

A fuzzy set (FS) is a pair (V, σ) where V is a non-empty set and σ : V → [0, 1] is a membership function. For two FS σ₁ and σ₂, we define $\begin{matrix} (σ_{1} \land σ_{2}) (x) = min {σ_{1} (x), σ_{2} (x)} \\ (σ_{1} \lor σ_{2}) (x) = max {σ_{1} (x), σ_{2} (x)} \\ σ_{1} \subseteq σ_{2} \Leftrightarrow σ_{1} (x) \leq σ_{2} (x), \forall x \in V . \end{matrix}$ The complement of σ denoted by $\bar{σ}$ , is defined by $\bar{σ} (x) = 1 - σ (x), \forall x \in V .$ By an interval number we mean a closed subinterval [a, b] of [0, 1], where 0 ≤ a ≤ b ≤ 1. The interval number [a, a] is denoted by a. The set of interval numbers is denoted by D [0, 1]. For two interval number c₁ = [a₁, b₁] and c₂ = [a₂, b₂], we define $\begin{matrix} r min {c_{1}, c_{2}} = [min {a_{1}, a_{2}}, min {b_{1}, b_{2}}] \\ c_{1} \geq c_{2} \Leftrightarrow a_{1} \geq a_{2} and b_{1} \geq b_{2} . \end{matrix}$ The complement of c ∈ D [0, 1], denoted by $\bar{c}$ , is defined by $\bar{c} = [1 - b, 1 - a] .$

A function μ : V → D [0, 1] is called an interval-valued fuzzy set (IVFS) in V. For x ∈ V, μ (x) = [μ^L (x) , μ^U (x)] is called the membership degree of element x, where μ^L, μ^U : V → [0, 1]. For every μ, ν ∈ D [0, 1], we define $μ \subseteq ν \Leftrightarrow μ (x) \leq ν (x), \forall x \in V .$ The complement of $\bar{μ}$ of μ ∈ D [0, 1] is defined by $\bar{μ} (x) = [1 - μ^{U} (x), 1 - μ^{L} (x)], \forall x \in V .$ Let $A = {[μ_{i}^{L} (x), μ_{i}^{U} (x)] \subseteq D [0, 1] | x \in V, i \in Ω}$ be a set of all IVFSs in V, then

$\begin{matrix} ⋃_{i \in Ω} [μ_{i}^{L} (x), μ_{i}^{U} (x)] = & [sup_{i \in Ω} μ_{i}^{L} (x), sup_{i \in Ω} μ_{i}^{U} (x)] \\ ⋂_{i \in Ω} [μ_{i}^{L} (x), μ_{i}^{U} (x)] = & [inf_{i \in Ω} μ_{i}^{L} (x), inf_{i \in Ω} μ_{i}^{U} (x)], \\ for all x \in V, \end{matrix}$ By a cubic set (CS) in V we mean a structure $A = {〈 x, [μ^{L} (x), μ^{U} (x)], σ (x) 〉 | x \in V},$ in which [μ^L, μ^U] is an IVFS in V and σ is a FS in V. A CS A = {〈x, [μ^L (x) , μ^U (x)] , σ (x) 〉| x ∈ V} is simply denoted by A = 〈 [μ^L, μ^U] , σ〉. A CS A in which [μ^L (x) , μ^U (x)] = [0, 0] and σ (x) =1 (resp. [μ^L (x) , μ^U (x)] = [1, 1] and σ (x) =0) for x ∈ V is denoted by 0 (resp. 1).

A CS A = 〈 [μ^L, μ^U] , σ〉 in V is said to be an internal cubic set if μ^L (x) ≤ σ (x) ≤ μ^U (x), and said to an external cubic set if σ (x) ∉ [μ^L, μ^U], for all x ∈ V.

The complement of A = 〈 [μ^L, μ^U] , σ〉 is defined to be the CS $\bar{A} = {〈 x, [1 - μ^{U} (x), 1 - μ^{L} (x)], 1 - σ (x) 〉 | x \in V} .$ A graph is a pair G^* = (V, E), where V is a set of vertices of G^* and E is a set of edges of G^*. The set D ⊆ V is called a dominating set (DS) in G^* if, for every y ∈ V - D, there is some x ∈ D such that xy ∈ E. The minimum cardinality DSs in G^* is said the domination number of G^* and is denoted by γ.

An FG G = (φ, ψ) on underlying graph G^* = (V, E) consists of two FS φ and ψ on V and E, respectively, such that $ψ (x, y) \leq φ (x) \land φ (y), \forall (x, y) \in E .$ We call φ the fuzzy vertex function of V, and ψ the fuzzy edge function of E. Note that ψ is a symmetric fuzzy relation on φ. We use the notation xy for an element (x, y) ∈ E. The order of a FG is defined by p = ∑_x∈Vφ (x). The degree of a vertex x in G is defined by deg(x) = ∑_xy∈Eψ (xy).

Definition 2.1. [33] Let V≠ ∅ be a finite set. By a CG over V, we mean a pair $G = (A, B)$ in which $A = 〈 [μ_{A}^{L}, μ_{A}^{U}], σ_{A} 〉$ is a CS in V and $B = 〈 [μ_{B}^{L}, μ_{B}^{U}], σ_{B} 〉$ is a CS in V × V such that for all xy ∈ V × V $\begin{matrix} μ_{B}^{L} (xy) \leq μ_{A}^{L} (x) \land μ_{A}^{L} (y), \\ μ_{B}^{U} (xy) \leq μ_{A}^{U} (x) \land μ_{A}^{U} (y), \\ σ_{B} (xy) \leq σ_{A} (x) \land σ_{A} (y) . \end{matrix}$ Let $\begin{matrix} [μ_{A}^{* L}, μ_{A}^{* U}] = {x \in V | [μ_{A}^{L} (x), μ_{A}^{U} (x)] > [0, 0]}, \\ σ_{A}^{*} = {x \in V | σ_{A} (x) > 0}, \\ [μ_{B}^{* L}, μ_{B}^{* U}] = {xy \in V \times V | [μ_{B}^{L} (xy), μ_{A}^{U} (xy)] > [0, 0]}, \\ σ_{B}^{*} = {xy \in V \times V | σ_{B} (xy) > 0}, \end{matrix}$ and $V^{*} = [μ_{A}^{* L}, μ_{A}^{* U}] \cap σ_{A}^{*}$ , $E^{*} = [μ_{B}^{* L}, μ_{B}^{* U}] \cap σ_{B}^{*}$ . Then, $G^{*} = (V^{*}, E^{*})$ is a underlying crisp graph of $G$ .

Note:Although the cubic graph in graph theory is also referred to as 3-regular graphs, but the meaning of the CG throughout this article means a cubic fuzzy graph consisting of an IVFG and an FG.

Definition 2.2. [33] A CG $G = (A, B)$ over V^* is said to be complete if for all xy ∈ V^* × V^* $\begin{matrix} μ_{B}^{L} (xy) = μ_{A}^{L} (x) \land μ_{A}^{L} (y), \\ μ_{B}^{U} (xy) = μ_{A}^{U} (x) \land μ_{A}^{U} (y), \\ σ_{B} (xy) = σ_{A} (x) \land σ_{A} (y) . \end{matrix}$

Definition 2.3. Let $G = (A, B)$ be a CG. Then vertex cardinality and edge cardinality of $G$ are defined by $\begin{matrix} P = \sum_{x \in V^{*}} (\frac{1 - μ_{A}^{L} (x) + μ_{A}^{U} (x) + σ_{A} (x)}{3}) \\ Q = \sum_{xy \in E^{*}} (\frac{1 - μ_{B}^{L} (xy) + μ_{B}^{U} (xy) + σ_{B} (xy)}{3}) . \end{matrix}$

Definition 2.4. Let $G = (A, B)$ be a CG and S ⊆ V. Then cardinality of S is defined by $\begin{matrix} | S | = \sum_{x \in S} (\frac{1 - μ_{A}^{L} (x) + μ_{A}^{U} (x) + σ_{A} (x)}{3}) . \end{matrix}$

Definition 2.5. [33] Let $G = (A, B)$ be a CG over V^*. A cubic path in $G$ is a sequence P_c : x₁x₂ ⋯ x_r of distinct vertics of V such that $[μ_{B}^{L} (x_{c - 1} x_{c}), μ_{B}^{U} (x_{c - 1} x_{c})] >] 0, 0]$ and σ_B (x_c-1x_c) >0 for c = 1, 2, ⋯ , r. We say that r is the length of the cubic path P_c, and P_c : x₁x₂ ⋯ x_r is a cubic path between x₀ and x_r.

The strength of P_c is defined by $\begin{matrix} S (P_{c}) = \\ 〈 ⋂_{i = 1}^{r} [μ_{B}^{L} (x_{i - 1} x_{i}), μ_{B}^{U} (x_{i - 1} x_{i})], ⋀_{i = 1}^{r} σ_{B} (x_{i - 1} x_{i}) 〉 . \end{matrix}$ The strength of connectedness between vertex x and vertex y is denoted by $〈 [μ_{B}^{\infty L} (xy), μ_{B}^{\infty U} (xy)], σ_{B}^{\infty} (xy) 〉$ and it is the maximum of the strengths of all cubic paths between x and y.

Definition 2.6. Let $G = (A, B)$ be a CG on V^*. Then an edge xy ∈ E^* is said to be cubic strong edge in $G$ if $\begin{matrix} μ_{B}^{L} (xy) \geq μ_{B}^{\infty L} (xy), μ_{B}^{U} (xy) \geq μ_{B}^{\infty U} (xy), \\ σ_{B} (xy) \geq σ_{B}^{\infty} (xy) . \end{matrix}$ If every xy ∈ E^* be strong, then $G$ is said to be strong.

Table 1
Some basic notations

Notation Meaning

CS Cubic Set

FS Fuzzy Set

FG Fuzzy Graph

IFG Intuitionistic Fuzzy Graph

IVFG Interval Valued Fuzzy Graph

CG Cubic Graph

DS Dominating Set

IVFS Interval Valued Fuzzy Set

ICS Independent Cubic Set

Notation	Meaning
CS	Cubic Set
FS	Fuzzy Set
FG	Fuzzy Graph
IFG	Intuitionistic Fuzzy Graph
IVFG	Interval Valued Fuzzy Graph
CG	Cubic Graph
DS	Dominating Set
IVFS	Interval Valued Fuzzy Set
ICS	Independent Cubic Set

3 Domination in cubic graphs

In this section, we define DS in a CG and examine some of its properties.

Definition 3.1. Let $G = (A, B)$ be a CG over V^* and x, y ∈ V^*, then x dominates y, if there exists a cubic strong edge between x and y.

Definition 3.2. Let $G = (A, B)$ be a CG over V^*. S ⊆ V^* is called a DS in $G$ if for every y ∉ S, there exists x ∈ S such that x dominates y. The minimum cardinality of all DSs in $G$ is called the domination number of $G$ and denoted by $γ (G)$ or by simply γ. In this case, S is called a γ-set.

Example 3.3. Consider a CG $G = (A, B)$ as shown in Figure 1.

Fig. 1

CG $G = (A, B)$ .

All edges are strong except ab and bd. The DSs for Figure 1 are as follows: $\begin{matrix} S_{1} = {a, c}, S_{2} = {b, e}, S_{3} = {c, e}, \\ S_{4} = {a, b}, S_{5} = {b, d} . \end{matrix}$ After calculating the cardinality of S₁, S₂, S₃, S₄, S₅, we obtain $\begin{matrix} | S_{1} | = 1.2, | S_{2} | = 1.36, | S_{3} | = 1.33, \\ | S_{4} | = 1.23, | S_{5} | = 1.13 . \end{matrix}$ It is clear that S₅ has the minimum cardinality among other DSs. So, γ = 1.13.

Definition 3.4. A DS S of CG $G$ is called minimal DS if for every vertex x ∈ S, the set S - {x} is not a DS, i.e., no proper subset of S is a DS of $G$ .

Remark 3.5. The domination is a symmetric relation an V^*.

Definition 3.6. Let $G = (A, B)$ be a CG. A cubic strong neighbor of x, is defined as: $N (x) = {y \in V | xy is a cubic strong edge} .$ Furthermore, $N [x] = N (x) \cup {x}$ is called a closed cubic strong neighbor of x. For non-empty set of S, we have $N (S) = {N (x) | x \in S}, N [S] = N (S) \cup S .$

Definition 3.7. The cubic strong neighborhood degree of vertex x is defined as: $\begin{matrix} D N (x) = \\ 〈 [\sum_{w \in N (x)} μ^{L} (w), \sum_{w \in N (x)} μ^{U} (w)], \sum_{w \in N (x)} σ (w) 〉, \end{matrix}$ and the minimum and maximum cardinality cubic strong neighborhood of $G$ are denoted by $δ_{N}$ and $Δ_{N}$ , respectively.

Example 3.8. Let $G = (A, B)$ be a CG as shown in Figure 2. All edges are strong. We have

$\begin{matrix} N (a) = {d}, N (b) = {c, d}, N (c) = {b, d}, \\ N (d) = {a, b, c} . \end{matrix}$ Thus, $\begin{matrix} D N (a) = 〈 [0.5, 0.9], 0.3 〉, D N (b) = 〈 [0.6, 1.3], 0.7 〉, \\ D N (c) = 〈 [0.8, 1.5], 0.9 〉, D N (d) = 〈 [0.8, 1.7], 1.7 〉 . \end{matrix}$ So, $δ_{N} = | 〈 [0.5, 0.9], 0.3 〉 | = 0.57$ , and $Δ_{N} = | 〈 [0.8, 1.7], 1.7 〉 | = 1.2$ .

Fig. 2

A strong CG.

Definition 3.9. In a CG $G = (A, B)$ , a vertex x ∈ V is called an isolated vertex if $N (x) = \emptyset$ , i.e., for any y ∈ V where x ≠ y, xy is not a cubic strong edge.

Theorem 3.10. If $G = (A, B)$ be a CG without isolated vertices and S is the minimal DS in $G$ , then V - S is a DS.

Proof. Since $G$ has no isolated vertex, so every vertex x ∈ V must be dominated by at least one vertex in S - {x}. This is in contradiction with the minimality of S. Hence, V - S is a DS. □

Theorem 3.11. A DS S of a CG $G = (A, B)$ is a minimal DS if and only if for each z ∈ S one of the following two conditions holds.

$i) N (z) \cap S = \emptyset$ .

ii) There is a vertex w ∈ V^* - S such that $N (w) \cap S = {z}$ .

Proof. Let S be a minimal DS of $G$ . Then, for every vertex z ∈ S, S - {z} is not a DS and there exists w ∈ V^* - (S - {z}) which is not dominated by any vertex in S - {z}. If w = z, then (i) holds.

If w ≠ z, then w is not dominated by S - {z}, but is dominated by S, i.e., w is dominated only by z in S. Thus, $N (w) \cap S = {z}$

Conversely, let S be a DS and for every vertex z ∈ S, one of two conditions holds. Suppose S be not a minimal DS. Then there exists a vertex z ∈ S such that S - {z} is a DS. Therefore, z is dominated by at least on vertex in S - {z} which implies that the condition (ii) does not hold. This is a contradiction. Thus S must be a minimal DS. □

Corollary 3.12. If $G = (A, B)$ be a CG without isolated vertex, then $γ \leq \frac{P}{2} .$

Proof. Let S be a minimal DS of $G$ . So, V^* - S is a DS too. Since P = |V^*| = |S| + |V^* - S|, at least one of the sets S or V^* - S, has the cardinality less than or equal $\frac{P}{2}$ .

Theorem 3.13. If $G = (A, B)$ be CG without isolated vertex, then $γ \leq P - Δ_{N} and γ \leq P - δ_{N} .$

Proof. Let u be a vertex in CG $G = (A, B)$ . Assume that $D N (u) = Δ_{N}$ . Let $V^{*} - N (u)$ is a DS of $G$ so that $γ \leq | V^{*} - N (u) | = P - Δ_{N} .$ Similarly, since $δ_{N} \leq Δ_{N}$ . Hence, $γ \leq P - δ_{N}$ . □

Definition 3.14. Two vertices in a CG are said to be independent if there is no cubic strong edge between them. A subset F ⊆ V^* is said to be independent cubic set (ICS) if $\begin{matrix} μ_{B}^{L} (xy) < μ_{A}^{L} (x) \land μ_{A}^{L} (y) \\ μ_{B}^{U} (xy) < μ_{A}^{U} (x) \land μ_{A}^{U} (y) \\ σ_{B} (xy) < σ_{A} (x) \land σ_{A}^{L} (y), for all x, y \in F . \end{matrix}$ A ICS F is said to be maximal if for every w ∈ V^* - F, F ∪ {w} is not a ICS.

Theorem 3.15. A ICS is a maximal ICS of CG $G = (A, B)$ if and only if it is both ICS and DS.

Proof. Let F be a maximal ICS in a CG. So, for every vertex z ∈ V^* - F, the set F ∪ {z} is not independent. For every vertex z ∈ V^* - F, there is a vertex t ∈ F such that t is a strong neighbor to z. So, F is a DS. Hence, F is both DS and ICS.

Conversely, assume F is both ICS and DS. Suppose F is not maximal ICS, then there exists a vertex z ∈ V^* - F, the set F ∪ {z} is ICS. If F ∪ {z} is ICS, then no vertex in F is strong neighbor to z. Hence, F can not be a DS, which is a contradiction. Hence, F is a maximal ICS. □

Theorem 3.16. Every maximal ICS in a CG $G = (A, B)$ is a minimal DS.

Proof. Let F be a maximal ICS in a CG. By Theorem 3.15, F is a DS. Suppose F is not a minimal DS, then there exists at least one vertex z ∈ F for which F - {z} is a DS. But if F - {z} dominates V^* - (F - {z}), then at least one vertex in F - {z} must be strong neighbor to z. This is contradiction. So, F must be a minimal DS. □

Definition 3.17. A DS S in a CG $G = (A, B)$ is said to be independent dominating set (IDS) in $G$ if S is ICS.

An IDS S in a CG $G$ is said to be maximal IDS, if for every z ∈ V^* - S, the set S ∪ {z} is not IDS. The minimum cardinality among all maximal IDSs is called lower independent dominating number of $G$ , and it is denoted by $γ_{i} (G)$ or simply γ_i.

The maximum cardinality among all maximal IDSs is called upper independent dominating number of $G$ , and it is denoted by $Γ_{i} (G)$ or simply Γ_i.

Example 3.18. The IDSs in Figure 2, are S₁ = {a, b} and S₂ = {a, c}. S₂ is maximal IDS of minimum cardinality γ_i = 1.24, and S₁ is a maximal IDS of maximum cardinality Γ_i = 1.3.

Remark 3.19. If $G$ be a CG, then γ ≤ γ_i .

Theorem 3.20. Let S be a γ-set of a CG. If the subgraph 〈S〉 induced by S has isolated vertices, then γ = γ_i.

Proof. Since S is a γ-set of a CG, So, S is a DS. Suppose that the subgraph 〈S〉 induced by S has isolated vertices. Hence, all vertices of S are independent. Thus, γ = γ_i.□

Corollary 3.21. If $G = (A, B)$ be a complete-CG, then γ = γ_i.

Theorem 3.22. Let $G = (A, B)$ be a connected-CG without isolated vertices. If G^* is not cycle or does not have an induced cycle subgraph, then $\begin{matrix} γ \leq P - Γ_{i} \leq P - γ_{i} . \end{matrix}$

Proof. Let G be a connected-CG without isolated vertices, such that $G^{*}$ is not a cycle and S be a maximal ICS in $G$ . Then V^* - S is a DS in $G$ . Hence, $γ \leq | V^{*} - S | = P - Γ_{i} .$ Similarly, since γ_i ≤ Γ_i. Hence, γ ≤ P - γ_i.□

4 Domination on operations in cubic graphs

In this section, we examine the domination of operations in CG.

Definition 4.1. Let $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ be two CGs of the underlying graphs $G_{1}^{*}$ and $G_{2}^{*}$ , respectively. The cartesian product of $G_{1}$ and $G_{2}$ is denoted by $G_{1} \times G_{2} = (A_{1} \times A_{2}, B_{1} \times B_{2})$ and is defined as follows:

$\begin{matrix} (i) {\begin{matrix} [μ_{A_{1}}^{L} \times μ_{A_{2}}^{L}, μ_{A_{1}}^{U} \times μ_{A_{2}}^{U}] (x_{1}, x_{2}) = [μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (x_{2}), μ_{A_{1}}^{U} (x_{1}) \land μ_{A_{2}}^{U} (x_{2})], \\ (σ_{A_{1}} \times σ_{A_{2}}) (x_{1}, x_{2}) = σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (x_{2}), \forall (x_{1}, x_{2}) \in V^{*} = V_{1}^{*} \times V_{2}^{*} . \end{matrix} \\ (ii) {\begin{matrix} [μ_{B_{1}}^{L} \times μ_{B_{2}}^{L}, μ_{B_{1}}^{U} \times μ_{B_{2}}^{U}] ((x_{1}, x_{2}) (x_{1}, y_{2})) = [μ_{A_{1}}^{L} (x_{1}) \land μ_{B_{2}}^{L} (x_{2} y_{2}), μ_{A_{1}}^{U} (x_{1}) \land μ_{B_{2}}^{U} (x_{2} y_{2})], \\ (σ_{B_{1}} \times σ_{B_{2}}) ((x_{1}, x_{2}) (x_{1}, y_{2})) = σ_{A_{1}} (x_{1}) \land σ_{B_{2}} (x_{2} y_{2}), \forall x_{1} \in V_{1}^{*}, x_{2} y_{2} \in E_{2}^{*} . \end{matrix} \\ (iii) {\begin{matrix} [μ_{B_{1}}^{L} \times μ_{B_{2}}^{L}, μ_{B_{1}}^{U} \times μ_{B_{2}}^{U}] ((x_{1}, x_{2}) (y_{1}, x_{2})) = [μ_{B_{1}}^{L} (x_{1} y_{1}) \land μ_{A_{2}}^{L} (x_{2}), μ_{B_{1}}^{U} (x_{1} y_{1}) \land μ_{A_{2}}^{U} (x_{2})], \\ (σ_{B_{1}} \times σ_{B_{2}}) ((x_{1}, x_{2}) (y_{1}, x_{2})) = σ_{B_{1}} (x_{1} y_{1}) \land σ_{A_{2}} (x_{2}), \forall x_{2} \in V_{2}^{*}, x_{1} y_{1} \in E_{1}^{*} . \end{matrix} \end{matrix}$

Example 4.2. Consider two CGs $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ as shown in Figure 3. The cartesian product of the CGs $G_{1}$ and $G_{2}$ is shown in Figure 4.

Fig. 3

CGs $G_{1}$ and $G_{2}$ .

Fig. 4

CG $G_{1} \times G_{2}$ .

Remark 4.3. If a vertex x₁ dominates a vertex y₁ in $G_{1}$ and a vertex x₂ dominates a vertex y₂ in $G_{2}$ , then the avertex (x₁, x₂) dominates the vertex x₁y₂ as shown in Figure 4.

Theorem 4.4. Let $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ be CGs with DSs S₁ and S₂, respectively. Then $V_{1}^{*} \times S_{2}$ and $S_{1} \times V_{2}^{*}$ are DSs of $G_{1} \times G_{2}$ .

Proof. Let $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ be CGs with DSs S₁ and S₂, respectively. So. every vertex in $G_{1}$ and $G_{2}$ be dominated by a vertex in S₁ and S₂, respectively. Case (i): $x \in V_{1}^{*}$ and $x_{2} y_{2} \in E_{2}^{*}$ . If the edge $x_{2} y_{2} \in E_{2}^{*}$ be cubic strong edge in $G_{2}$ , then $\begin{matrix} (μ_{B_{1}}^{L} \times μ_{B_{2}}^{L}) ((x_{1}, x_{2}) (x_{1}, y_{2})) \\ = μ_{A_{1}}^{L} (x_{1}) \land μ_{B_{2}}^{L} (x_{2} y_{2}) \\ = μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (x_{2}) \land μ_{A_{2}}^{L} (y_{2}) \\ = μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (x_{2}) \land μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (y_{2}) \\ = (μ_{A_{1}}^{L} \times μ_{A_{2}}^{L}) (x_{1}, x_{2}) \land (μ_{A_{1}}^{L} \times μ_{A_{2}}^{L}) (x_{1}, y_{2}) . \end{matrix}$ Similarly, $(μ_{B_{1}}^{U} \times μ_{B_{2}}^{U}) ((x_{1}, x_{2}) (x_{1}, y_{2})) = (μ_{A_{1}}^{U} \times μ_{A_{2}}^{U}) (x_{1}, x_{2}) \land (μ_{A_{1}}^{U} \times μ_{A_{2}}^{U}) (x_{1}, y_{2})$ . Also, $\begin{matrix} (σ_{B_{1}} \times σ_{B_{2}}) ((x_{1}, x_{2}) (x_{1}, y_{2})) \\ = σ_{A_{1}} (x_{1}) \land σ_{B_{2}} (x_{2} y_{2}) \\ = σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (x_{2}) \land σ_{A_{2}} (y_{2}) \\ = σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (x_{2}) \land σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (y_{2}) \\ = (σ_{A_{1}} \times σ_{A_{2}}) (x_{1}, x_{2}) \land (σ_{A_{1}} \times σ_{A_{2}}) (x_{1}, y_{2}) . \end{matrix}$ This gives results that (x₁, x₂) (x₁, y₂) is a cubic strong edge in $G_{1} \times G_{2}$ whose end vertices belong to $V_{1}^{*} \times S_{2}$ . Since $x_{2} y_{2} \in E_{2}^{*}$ is a cubic strong edge such that x₂ ∈ S₂ or y₂ ∈ S₂. Hence, the set $V_{1}^{*} \times S_{2}$ is dominated the all vertices in $G_{1} \times G_{2}$ .

case (ii): $x_{2} \in V_{2}^{*}$ , and $x_{1} y_{1} \in E_{1}^{*}$ . If the edge $x_{1} y_{1} \in E_{1}^{*}$ be cubic strong edge in $G_{1}$ , then $\begin{matrix} (μ_{B_{1}}^{L} \times μ_{B_{2}}^{L}) ((x_{1}, x_{2}) (y_{1}, x_{2})) \\ = μ_{B_{1}}^{L} (x_{1} y_{1}) \land μ_{A_{2}}^{L} (x_{2}) \\ = μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{1}}^{L} (y_{1}) \land μ_{A_{2}}^{L} (x_{2}) \\ = μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (x_{2}) \land μ_{A_{1}}^{L} (y_{1}) \land μ_{A_{2}}^{L} (x_{2}) \\ = (μ_{A_{1}}^{L} \times μ_{A_{2}}^{L}) (x_{1}, x_{2}) \land (μ_{A_{1}}^{L} \times μ_{A_{2}}^{L}) (y_{1}, x_{2}) . \end{matrix}$

Similarly, $(μ_{B_{1}}^{U} \times μ_{B_{2}}^{U}) ((x_{1}, x_{2}) (y_{1}, x_{2})) = (μ_{A_{1}}^{U} \times μ_{A_{2}}^{U}) (x_{1}, x_{2}) \land (μ_{A_{1}}^{U} \times μ_{A_{2}}^{U}) (y_{1}, x_{2})$ . Also, $\begin{matrix} (σ_{B_{1}} \times σ_{B_{2}}) ((x_{1}, x_{2}) (y_{1}, x_{2})) \\ = σ_{B_{1}} (x_{1} y_{1}) \land σ_{A_{2}} (x_{2}) \\ = σ_{A_{1}} (x_{1}) \land σ_{A_{1}} (y_{1}) \land σ_{A_{2}} (x_{2}) \\ = σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (x_{2}) \land σ_{A_{1}} (y_{1}) \land σ_{A_{2}} (x_{2}) \\ = (σ_{A_{1}} \times σ_{A_{2}}) (x_{1}, x_{2}) \land (σ_{A_{1}} \times σ_{A_{2}}) (y_{1}, x_{2}) . \end{matrix}$ Therefore, (x₁, x₂) (y₁, x₂) is a cubic strong edge in $G_{1} \times G_{2}$ whose end vertices belong to $S_{1} \times V_{2}^{*}$ . Since $x_{1} y_{1} \in E_{1}^{*}$ is a cubic strong edge such that x₁ ∈ S₁ or y₁ ∈ S₁. So $S_{1} \times V_{2}^{*}$ is DS in $G_{1} \times G_{2}$ .□

Corollary 4.5. If S₁ and S₂ are minimum DSs of CGs $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ respectively, then $γ (G_{1} \times G_{2}) \leq | S_{1} \times V_{2}^{*} | \land | V_{1}^{*} \times S_{2} | .$

Example 4.6. In Example 4.2, we have $\begin{matrix} S_{1} = {x_{1}}, S_{2} = {y_{2}}, | S_{1} \times V_{2}^{*} | = 1.14, \\ | V_{1}^{*} \times S_{2} | = 1.10, γ (G_{1} \times G_{2}) = 1.10 . \end{matrix}$ Clearly, 1.10 ≤ min {1.14, 1.10}.

Definition 4.7. Let $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ be two CGs of the underlying graphs $G_{1}^{*}$ and $G_{2}^{*}$ , respectively. The composition of $G_{1}$ and $G_{2}$ is denoted by $G_{1} \circ G_{2} = (A_{1} \circ A_{2}, B_{1} \circ B_{2})$ and is defined as follows:

$\begin{matrix} (i) {\begin{matrix} [μ_{A_{1}}^{L} \circ μ_{A_{2}}^{L}, μ_{A_{1}}^{U} \circ μ_{A_{2}}^{U}] (x_{1}, x_{2}) = [μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (x_{2}), μ_{A_{1}}^{U} (x_{1}) \land μ_{A_{2}}^{U} (x_{2})] \\ (σ_{A_{1}} \circ σ_{A_{2}}) (x_{1}, x_{2}) = σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (x_{2}) \\ \forall (x_{1}, x_{2}) \in V^{*} = V_{1}^{*} \times V_{2}^{*}, \end{matrix} \\ (ii) {\begin{matrix} [μ_{B_{1}}^{L} \circ μ_{B_{2}}^{L}, μ_{B_{1}}^{U} \circ μ_{B_{2}}^{U}] ((x_{1}, x_{2}) (x_{1}, y_{2})) = [μ_{A_{1}}^{L} (x_{1}) \land μ_{B_{2}}^{L} (x_{2} y_{2}), μ_{A_{1}}^{U} (x_{1}) \land μ_{B_{2}}^{U} (x_{2} y_{2})] \\ (σ_{B_{1}} \circ σ_{B_{2}}) ((x_{1}, x_{2}) (x_{1}, y_{2})) = σ_{A_{1}} (x_{1}) \land σ_{B_{2}} (x_{2} y_{2}) \\ \forall x_{1} \in V_{1}^{*}, x_{2} y_{2} \in E_{2}^{*}, \end{matrix} \\ (iii) {\begin{matrix} [μ_{B_{1}}^{L} \circ μ_{B_{2}}^{L}, μ_{B_{1}}^{U} \circ μ_{B_{2}}^{U}] ((x_{1}, x_{2}) (y_{1}, x_{2})) = [μ_{B_{1}}^{L} (x_{1} y_{1}) \land μ_{A_{2}}^{L} (x_{2}), μ_{B_{1}}^{U} (x_{1} y_{1}) \land μ_{A_{2}}^{U} (x_{2})] \\ (σ_{B_{1}} \circ σ_{B_{2}}) ((x_{1}, x_{2}) (y_{1}, x_{2})) = σ_{B_{1}} (x_{1} y_{1}) \land σ_{A_{2}} (x_{2}) \\ \forall x_{2} \in V_{2}^{*}, x_{1} y_{1} \in E_{1}^{*}, \end{matrix} \\ (iv) {\begin{matrix} [μ_{B_{1}}^{L} \circ μ_{B_{2}}^{L}, μ_{B_{1}}^{U} \circ μ_{B_{2}}^{U}] ((y_{1}, y_{2}) (x_{1}, x_{2})) = [μ_{A_{2}}^{L} (y_{2}) \land μ_{A_{2}}^{L} (x_{2}) \land μ_{B_{1}}^{L} (x_{1} y_{1}), \\ μ_{A_{2}}^{U} (y_{1}) \land μ_{A_{2}}^{U} (x_{2}) \land μ_{B_{1}}^{U} (x_{1} y_{1})] \\ (σ_{B_{1}} \circ σ_{B_{2}}) ((y_{1}, y_{2}) (x_{1}, x_{2})) = σ_{A_{2}} (y_{2}) \land σ_{A_{2}} (x_{2}) \land σ_{B_{1}} (x_{1} y_{1}) \\ \forall x_{2}, y_{2} \in V_{2}^{*}, x_{2} \neq y_{2}, x_{1} y_{1} \in E_{1}^{*} . \end{matrix} \end{matrix}$

Theorem 4.8. Let S₁ and S₂ be DSs of the CGs $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ , respectively. Then, S₁ × S₂ is a DS of $G_{1} \circ G_{2}$ .

Proof. Let (x, y) ∉ S₁ × S₂, Then x ∉ S₁ or y ∉ S₂.

Case (i): x ∉ S₁ and y ∈ S₂. Let x₁ ∈ S₁ dominates x. So, $\begin{matrix} μ_{B_{1}}^{L} ({xx}_{1}) = μ_{A_{1}}^{L} (x) \land μ_{A_{1}}^{L} (x_{1}), \\ μ_{B_{1}}^{U} ({xx}_{1}) = μ_{A_{1}}^{U} (x) \land μ_{A_{1}}^{U} (x_{1}), \\ σ_{B_{1}} ({xx}_{1}) = σ_{A_{1}} (x) \land σ_{A_{1}} (x_{1}) . \end{matrix}$ Therefore, (x₁, y) ∈ S₁ × S₂ and we have $\begin{matrix} (μ_{B_{1}}^{L} \circ μ_{B_{2}}^{L}) ((x, y) (x_{1}, y)) \\ = μ_{B_{1}}^{L} ({xx}_{1}) \land μ_{A_{2}}^{L} (y) \\ = μ_{A_{1}}^{L} (x) \land μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (y) \\ = μ_{A_{1}}^{L} (x) \land μ_{A_{2}}^{L} (y) \land μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (y) \\ = (μ_{A_{1}}^{L} \circ μ_{A_{2}}^{L}) (x, y) \land (μ_{A_{1}}^{L} \circ μ_{A_{2}}^{L}) (x_{1}, y) . \end{matrix}$ Similarly, $(μ_{B_{1}}^{U} \circ μ_{B_{2}}^{U}) ((x, y) (x_{1}, y)) = (μ_{A_{1}}^{U} \circ μ_{A_{2}}^{U}) (x, y) \land (μ_{A_{1}}^{U} \circ μ_{A_{2}}^{U}) (x_{1}, y)$ . Also, $\begin{matrix} (σ_{B_{1}} \circ σ_{B_{2}}) ((x, y) (x_{1}, y)) \\ = σ_{B_{1}} ({xx}_{1}) \land σ_{A_{2}} (y) \\ = σ_{A_{1}} (x) \land σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (y) \\ = σ_{A_{1}} (x) \land σ_{A_{2}} (y) \land σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (y) \\ = (σ_{A_{1}} \circ σ_{A_{2}}) (x, y) \land (σ_{A_{1}} \circ σ_{A_{2}}) (x_{1}, y) . \end{matrix}$ Thus, (x₁, y) dominates (x, y) in $G_{1} \circ G_{2}$ .

Case (ii): x ∈ S₁ and y ∉ S₂. Let y₁ ∈ S₂ dominates y. Then, (x, y₁) ∈ S₁ × S₂ and

$\begin{matrix} (μ_{B_{1}}^{L} \circ μ_{B_{2}}^{L}) ((x, y) (x, y_{1})) \\ = μ_{A_{1}}^{L} (x) \land μ_{B_{2}}^{L} ({yy}_{1}) \end{matrix}$ $\begin{matrix} = μ_{A_{1}}^{L} (x) \land μ_{A_{2}}^{L} (y) \land μ_{A_{2}}^{L} (y_{1}) \\ = μ_{A_{1}}^{L} (x) \land μ_{A_{2}}^{L} (y) \land μ_{A_{1}}^{L} (x) \land μ_{A_{2}}^{L} (y_{1}) \\ = (μ_{A_{1}}^{L} \circ μ_{A_{2}}^{L}) (x, y) \land (μ_{A_{1}}^{L} \circ μ_{A_{2}}^{L}) (x, y_{1}) . \end{matrix}$ Similarly, $(μ_{B_{1}}^{U} \circ μ_{B_{2}}^{U}) ((x, y) (x, y_{1})) = (μ_{A_{1}}^{U} \circ μ_{A_{2}}^{U}) (x, y) \land (μ_{A_{1}}^{U} \circ μ_{A_{2}}^{U}) (x, y_{1})$ . $\begin{matrix} (σ_{B_{1}} \circ σ_{B_{2}}) ((x, y) (x_{1}, y)) \\ = σ_{A_{1}} (x) \land σ_{B_{2}} ({yy}_{1}) \\ = σ_{A_{1}} (x) \land σ_{A_{2}} (y) \land σ_{A_{2}} (y_{1}) \\ = σ_{A_{1}} (x) \land σ_{A_{2}} (y) \land σ_{A_{1}} (x) \land σ_{A_{2}} (y_{1}) \\ = (σ_{A_{1}} \circ σ_{A_{2}}) (x, y) \land (σ_{A_{1}} \circ σ_{A_{2}}) (x, y_{1}) . \end{matrix}$ Thus, (x, y₁) dominates (x, y) in $G_{1} \circ G_{2}$ .

Case (iii): x ∉ S₁ and y ∉ S₂. Let x₁ ∈ S₁, y₁ ∈ S₂ such that x₁ dominates x in $G_{1}$ and y₁ dominates y in $G_{2}$ . Then, (x₁, y₁) ∈ S₁ × S₂. We have $\begin{matrix} (μ_{B_{1}}^{L} \circ μ_{B_{2}}^{L}) ((x, y) (x_{1}, y_{1})) \\ = μ_{A_{2}}^{L} (y) \land μ_{A_{2}}^{L} (y_{1}) \land μ_{B_{1}}^{L} ({xx}_{1}) \\ = μ_{A_{2}}^{L} (y) \land μ_{A_{2}}^{L} (y_{1}) \land μ_{A_{1}}^{L} (x) \land μ_{A_{1}}^{L} (x_{1}) \\ = (μ_{A_{1}}^{L} \circ μ_{A_{2}}^{L}) (x, y) \land (μ_{A_{1}}^{L} \circ μ_{A_{2}}^{L}) (x_{1}, y_{1}) . \end{matrix}$ Similarly, $(μ_{B_{1}}^{U} \circ μ_{B_{2}}^{U}) ((x, y) (x_{1}, y_{1})) = (μ_{A_{1}}^{U} \circ μ_{A_{2}}^{U}) (x, y) \land (μ_{A_{1}}^{U} \circ μ_{A_{2}}^{U}) (x_{1}, y_{1})$ . $\begin{matrix} (σ_{B_{1}} \circ σ_{B_{2}}) ((x, y) (x_{1}, y)) \\ = σ_{A_{2}} (y) \land σ_{A_{2}} (y_{1}) \land σ_{B_{1}} ({xx}_{1}) \\ = σ_{A_{2}} (y) \land σ_{A_{2}} (y_{1}) \land σ_{A_{1}} (x) \land σ_{A_{1}} (x_{1}) \\ = (σ_{A_{1}} \circ σ_{A_{2}}) (x, y) \land (σ_{A_{1}} \circ σ_{A_{2}}) (x_{1}, y_{1}) . \end{matrix}$ Thus, (x₁, y₁) dominates (x, y) in $G_{1} \circ G_{2}$ . Therefore, S₁ × S₂ is a DS of $G_{1} \circ G_{2}$ . □

Corollary 4.9. If S₁ and S₂ are γ-sets of CGs $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ , respectively, then $γ (G_{1} \circ G_{2}) \leq | S_{1} \times S_{2} | .$

Example 4.10. Consider two CGs as shown in Figure 3. The composition of $G_{1}$ and $G_{2}$ is shown inFigure 5. We have $S_{1} = {x_{1}}, S_{2} = {y_{2}}, | S_{1} \times S_{2} | = | {(x_{1} y_{2})} | = 0.57 .$ Since {y₁y₂} is γ-set in $G_{1} \circ G_{2}$ , so $γ (G_{1} \circ G_{2}) = 0.53 \leq 0.57 = | S_{1} \times S_{2} |$ .

Fig. 5

CG $G_{1} \circ G_{2}$ .

Definition 4.11. Let $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ are two CGs with underlying $G_{1}^{*}$ and $G_{2}^{*}$ , respectively, where $V_{1}^{*} \cap V_{2}^{*} = \emptyset$ . The direct product (simply product) $G_{1}$ and $G_{2}$ is denoted by $G_{1} \cdot G_{2} = (A_{1} \cdot A_{2}, B_{1} \cdot B_{2})$ and is defined as follows:

$\begin{matrix} (i) {\begin{matrix} [μ_{A_{1}}^{L} \cdot μ_{A_{2}}^{L}, μ_{A_{1}}^{U} \cdot μ_{A_{2}}^{U}] (x_{1}, x_{2}) = [μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (x_{2}), μ_{A_{1}}^{U} (x_{1}) \land μ_{A_{2}}^{U} (x_{2})] \\ (σ_{A_{1}} \cdot σ_{A_{2}}) (x_{1}, x_{2}) = σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (x_{2}) \\ \forall (x_{1}, x_{2}) \in V^{*} = V_{1}^{*} \times V_{2}^{*}, \end{matrix} \\ (ii) {\begin{matrix} [μ_{B_{1}}^{L} \cdot μ_{B_{2}}^{L}, μ_{B_{1}}^{U} \cdot μ_{B_{2}}^{U}] ((x_{1}, x_{2}) (x_{1}, y_{2})) = [μ_{B_{1}}^{L} (x_{1} y_{1}) \land μ_{B_{2}}^{L} (x_{2} y_{2}), μ_{B_{1}}^{U} (x_{1} y_{1}) \land μ_{B_{2}}^{U} (x_{2} y_{2})] \\ (σ_{B_{1}} \cdot σ_{B_{2}}) ((x_{1}, x_{2}) (x_{1}, y_{2})) = σ_{B_{1}} (x_{1} y_{1}) \land σ_{B_{2}} (x_{2} y_{2}) \\ \forall x_{1} y_{1} \in E_{1}^{*}, x_{2} y_{2} \in E_{2}^{*} . \end{matrix} \end{matrix}$

Theorem 4.12. Let S₁ and S₂ are two DSs for $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ , respectively. Then $V_{1}^{*} \times S_{2}$ or $S_{1} \times V_{2}^{*}$ is DS of $G_{1} \cdot G_{2}$ .

Proof. Let $G_{1}$ and $G_{2}$ are two CGs with DSs S₁ and S₂, respectively. For edge (x₁, x₂) (y₁, y₂) where $x_{1} y_{1} \in E_{1}^{*}$ and $x_{2} y_{2} \in E_{2}^{*}$ , we have

$\begin{matrix} (μ_{B_{1}}^{L} \cdot μ_{B_{2}}^{L}) ((x_{1}, x_{2}) (y_{1}, y_{2})) \\ = μ_{B_{1}}^{L} (x_{1} y_{1}) \land μ_{B_{2}}^{L} (x_{2} y_{2}) \\ = μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{1}}^{L} (y_{1}) \land μ_{A_{2}}^{L} (x_{2}) \land μ_{A_{2}}^{L} (y_{2}) \\ = (μ_{A_{1}}^{L} \cdot μ_{A_{2}}^{L}) (x_{1}, x_{2}) \land (μ_{A_{1}}^{L} \cdot μ_{A_{2}}^{L}) (y_{1}, y_{2}) . \end{matrix}$ Similarly, $(μ_{B_{1}}^{U} \cdot μ_{B_{2}}^{U}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = (μ_{A_{1}}^{U} \cdot μ_{A_{2}}^{U}) (x_{1}, x_{2}) \land (μ_{A_{1}}^{U} \cdot μ_{A_{2}}^{U}) (y_{1}, y_{2})$ . $\begin{matrix} (σ_{B_{1}} \cdot σ_{B_{2}}) ((x_{1}, x_{2}) (y_{1}, y_{2})) \\ = σ_{B_{1}} (x_{1} y_{1}) \land σ_{B_{2}} (x_{2} y_{2}) \\ = σ_{A_{1}} (x_{1}) \land σ_{A_{1}} (y_{1}) \land σ_{A_{2}} (x_{2}) \land σ_{A_{2}} (y_{2}) \\ = (σ_{A_{1}} \cdot σ_{A_{2}}) (x_{1}, x_{2}) \land (σ_{A_{1}} \cdot σ_{A_{2}}) (y_{1}, y_{2}) . \end{matrix}$ Therefore, (x₁, x₂) (y₁, y₂) is cubic strong edge whose end vertices belong to $V_{1}^{*} \times S_{2}$ or $S_{1} \times V_{2}^{*}$ . Thus, $V_{1}^{*} \times S_{2}$ or $S_{1} \times V_{2}^{*}$ is DS in $G_{1} \cdot G_{2}$ . □

Corollary 4.13. If S₁ and S₂ are γ-sets of CGs $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ , respectively, then $γ (G_{1} \cdot G_{2}) \leq | V_{1}^{*} \times S_{2} | \land | S_{1} \times V_{2}^{*} | .$

Example 4.14. Consider two CGs as shown in Figure 3. The direct product of $G_{1}$ and $G_{2}$ is shown inFigure 6. We have $\begin{matrix} S_{1} = {x_{1}}, S_{2} = {y_{2}}, \\ V_{1}^{*} \times S_{2} = {x_{1} y_{2}, y_{1} y_{2}}, \end{matrix}$

$\begin{matrix} S_{1} \times V_{2}^{*} = {x_{1} x_{2}, x_{2} y_{2}}, \\ γ (G_{1} \cdot G_{2}) = 1.10, \\ | V_{1}^{*} \times S_{2} | = 1.10, | S_{1} \times V_{2}^{*} | = 1.14 . \end{matrix}$ Thus 1.10 ≤ min {1.10, 1.14}.

Definition 4.15. Let $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ be two CGs on $G_{1}^{*}$ and $G_{2}^{*}$ , respectively. Then, the semi-strong product of $G_{1}$ and $G_{2}$ is denoted by $G_{1} • G_{2} = (A_{1} • A_{2}, B_{1} • B_{2})$ and is defined as follows:

$\begin{matrix} (i) {\begin{matrix} [μ_{A_{1}}^{L} • μ_{A_{2}}^{L}, μ_{A_{1}}^{U} • μ_{A_{2}}^{U}] (x_{1}, x_{2}) = [μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (x_{2}), μ_{A_{1}}^{U} (x_{1}) \land μ_{A_{2}}^{U} (x_{2})] \\ (σ_{A_{1}} • σ_{A_{2}}) (x_{1}, x_{2}) = σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (x_{2}) \\ \forall (x_{1}, x_{2}) \in V^{*} = V_{1}^{*} \times V_{2}^{*}, \end{matrix} \\ (ii) {\begin{matrix} [μ_{B_{1}}^{L} • μ_{B_{2}}^{L}, μ_{B_{1}}^{U} • μ_{B_{2}}^{U}] ((x_{1}, x_{2}) (x_{1}, y_{2})) = [μ_{A_{1}}^{L} (x_{1}) \land μ_{B_{2}}^{L} (x_{2} y_{2}), μ_{A_{1}}^{U} (x_{1}) \land μ_{B_{2}}^{U} (x_{2} y_{2})] \\ (σ_{B_{1}} • σ_{B_{2}}) ((x_{1}, x_{2}) (x_{1}, y_{2})) = σ_{A_{1}} (x_{1}) \land σ_{B_{2}} (x_{2} y_{2}) \\ \forall x_{1} \in V_{1}^{*}, x_{2} y_{2} \in E_{2}^{*}, \end{matrix} \\ (iii) {\begin{matrix} [μ_{B_{1}}^{L} • μ_{B_{2}}^{L}, μ_{B_{1}}^{U} • μ_{B_{2}}^{U}] ((x_{1}, x_{2}) (y_{1}, y_{2})) = [μ_{B_{1}}^{L} (x_{1} y_{1}) \land μ_{B_{2}}^{L} (x_{2} y_{2}), μ_{B_{1}}^{U} (x_{1} y_{1}) \land μ_{B_{2}}^{U} (x_{2} y_{2})] \\ (σ_{B_{1}} • σ_{B_{2}}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = σ_{B_{1}} (x_{1} y_{1}) \land σ_{B_{2}} (x_{2} y_{2}) \\ \forall x_{1} y_{1} \in E_{1}^{*}, x_{2} y_{2} \in E_{2}^{*}, \end{matrix} \end{matrix}$

Fig. 6

Direct product $G_{1}$ and $G_{2}$ .

Theorem 4.16 Let S₁ and S₂ be DSs of two CGs $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ , respectively. Then $V_{1}^{*} \times S_{2}$ is the minimum DS of $G_{1} • G_{2}$ .

Proof. Let $G_{1}$ and $G_{2}$ be two CGs with DSs S₁ and S₂, respectively. So every vertex in $G_{1}$ and $G_{2}$ be dominated by a vertex in S₁ and S₂, respectively. Consider edge $(x_{1}, x_{2}) (y_{1}, y_{2}) \in G_{1} • G_{2}$ .

Case (i): $\begin{matrix} (μ_{B_{1}}^{L} • μ_{B_{2}}^{L}) ((x_{1}, x_{2}) (y_{1}, y_{2})) \\ = μ_{A_{1}}^{L} (x_{1}) \land μ_{B_{1}}^{L} (x_{2} y_{2}) \\ = μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (x_{2}) \land μ_{A_{2}}^{L} (y_{2}) \\ = μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (x_{2}) \land μ_{A_{1}}^{L} (y_{1}) \land μ_{A_{2}}^{L} (y_{2}) \\ = (μ_{A_{1}}^{L} • μ_{A_{2}}^{L}) (x_{1}, x_{2}) \land (μ_{A_{1}}^{L} • μ_{A_{2}}^{L}) (y_{1}, y_{2}) . \end{matrix}$ Similarly, $(μ_{B_{1}}^{U} • μ_{B_{2}}^{U}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = (μ_{A_{1}}^{U} • μ_{A_{2}}^{U}) (x_{1}, x_{2}) \land (μ_{A_{1}}^{U} • μ_{A_{2}}^{U}) (y_{1}, y_{2})$ .

Also,

$\begin{matrix} (σ_{B_{1}} • σ_{B_{2}}) ((x_{1}, x_{2}) (y_{1}, y_{2})) \\ = σ_{A_{1}} (x_{1}) \land σ_{B_{2}} (x_{2} y_{2}) \\ = σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (x_{2}) \land σ_{A_{2}} (y_{2}) \\ = σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (x_{2}) \land σ_{A_{1}} (y_{1}) \land σ_{A_{2}} (y_{2}) \\ = (σ_{A_{1}} • σ_{A_{2}}) (x_{1}, x_{2}) \land (σ_{A_{1}} • σ_{A_{2}}) (y_{1}, y_{2}) . \end{matrix}$ Therefore, (x₁, x₂) (y₁, y₂) is cubic strong edge whose end vertices belong to $V_{1}^{*} \times S_{2}$ .

Case (ii): Let $x_{1} y_{1} \in E_{1}^{*}$ and $x_{2} y_{2} \in E_{2}^{*}$ . Let $x_{1} y_{1} \in E_{1}^{*}$ and $x_{2} y_{2} \in E_{2}^{*}$ be cubic strong edges in $G_{1}$ and $G_{2}$ , respectively. Then $\begin{matrix} (μ_{B_{1}}^{L} • μ_{B_{2}}^{L}) ((x_{1}, x_{2}) (y_{1}, y_{2})) \\ = μ_{B_{1}}^{L} (x_{1} y_{1}) \land μ_{B_{2}}^{L} (x_{2} y_{2}) \end{matrix}$

$\begin{matrix} = μ_{A_{1}}^{L} (x_{1}) \land μ_{A_{2}}^{L} (x_{2}) \land μ_{A_{1}}^{L} (y_{1}) \land μ_{A_{2}}^{L} (y_{2}) \\ = (μ_{A_{1}}^{L} • μ_{A_{2}}^{L}) (x_{1}, x_{2}) \land (μ_{A_{1}}^{L} • μ_{A_{2}}^{L}) (y_{1}, y_{2}) . \end{matrix}$ Similarly, $(μ_{B_{1}}^{U} • μ_{B_{2}}^{U}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = (μ_{A_{1}}^{U} • μ_{A_{2}}^{U}) (x_{1}, x_{2}) \land (μ_{A_{1}}^{U} • μ_{A_{2}}^{U}) (y_{1}, y_{2})$ .

Also, $\begin{matrix} (σ_{B_{1}} • σ_{B_{2}}) ((x_{1}, x_{2}) (y_{1}, y_{2})) \\ = σ_{B_{1}} (x_{1} y_{1}) \land σ_{B_{2}} (x_{2} y_{2}) \\ = σ_{A_{1}} (x_{1}) \land σ_{A_{2}} (x_{2}) \land σ_{A_{1}} (y_{1}) \land σ_{A_{2}} (y_{2}) \\ = (σ_{A_{1}} • σ_{A_{2}}) (x_{1}, x_{2}) \land (σ_{A_{1}} • σ_{A_{2}}) (y_{1}, y_{2}) . \end{matrix}$ So, (x₁, x₂) (y₁, y₂) is cubic strong edge whit end vertices belong to $V_{1}^{*} \times S_{2}$ . Thus, $V_{1}^{*} \times S_{2}$ is the minimum Ds of $G_{1} • G_{2}$ . □

Corollary 4.17. If S₁ and S₂ be two DSs of CGs $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ , respectively, then $γ (G_{1} • G_{2}) = | V_{1}^{*} \times S_{2} | .$

Example 4.18. Consider two CGs $G_{1}$ and $G_{2}$ as shown in Figure 3. The semi-strong product of $G_{1}$ and $G_{2}$ is shown in Figure 7.

Fig. 7

Semi-strong product two CGs $G_{1}$ and $G_{2}$ .

In here, we have $V_{1}^{*} = {x_{1}, y_{1}}, S_{2} = {y_{2}}, V_{1}^{*} \times S_{2} = {x_{1} y_{2}, y_{1} y_{2}} .$ So, $γ (G_{1} • G_{2}) = | V_{1}^{*} \times S_{2} | = 0.57 + 0.53 = 1.10$ .

Definition 4.19. Let $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ be two CGs on $G_{1}^{*}$ and $G_{2}^{*}$ , respectively. The union of two CGs $G_{1}$ and $G_{2}$ is denoted by $G_{1} \cup G_{2} = (A_{1} \cup A_{2}, B_{1} \cup B_{2})$ and is defined by:

$\begin{matrix} (i) [μ_{? A_{1}}^{L} \cup μ_{A_{2}}^{L}, μ_{? A_{1}}^{U} \cup μ_{A_{2}}^{U}] (x) = {\begin{matrix} [μ_{? A_{1}}^{L} (x), μ_{A_{1}}^{U} (x)] & If x \in V_{1}^{*} - V_{2}^{*}, \\ [μ_{? A_{2}}^{L} (x), μ_{A_{2}}^{U} (x)] & If x \in V_{2}^{*} - V_{1}^{*}, \\ [μ_{? A_{1}}^{L} (x) \lor μ_{A_{2}}^{L} (x), μ_{? A_{1}}^{U} (x) \lor μ_{A_{2}}^{U} (x)] & If x \in V_{1}^{*} \cap V_{2}^{*} . \end{matrix} \\ (ii) (σ_{? A_{1}} \cup σ_{A_{2}}) (x) = {\begin{matrix} σ_{A_{1}} (x) & If x \in V_{1}^{*} - V_{2}^{*}, \\ σ_{A_{2}} (x) & If x \in V_{2}^{*} - V_{1}^{*}, \\ σ_{A_{1}} (x) \lor σ_{A_{2}} (x) & If x \in V_{1}^{*} \cap V_{2}^{*} . \end{matrix} \\ (iii) [μ_{? B_{1}}^{L} \cup μ_{B_{2}}^{L}, μ_{? B_{1}}^{U} \cup μ_{B_{2}}^{U}] (xy) = {\begin{matrix} [μ_{? B_{1}}^{L} (xy), μ_{B_{1}}^{U} (xy)] & If xy \in E_{1}^{*} - E_{2}^{*}, \\ [μ_{? B_{2}}^{L} (xy), μ_{B_{2}}^{U} (xy)] & If xy \in E_{2}^{*} - E_{1}^{*}, \\ [μ_{? B_{1}}^{L} (xy) \lor μ_{B_{2}}^{L} (xy), μ_{? B_{1}}^{U} (xy) \lor μ_{B_{2}}^{U} (xy)] & If xy \in E_{1}^{*} \cap E_{2}^{*} . \end{matrix} \end{matrix}$ $\begin{matrix} (iv) (σ_{B_{1}} \cup σ_{B_{2}}) (xy) = {\begin{matrix} σ_{B_{1}} (xy) If xy \in E_{1}^{*} - E_{2}^{*}, \\ σ_{B_{2}} (xy) If xy \in E_{2}^{*} - E_{1}^{*}, \\ σ_{B_{1}} (xy) \lor σ_{B_{2}} (xy) If xy \in E_{1}^{*} \cap E_{2}^{*} . \end{matrix} \end{matrix}$

Remark 4.20. Since the DS S of $G_{1} \cup G_{2}$ is the form S = S₁ ∪ S₂, where S₁ is the DS of $G_{1}$ and S₂ is the DS of $G_{2}$ , then $γ (G_{1} \cup G_{2}) = γ (G_{1}) + γ (G_{2}) .$

Example 4.21. Consider two CGs $G_{1}$ and $G_{2}$ as shown in Figure 3. It is clear that $γ (G_{1} \cup G_{2}) = γ (G_{1}) + γ (G_{2}) = 0.57 + 0.53 = 1.10 .$

Definition 4.22. Let $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ be two CGs of $G_{1}^{*}$ and $G_{2}^{*}$ , respectively. The join of two CGs $G_{1}$ and $G_{2}$ is denoted by $G_{1} + G_{2} = (A_{1} + A_{2}, B_{1} + B_{2})$ and is defined as follows:

$\begin{matrix} (i) {\begin{matrix} [μ_{A_{1}}^{L} + μ_{A_{2}}^{L}, μ_{A_{1}}^{U} + μ_{A_{2}}^{U}] (x) = [μ_{A_{1}}^{L} \cup μ_{A_{2}}^{L}, μ_{A_{1}}^{U} \cup μ_{A_{2}}^{U}] (x), \\ (σ_{A_{1}} + σ_{A_{2}}) (x) = (σ_{A_{1}} \cup σ_{A_{2}}) (x), \\ for x \in V_{1}^{*} \cup V_{2}^{*}, \end{matrix} \\ (ii) {\begin{matrix} [μ_{B_{1}}^{L} + μ_{B_{2}}^{L}, μ_{B_{1}}^{U} + μ_{B_{2}}^{U}] (xy) = [μ_{B_{1}}^{L} \cup μ_{B_{2}}^{L}, μ_{B_{1}}^{U} \cup μ_{B_{2}}^{U}] (xy), \\ (σ_{B_{1}} + σ_{B_{2}}) (xy) = (σ_{B_{1}} \cup σ_{B_{2}}) (xy), \\ for xy \in E_{1}^{*} \cap E_{2}^{*}, \end{matrix} \\ (iii) {\begin{matrix} [μ_{B_{1}}^{L} + μ_{B_{2}}^{L}, μ_{B_{1}}^{U} + μ_{B_{2}}^{U}] (xy) = [μ_{A_{1}}^{L} (x) \land μ_{A_{1}}^{L} (y), μ_{A_{2}}^{U} (x) \land μ_{A_{2}}^{U}] (y), \\ (σ_{B_{1}} + σ_{B_{2}}) (xy) = σ_{A_{1}} (x) \land σ_{A_{2}} (y), \end{matrix} \end{matrix}$ for xy belong to the set of all edges joining the vertices of $V_{1}^{*}$ and $V_{2}^{*}$ .

Theorem 4.23. Let $G_{1} = (A_{1}, B_{1})$ and $G_{2} = (A_{2}, B_{2})$ be two CG on $V_{1}^{*}$ and $V_{2}^{*}$ , respectively. If $V_{1}^{*} \cap V_{2}^{*} = \emptyset$ , then $γ (G_{1} + G_{2}) = γ (G_{1}) \land γ (G_{2}) .$

Proof. Since every vertex of $V_{1}^{*}$ dominates by every vertex of $V_{2}^{*}$ in $G_{1} + G_{2}$ . Any DS in $G_{1} + G_{2}$ is either a subset of $V_{1}^{*}$ or a subset of $V_{2}^{*}$ . Thus, $γ (G_{1} + G_{2}) = γ (G_{1}) \land γ (G_{2}) .$

Example 4.24. Consider two CGs $G_{1}$ and $G_{2}$ as shown in Figure 3. The join of $G_{1}$ and $G_{2}$ is shown inFigure 8.

Fig. 8

The joining of two CGs $G_{1}$ and $G_{2}$ .

It is clear, S₁ = {x₁} and S₂ = {x₂} are γ-sets of $G_{1}$ and $G_{2}$ , respectively. Therefore, $γ (G_{1} + G_{2}) = min {| S_{1} |, | S_{2} |} = min {0.57, 0.53} = 0.53 .$

5 Application

Escape stairs are a kind of emergency exit in case of fire or emergency conditions that are installed outside or inside the building (outside the main building environment). The design and construction of escape stairs has a direct relationship with the number of floors of the building is known as the only way to increase the people’speed and ease when leaving the building in times of danger, especially fire.

All corridors that are available as an exit to evacuate people with more than 30 people, should be considered by a structure with at least 1-hour fire resistance from other parts of the building, and the doors that open to them must have a fire protection time of at least 20 minutes. A flow rate of 1.5 persons/m/s is assumed as the convention for the Building Standard Law of Japan. However, when a corridor connected to a door is crowded, the flow rate at the door decreases. In effect, when evacuees pass through an evacuation route, the flow rate can be said to be 1.5 persons/m/s until a density threshold is reached; after that, queuing begins and the door flow declines.

The maximum travel distance measured from the most remote point of a bedroom of a worker’s dormitory to the nearest exit staircase, in accordance to Table 2.2A of the Fire Code, should not exceed 60m (https.//www.Scdf.gov.sg/docs/default-source/scolf- library/ fssd-downloads/ hb v6 ch2.pdf). Everyone is faced with two choices in times of danger, whether to use the stairs between the floors to exit or the escape stairs. It is usually difficult for certain patients to leave and be admitted to escaping stairs, in which case using an elevator is a good option. In this study, escape stairs and stairs between floors in Moheb Mehr 5-story hospital were considered as the study points. Because the largest volume of evacuation in times of danger is through these points, we consider these points as vertices of a cubic graph.

The most important exit parameters are the time and distance to reach the nearest exit stairs and the number of crowd in front of the stairs. Because the probability of reaching the nearest exit stairs at any time, according to the desired time and distance, and the number of crowd in front of the exit stairs are unknown values, so we are faced with fuzzy values. Since calculating the time and distance to the nearest exit is a little more difficult than calculating the crowd in front of the exit stairs at any given moment, therefore, we denote these values by the interval-valued fuzzy number and the number of crowd by a fuzzy number according to Tables 2, 3, and 4.

Table 2
Cubic values of stairs

Floors Cubic Values Fire Exits Cubic Values

F5 〈[0.8, 0.9] , 0.2〉 E5 〈[0.6, 0.7] , 0.1〉

F4 〈[0.7, 0.8] , 0.3〉 E4 〈[0.5, 0.6] , 0.2〉

F3 〈[0.6, 0.7] , 0.5〉 E3 〈[0.4, 0.5] , 0.4〉

F2 〈[0.5, 0.6] , 0.7〉 E2 〈[0.3, 0.4] , 0.6〉

F1 〈[0.4, 0.5] , 0.9〉 E1 〈[0.2, 0.3] , 0.8〉

GF 〈[0.3, 0.4] , 1〉 GE 〈[0.1, 0.2] , 1〉

Floors	Cubic Values	Fire Exits	Cubic Values
F5	〈[0.8, 0.9] , 0.2〉	E5	〈[0.6, 0.7] , 0.1〉
F4	〈[0.7, 0.8] , 0.3〉	E4	〈[0.5, 0.6] , 0.2〉
F3	〈[0.6, 0.7] , 0.5〉	E3	〈[0.4, 0.5] , 0.4〉
F2	〈[0.5, 0.6] , 0.7〉	E2	〈[0.3, 0.4] , 0.6〉
F1	〈[0.4, 0.5] , 0.9〉	E1	〈[0.2, 0.3] , 0.8〉
GF	〈[0.3, 0.4] , 1〉	GE	〈[0.1, 0.2] , 1〉

Table 3

Cubic values among stairs

	GF	F1	F2	F3	F4	F5
GF		〈[0.3, 0.4] , 0.9〉
F1	〈[0.3, 0.4] , 0.9〉		〈[0.4, 0.5] , 0.7〉
F2		〈[0.4, 0.5] , 0.7〉		〈[0.5, 0.6] , 0.5〉
F3			〈[0.5, 0.6] , 0.5〉		〈[0.6, 0.7] , 0.3〉
F4				〈[0.6, 0.7] , 0.3〉		〈[0.7, 0.8] , 0.2〉
F5					〈[0.7, 0.8] , 0.2〉
GE	〈[0.1, 0.2] , 0.1〉
E1		〈[0.2, 0.3] , 0.8〉
E2			〈[0.3, 0.4] , 0.6〉
E3				〈[0.4, 0.5] , 0.4〉
E4					〈[0.5, 0.6] , 0.2〉
E5						〈[0.6, 0.7] , 0.1〉

Table 4

Cubic values among stairs

	GE	E1	E2	E3	E4	E5
GF	〈[0.1, 0.2] , 1〉
F1		〈[0.6, 0.3] , 0.8〉
F2			〈[0.3, 0.4] , 0.6〉
F3				〈[0.4, 0.5] , 0.4〉
F4					〈[0.5, 0.6] , 0.2〉
F5						〈[0.6, 0.7] , 0.1〉
GE		〈[0.1, 0.2] , 1〉
E1	〈[0.1, 0.2] , 0.8〉		〈[0.2, 0.3] , 0.6〉
E2		〈[0.2, 0.3] , 0.6〉		〈[0.3, 0.4] , 0.4〉
E3			〈[0.3, 0.4] , 0.4〉		〈[0.4, 0.5] , 0.2〉
E4				〈[0.4, 0.5] , 0.2〉		〈[0.5, 0.6] , 0.1〉
E5					〈[0.5, 0.6] , 0.1〉

It is natural that in the lower floors, the crowds at the exits are more and the probability of reaching in the shortest time and distance is less. Because the only exit routes from the hospital are emergency exits and stairwells between floors, according to the construction principles of this type of building, all edges are considered strong. In this study, we seek to determine the domination in the emergency exit stairs of this hospital to establish the necessary facilities and equipment, and not just determine the emergency exit routes.

Considering the CG shown in Figure 10, F shows the internal stairs between the hospital floors and E represents the emergency stairs of the different floors. Minimum DSs are $\begin{matrix} S_{1} = {F 5, F 1, E 3, GE}, | S_{1} | = 2.02 \\ S_{2} = {E 5, E 1, F 3, GF}, | S_{2} | = 2.26 \\ S_{3} = {F 5, F 2, E 4, GE}, | S_{3} | = 1.86 \\ S_{4} = {F 4, GF, E 5, E 2}, | S_{4} | = 2.12 \\ S_{5} = {F 4, F 1, E 4, E 1}, | S_{5} | = 2.18 . \end{matrix}$ Therefore, S₃ is a γ-set.

Fig. 9

Moheb Mehr Hospital floors plan.

Fig. 10

CG floors and fire exits.

According to the map of the floors shown in Figure 9, the maximum volume of referrals and hospitalizations is in these floors. According to the studies performed in this research, the following suggestions are recommended.

• Install wider emergency exit doors at domination points.

•Deployment of relief forces and special equipment in these areas.

•Installation of monitoring systems.

•Create the least crowded sections on the upper floors.

6 Conclusion

Graph theory is widely used in the study and modeling of problems in various fields such as economics, operations research, transportation, automata theory, and so on. Fuzzy graphs are used to solve problems in which some aspects are ambiguous. Cubic graphs based on cubic sets have better flexibility in solving indeterminate problems. Because this graph uses both fuzzy values ??and interval values, which gives a better ability to model phenomena. Especially when it comes to making decisions and choosing the final option. Since this parameter corresponds to the domination in graphs, therefore, in this study we studied the domination in a cubic graph based on strong edges. We also examined this concept in terms of independent sets. Since many of the phenomena around us are hybrid, we also discussed the concept of domination on its fuzzy. Operations in a CG could be the field of new research on this graph in the future. As an application, we used domination on the emergency stairs of a hospital. In this study, we obtained specific results regarding emergency stairs.

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New concepts of domination in cubic graphs with application

Abstract

Keywords

1 Introduction

2 Preliminaries

Table 1 Some basic notations Notation Meaning CS Cubic Set FS Fuzzy Set FG Fuzzy Graph IFG Intuitionistic Fuzzy Graph IVFG Interval Valued Fuzzy Graph CG Cubic Graph DS Dominating Set IVFS Interval Valued Fuzzy Set ICS Independent Cubic Set

References

Table 1
Some basic notations

Notation Meaning

CS Cubic Set

FS Fuzzy Set

FG Fuzzy Graph

IFG Intuitionistic Fuzzy Graph

IVFG Interval Valued Fuzzy Graph

CG Cubic Graph

DS Dominating Set

IVFS Interval Valued Fuzzy Set

ICS Independent Cubic Set