Eternal 1-security number of a fuzzy graph with level J

Abstract

The eternal 1-secure set, strategically protects graphical structures against attacks by placing sentinels within dominating sets. This study adapts this concept to real-world scenarios, like emergency response systems, by addressing challenges such as roadblocks and traffic congestion. We introduce the secure set and the eternal 1-secure set for fuzzy graphs based on edge membership levels J. Our analysis examines the relationship between the security number of a fuzzy graph G at level J and the domination number of its spanning subgraph at level J, demonstrating the potential of the eternal 1-secure set to enhance security measures in dynamic environments.

Keywords

Fuzzy graph fuzzy domination number eternal 1-security number fuzzy path fuzzy cycle

1. Introduction

In many real-world scenarios, particularly in surveillance and protection applications, the role of network security is crucial. In graph theory, a secure set is fundamental, referring to a subset of vertices where every vertex outside the set is connected to at least a vertex within it. This ensures comprehensive surveillance or protection coverage within the graph. However, maintaining continuous surveillance or protection becomes challenging in dynamic environments where potential threats can move or change. To address this, an eternal secure set emerges, representing a set of vertices that remains secure under all possible movements of potential threats. Specifically, the notion of an eternal 1-secure set is of significant interest. Here, “1” signifies the number of moves allowed for potential threats. Thus, an eternal 1-secure set comprises vertices that remain secure even after a single move by potential threats.

1.1 Motivation

A.P. Burger first introduced the concept of the eternal 1-secure set of a graph [5]. In the eternal 1-secure set, sentinels are strategically placed within a dominating set $H$ of a graph $G$ , ensuring that for every vertex $y$ not in $H$ , there exists a vertex $x$ in $H$ such that the set $H^{'} = H - {u} \cup {v}$ remains a dominating set. The eternal 1-security number $σ_{1} (G)$ represents the least quantity of sentinels essential to shield the graphical structure against a succession of individual attacks, each countered by the movement of a single guard. However, in real-world scenarios, such as emergency response systems where guards may represent ambulances vertices represent landmarks, and edges represent the road connections between them, various obstacles like roadblocks, traffic congestion, or natural disasters may impede the use of certain roads. Consequently, adapting the path and adjusting the number of sentinels becomes crucial. Despite the extensive research in fuzzy graph theory, there remains a gap in exploring secure sets within fuzzy graphs. Existing literature primarily focuses on domination sets, leaving a significant gap in understanding the concept of secure sets and their potential applications. Therefore, this study addresses this gap by introducing and analyzing secure sets within fuzzy graphs. By leveraging real-time data to analyze the minimum number of ambulances needed for medical emergencies in Chennai, India, this study aims to provide practical insights into the application of secure sets while advancing theoretical understanding in fuzzy graph theory.

1.2 Literature review

Fuzzy graph theory has emerged as a significant area of study since its inception by Rosenfield in 1975 [6], coinciding with the introduction of fuzzy set theory a decade earlier. Notably, Kaufmann had already defined fuzzy graphs in 1973 [7], contributing to the early development of this field. This branch of graph theory has gained importance due to its diverse applications in network-related fields. Rosenfeld’s seminal work 1975 delved into the composition of fuzzy relations and explored equivalence properties in fuzzy graphs. He elaborated on paths, connectivity, path strength, and the strongest path. Additionally, Rosenfeld introduced definitions for cluster nodes, cut nodes, bridges, and fuzzy forests.

Mordeson made subsequent advancements in fuzzy graph theory in 1993, introducing fuzzy line graphs and discussing their isomorphic properties with corresponding line graphs [8]. In 1994, Mordeson extended his work by defining operations such as Cartesian product, joining on fuzzy graphs and composition of fuzzy graphs. Zadeh’s 1996 contribution in the book chapter “Fuzzy Logic, Neural Networks, and Soft Computing” further emphasized the significance of fuzzy sets and fuzzy systems [9]. The chapter, part of a compilation of some papers by Lotfi A. Zadeh, enriched the understanding of these topics. The pioneering work on the theory of domination within fuzzy graphs was initiated by Somasundaram and Somasundaram in 1988, marking a significant milestone in exploring fuzzy graph theory, where they defined domination and total domination [10]. Ghobadi and Soner (2000) contributed by defining inverse domination sets [11], while Munoz and Ortuno (2005) addressed the chromatic number of the fuzzy graphs, proposing a novel coloring problem and an algorithm for its solution [12].

In subsequent years, Gani and Akram (2014) introduced fuzzy labeling graphs and fuzzy magic labeling graphs [13]. They established the relationship between these graphs and discussed the properties of fuzzy bridges and fuzzy cut nodes. Gani and Prasanna Devi (2015) introduced 2-domination sets and 2-domination numbers in fuzzy graphs, proving their properties and relationships with domination sets [14]. They also demonstrated the presence of end vertices of an isolated edge in every 2-domination set. In 2013 Xavior gave the domination number on the fuzzy graph more clearly and explained an algorithm to compute the domination number of the fuzzy graph [1]. In 2019, Jiny introduced the concept of the fuzzy resolving number, followed by an elucidation in 2023 on the generation of realbasis fuzzy graphs and their associated properties [15]. In 2021, Chakraborty worked on enhancing decision-making processes in complex, uncertain environments by providing a more robust and nuanced approach to handling imprecise information. This can be particularly useful in risk assessment, medical diagnosis, and systems engineering, where precise decision-making is critical despite incomplete or ambiguous data [20]. In 2022, studies on trapezoidal neutrosophic numbers were extended to include an exponential operational law with a positive real base and trapezoidal neutrosophic weights [21]. Sanchari Bera (2022) explored the concept of domination in m-polar interval-valued fuzzy graphs (m-PIVFG). This study covered various aspects such as domination number, isolated vertices, total dominating sets, and independent sets. Bera applied these theoretical concepts to a practical case study, focusing on locating new facilities for managing COVID-19 emergency responses in West Bengal, India [22]. Uzma Ahmad (2023) expanded the domination theory of fuzzy graphs to rough fuzzy digraphs, demonstrating its application in decision-making problems. Ahmad illustrated how dominating sets can be used to select the optimal set of cities to supply a commodity across a country at minimal cost, enhancing resource distribution and cost efficiency [23]. Sivasankar (2023) introduced the concepts of bridge domination set and bridge domination number in fuzzy graphs to improve network stability. The study identifies various dominating sets using strong edges and examines key properties of bridge domination numbers with pertinent examples [24].

The “eternal $m$ -security number” concept denotes the number of sentinels required to manage a sequence of individual assaults through coordinated movements of multiple sentinels in a graphstructured framework. A strategically positioned configuration of sentinels satisfying this criterion is labeled as an “eternal $m$ -secure set.” Burger and colleagues [16, 17] originally introduced a dynamic domination concept, which was later coined “eternal security” in the research conducted by Goddard and colleagues [19]. This development has sparked diverse applications to bolster network security against potential attacks. Klostermeyer and colleagues introduced the notion of an eternal dominating set in graph theory [18]. Numerous researchers have delved into studying the eternal $m$ -security number in subsequent years. This paper is inspired by their contributions, aiming to extend the concept of the eternal m-security number to fuzzy graphs while introducing new parameters in the domination number of fuzzy graphs.

1.3 Novelties

In this study, we present three key innovations in fuzzy graph theory. Firstly, we introduce the concept of secure sets within fuzzy graphs, delineating them from traditional domination sets. Secondly, through rigorous analysis, we demonstrate that the secure number at level $J$ significantly diverges from the corresponding domination number for graphs at the same level. Finally, leveraging these findings, we pioneer the definition of eternal 1-security numbers for fuzzy graphs at level $J$ , marking the first instance of such a measure in the existing literature. We substantiated our findings by providing a real-time example, analyzing the minimum number of ambulances needed to address a medical emergency in a specific location in Chennai, India, based on real-time data.

1.4 Organization of the article

This article is organized as follows. Section 1 introduces the importance of the eternal m-security number in graph theory and the motivation for its application in fuzzy graph theory, along with a review of previous studies in the literature. Section 2 provides the necessary basic notations and definitions. Section 3 presents the concept of a secure set in fuzzy graphs, explaining how the domination number of the spanning subgraph of G with level J differs from the secure set parameter of a fuzzy graph at level J, illustrated with an example. Theorems in this section analyze when these two parameters coincide. Section 4 builds on this by introducing the Eternal 1-security number for a fuzzy graph at level J. It includes a real-world example that determines the minimum number of ambulances required to handle a sequence of emergencies across five locations by calculating the Eternal 1-security number of the fuzzy graph at various levels J, each representing different traffic intensities. Subsection 4.1 addresses the eternal 1-secure set for fuzzy paths and cycles at level J. Section 5 explores future research directions and limitations, and Section 6 concludes the article.

2. Preliminaries

Definition 1

In a fuzzy graph represented by $G (V, σ, μ)$ , the associated crisp graph is represented as $G (σ^{*}, μ^{*})$ . In this context, $σ^{*}$ is represented as the collection of vertices $x$ in $V$ where $σ (x)$ exceeds zero, while $μ^{*}$ is comprised of pairs $(x, y)$ in $V \times V$ where $μ (x, y)$ greater than zero [2]. This transformation establishes a clear-cut interpretation of the original fuzzy graph by delineating vertices with non-zero significance and delineating edges with non-zero connectivity, thereby facilitating a more tangible analysis of the graph’s structure and properties.

Definition 2

In a fuzzy graph denoted as $G (V, σ, μ)$ , a series of distinct vertices $x_{1}, x_{2}, \dots, x_{n}$ with $μ (x_{i - 1}, x_{i}) > 0$ for $i = 1, 2, \dots, n$ is defined as a fuzzy path $P$ of length $n$ . The sequential pairs within this series are identified as the edges of the path. If the fuzzy path $P$ satisfies the condition $x_{1} = x_{n}$ and has a length of at least 3; it is termed a fuzzy cycle.

Definition 3

If bijective function $w$ exists, mapping the vertices together with the edges of graph $G$ to [0,1] , allocating each node $a, b$ and edge $(a, b)$ a membership value whereby $μ^{ω} (a, b) < σ^{ω} (a) \land σ^{ω} (b)$ for every $a, b \in V$ it is termed as fuzzy labeling. A graph possessing such a labeling is referred to as a fuzzy labeling graph denoted as $G^{ω}$ . A cycle graph $G^{ω}$ is termed a fuzzy labeling cycle graph if it possesses fuzzy labeling. In a cycle graph $G$ , the fuzzy labeling cycle $G^{ω}$ contains precisely one weakest arc [3].

Definition 4

A fuzzy subgraph $S (V^{*}, τ, φ)$ is a partial subgraph of $G (V, σ, μ)$ if $V^{*} \subseteq V$ , $τ (x) ⩽ σ (x)$ , and $φ (x, y) ⩽ μ (x, y)$ for all $x, y \in V^{*}$ . When $V^{*} = V$ , the fuzzy subgraph is termed a spanning fuzzy subgraph of $G$ [2].

Furthermore, we define a spanning subgraph $S$ with level J for the fuzzy graph $G$ is the spanning subgraph such that $V = V^{*}, τ (x) = σ (x), φ (x, y) = μ (x, y) ⩾ J$ .

Definition 5

Consider a fuzzy graph $G (V, σ, μ)$ and let $D \subseteq V$ . Set $D$ is deemed a dominating set if, for every $x \in V - S$ , there exists $y \in D$ such that (i) $(u, v)$ forms a strong arc and (ii) $σ (x) ⩽ σ (y)$ . The domination number of the fuzzy graph $G$ is defined as the cardinality of the smallest dominating set.

To identify a dominating set within a given fuzzy graph $G$ , we follow these steps: First, determine the function $μ^{\infty} (x, y)$ for every pairs of vertices $x$ and $y$ in $G$ . Next, eliminate all weak edges from the graph, creating a modified graph, denoted as $G^{'}$ . Within this modified graph, identify the vertex $x$ with the highest value of $σ (x)$ . This vertex serves as a pivotal point for domination. Group together all vertices dominated by $x$ into a set labeled as $X$ . Next, derive a new graph $G^{'} - X$ by removing the vertices in $X$ along with their incident edges. Subsequently, repeat the process from step 3 to step 5, selecting new dominating vertices until isolated vertices remain. Ultimately, the amalgamation of vertices chosen in step 3 and the isolated vertices constitutes the dominating set of the original graph. This method systematically identifies the minimum subset of vertices necessary to dominate the entire graph [1].

Definition 6

A fuzzy graph $G = (σ, μ),$ which is connected is labeled as a fuzzy tree when it possesses a fuzzy spanning subgraph $H = (σ^{*}, μ *)$ that is a tree. This condition is met when for every arc $(u, v)$ not included in $H$ , the degree of connectivity $μ (u, v)$ is less than $μ^{\infty} (u, v)$ in $H$ [4].

Definition 7: Fuzzy Bridge

In a fuzzy graph $G (V, σ, μ) G (V, σ, μ)$ , a fuzzy bridge is an edge $(u, v) (u, v)$ with $μ (u, v) > 0 μ (u, v) > 0$ such that its removal increases the number of connected components in the associated crisp graph $G * (σ *, μ *) G * (σ *, μ *)$ . This indicates that $(u, v) (u, v)$ plays a crucial role in maintaining the connectivity of the graph.

Definition 8: Fuzzy Cut-Vertex

A vertex $v \in V$ in a fuzzy graph $G (V, σ, μ)$ is termed a fuzzy cut-vertex if the removal of $v$ and its incident edges increases the number of connected components in the associated crisp graph $G * (σ *, μ *)$ . This means that $v$ is essential for maintaining the overall connectivity of the graph.

Definition 9: Fuzzy Connectivity

The fuzzy connectivity of a fuzzy graph $G (V, σ, μ)$ is a measure of how strongly connected the graph is, considering the membership values of the vertices and edges. It can be defined as the minimum degree of connectivity among all pairs of vertices in the graph, taking into account their membership values.

Definition 10: Fuzzy Cut-Set

A fuzzy cut-set in a fuzzy graph $G (V, σ, μ)$ is a set of edges whose removal disconnects the graph into two or more components. The membership value of the cut-set is the minimum sum of the membership values of the edges in any such set.

3. Secure set of a fuzzy graph

This section presents the concept of a secure set within a fuzzy graph. Specifically, we define the secure set of a fuzzy graph $G (V, σ, μ)$ with level $J$ as the arrangement of sentinels within the domination set $H$ of $G_{J} (V, E_{J})$ . For every vertex $x$ not in $H$ , there is a vertex $y$ in $H$ such that the edge $(x, y)$ belongs to $E_{J}$ . The least number of sentinels needed to safeguard the graph $G_{J} (V, E_{J})$ against a single node attack is referred to as the security number of the fuzzy graph G with level $J$ represented as $γ_{J} (G)$ .

Figure 1.

The alpha cut of the fuzzy graph $G$ .

Table 1

Security number of fuzzy graph $G$ with varying level J

$J$ levels	Security number of the fuzzy graph G with level $J$	Domination number of the spanning subgraph of $G$ with level $J$
0.3	1	2
0.5	2	2
0.6	2	2
0.7	3	3

Example 1: Figure 1 provides a visual representation of the different alpha cuts observed within the fuzzy graph $G$ , offering insights into the distribution of membership values across vertices and edges. Complementing this visualization, Table 1 presents the security number of the fuzzy graph $G$ with level $J$ , alongside the domination number of the spanning subgraph of $G$ at each respective level. This tabulated data facilitates a nuanced understanding of the graph’s resilience under different perturbations and the efficiency of vertex selection strategies in achieving dominance.

Theorem 1: For a fuzzy labeling cycle $G^{ω}$ the security number of the fuzzy graph G with level $J$ and the Domination number of the spanning subgraph of $G$ with level J need not be the same.

Proof. Based on the definition of a fuzzy labeling cycle, it’s evident that it contains only one weakest arc, consequently leading to only one weakest arc in every fuzzy spanning subgraph derived from it. The security number of the fuzzy graph G with level $J$ is the domination number of the crisp graph $G_{J} (V, E_{J})$ , wherein any vertex from $V$ can be arbitrarily chosen as an element in the dominating set. However, In the scenario of a spanning fuzzy subgraph with level $J$ , the selection of domination set elements may differ from the crisp graph due to the condition $ρ (x) ⩽ ρ (y)$ for every $u \in D$ and $v \in V_{D}$ . Thus, the choice of domination set elements may vary as we are permitted to select the vertex with the highest membership value. Consequently, for a fuzzy labeling cycle $G^{ω}$ , it’s apparent that the security number of the fuzzy graph G with level $J$ and the domination number of the spanning subgraph of $G$ at level $J$ may not necessarily align.

In general, Table 1 clearly illustrates that the security number of the fuzzy graph G with level $J$ may differ from the domination number of the spanning subgraph of $G$ at level $J$ . In the following theorem, we assert that these two parameters become equal.

Theorem 2: Considers $G (V, σ, μ)$ is a fuzzy graph with $σ$ is a constant function and $μ$ is a one-one function then the secure set with level $J$ is the fuzzy dominating set of the spanning subgraph of $G$ with level $J$ .

Proof. Given that $G (V, σ, μ)$ is a fuzzy graph having $σ$ as a constant function and $μ$ as a one-to-one function, we aim to show that the secure set with level $J$ is the dominating set of the spanning subgraph of $G$ with level $J$ . Since $σ$ is constant, the spanning subgraph $S (V^{*}, τ, φ)$ with level $J$ will also have a constant membership function $τ (x) = σ (x) \forall x \in V^{*}$ and a connectivity function $φ$ that satisfies $φ (x, y) ⩾ J$ for each edge $(x, y)$ in $S$ . Moreover, $μ$ is a one-one function for each edge membership levels of $J = J_{1} ⩽ J_{2} ⩽ \dots J_{k}$ of $G,$ there will be only one weakest edge. Therefore the crisp graph of spanning subgraph $S (V^{*}, τ, φ)$ with level $J_{i}$ is same as the $G_{J_{i}} (E, J_{i})$ . which implies that the selection of domination set elements of $G_{J_{i}}$ will also form a fuzzy domination of the spanning subgraph $S (V^{*}, τ, φ)$ with level $J_{i}$ . Therefore, the theorem holds true, establishing the equivalence between the secure set with level $J$ and the dominating set spanning subgraph of $G$ with level $J$ in the context of the specified fuzzy graph with constant $σ$ -function and $μ$ is a one-to-one function.

Theorem 3: Consider $G (V, σ, μ)$ , a fuzzy graph with edge membership levels $J = J_{1} ⩽ J_{2} ⩽ \dots ⩽ J_{k}$ . Then, for each $i = 1, 2, \dots, k - 1$ , then $γ_{J_{i}} (G) ⩽ γ_{J_{i + 1}} (G)$ for each for each $i = 1, 2, \dots, k - 1$ . where $γ_{J_{i}} (G)$ is the security number of $G$ at level $J_{i}$ .

Proof. Let $G_{J_{i}} (V, E_{J_{i}})$ represent the crisp graph obtained from the fuzzy graph $G$ by retaining edges with membership values greater than or equal to $J_{i}$ . Let $D_{i}$ be the dominating set of $G_{J_{i}}$ with the smallest cardinality. In $G_{J_{i + 1}} (V, E_{J_{i + 1}})$ , edges with membership values less than $J_{i + 1}$ are removed. Consequently, at least one edge $e$ with a membership value of $J_{i}$ might be removed from $G_{J_{i}}$ . If both endpoints of $e$ are in the dominating set $D_{i}$ , then $γ_{J_{i}} (G) = γ_{J_{i + 1}} (G)$ . Similarly, if neither endpoint of $e$ is in $V - D_{i}$ , then $γ_{J_{i}} (G) = γ_{J_{i + 1}} (G)$ .

In the scenario where one endpoint $x$ of $e$ is in $D_{i}$ while the other, denoted as $y$ , is not, two possibilities arise: either the removal of $e$ leaves $y$ without an adjacent vertex in $D_{i}$ , or there exists another vertex in $D_{i}$ that dominates $y$ . In the first case, $y$ belongs to $D_{i + 1}$ , and $γ_{J_{i}} (G) < γ_{J_{i + 1}} (G)$ . And in the second case $γ_{J_{i}} (G) = γ_{J_{i + 1}} (G)$ . Hence, $γ_{J_{i}} (G) ⩽ γ_{J_{i + 1}} (G)$ .

Figure 2.

Domination number of a fuzzy graph: Step-by-step process.

4. Eternal 1-security number for a fuzzy graph with level

𝑱

This section presents the eternal 1-security number for the fuzzy graph $G$ with level $J$ . Consider the placement of a sentinel on each vertex of a secure set $H$ of a graph $G_{J} (V, E_{J})$ . If for each vertex $y \in H$ , there exists an adjacent vertex $x$ such that the resulting set $H_{1} = H - y \cup^{x}$ is the secure set of the fuzzy graph, then $G_{J}$ is deemed 1-secure with level $J$ . The fuzzy graph $G$ is said to be eternally 1-secure with level $J$ if, for any sequence $x_{1}, x_{2}, \dots, x_{k}$ of vertices, there exists a sequence of sentinels $y_{1}, y_{2}, \dots, y_{k}$ where $y_{i} \in H_{i - 1}$ and $y_{i}$ is equal to or neighboring to $x_{i}$ , in a way that each set $H_{i} = H_{i - 1} - y_{i} \cup_{i}^{x}$ is a secure set. The minimum cardinality of the eternal 1-secure set is termed the eternal 1-security number of the fuzzy graph with level $J$ denoted as $σ_{J_{1}} (G)$ . In other words the eternal 1-security number of the fuzzy graph with level $J$ is the eternal 1-security number of the crisp graph $G_{J}$ .

Figure 3.

Google maps displays traffic levels around landmarks in the evening.

Example 2: To ensure efficient ambulance services in the vicinity of Guindy, Chennai, India, we aim to determine the least number of ambulances required to handle the sequency of emergency situations effectively across five nearby locations. This entails calculating the eternal 1-security number of the fuzzy graph at various levels $J$ , each representing the traffic flow intensity. By doing so, we can ascertain the optimal number of ambulances needed to address emergencies promptly under different traffic conditions. In the spectrum of traffic conditions, traffic levels can be categorized into six distinct levels, each represented by a value between 1 and 0. At the highest end of the scale is “free flow,” designated as 1, indicating smooth and uninterrupted traffic flow. As traffic intensity increases, the level transitions to “light traffic,” assigned a value of approximately 0.8, indicating minimal congestion and minor slowdowns. Following this, “moderate traffic” represents a level with slightly heavier congestion, assigned a value of approximately 0.6. Subsequently, “heavy traffic” and “gridlock” correspond to values nearing 0.4 and 0.2, respectively, indicating significant congestion, slowdowns, and a complete standstill in the case of gridlock. Finally, “severe congestion” represents the lowest point on the scale, denoted by a value of 0, where traffic experiences extensive delays and disruptions. Figure 3 displays the evening traffic levels observed between the junctions obtained from Google Maps. Similarly, the other traffic levels are observed and below is the adjacency matrix representing the traffic flow around the landmarks during the morning (7AM–9AM), afternoon period (12PM–2PM), and evening rush (7PM–9PM).

Figure 4.

Methodology of finding the minimal ambulance required to serve the emergency.

The methodology utilized to find the eternal 1-security of the fuzzy graph for a sequence of single emergencies is outlined in the flowchart below in Fig. 4.

Morning, Adjacency Matrix

$A_{i, j} = [\begin{matrix} 0 & 0.6 & 0 & 0.8 & 0.6 \\ 0.6 & 0 & 0.8 & 0.8 & 0.8 \\ 0 & 0.8 & 0 & 0 & 0 \\ 0.8 & 0.8 & 0 & 0 & 1 \\ 0.6 & 0.8 & 0 & 1 & 0 \end{matrix}]$

Afternoon, Adjacency Matrix

$B_{i, j} = [\begin{matrix} 0 & 1 & 0 & 0.8 & 1 \\ 1 & 0 & 0.6 & 0.8 & 1 \\ 0 & 0.6 & 0 & 0 & 0 \\ 0.8 & 0.8 & 0 & 0 & 0.8 \\ 1 & 1 & 0 & 0.8 & 0 \end{matrix}]$

Evening, Adjacency Matrix

$B_{i, j} = [\begin{matrix} 0 & 0.6 & 0 & 0.8 & 0.6 \\ 0.6 & 0 & 0.4 & 0.2 & 0.4 \\ 0 & 0.4 & 0 & 0 & 0 \\ 0.8 & 0.2 & 0 & 0 & 0.6 \\ 0.6 & 0.4 & 0 & 0.6 & 0 \end{matrix}]$

Figure 5.

Eternal 1-security number across morning, afternoon, and evening.

Figure 5 depicts the eternal 1-security number across morning, afternoon, and evening time frames, represented by a frequency polygon. This graph illustrates that the eternal 1-security number remains consistent until the free flow level reduces to 0.4. When we expect the free flow below this threshold, more ambulances are necessary to manage the sequence of emergencies. However, as the traffic volume rises, the required number of ambulances increases accordingly. During the afternoon period, a maximum of three ambulances suffice to address the emergency scenarios.

Theorem 4: Let $G (V, σ, μ)$ be a fuzzy graph with edge membership levels $J = J_{1} ⩽ J_{2} ⩽ \dots ⩽ J_{k}$ . For each $i = 1, 2, \dots, k - 1$ , it holds that $σ_{J_{i}, 1} (G) ⩽ σ_{J_{i + 1, 1}} (G)$ , where $σ_{J_{i}, 1} (G)$ denotes the eternal 1-security number of $G$ at level $J_{i}$ .

Proof. According to the definition of the eternal 1-security number $σ_{J_{1}} (G)$ of the fuzzy graph $G$ with level $J$ , it represents the least cardinality of the secure set $H_{i}$ of $G_{J}$ . This set ensures that after each movement, the resulting position of the sentinels in $H_{i + 1}$ still forms a secure set of $G_{J}$ and $γ_{J_{i}} (G) ⩽ γ_{J_{i + 1}} (G)$ for each for each $i = 1, 2, \dots, k - 1$ . Therefore, by Theorem 3.3, it follows that $σ_{J_{i}, 1} (G) ⩽ σ_{J_{i + 1, 1}} (G)$ .

4.1 Eternal 1-security of the number of a fuzzy path

P_{n}

with level J

In this section, we aim to extend the concept of the eternal 1-security number to a fuzzy path. Let’s consider a fuzzy path comprising five vertices labeled as $x_{1}, x_{2}, \dots, x_{5}$ . Within each alpha cut $G_{J}$ of the path, there may exist a possibility of one or more edges being removed, altering the path’s structure.

The potential vertex partitions of the path $P_{5}$ are as follows: 5, 4 $+$ 1, 3 $+$ 2, 3 $+$ 1 $+$ 1, 2 $+$ 2 $+$ 1, 2 $+$ 1 $+$ 1 $+$ 1, 1 $+$ 1 $+$ 1 $+$ 1 $+$ 1. Goddard et al. [5] demonstrated that the eternal 1-security number of a crisp path $P_{n}$ is $⌈ \frac{n}{2} ⌉$ . Consequently, we deduce that a minimum of $⌈ \frac{5}{2} ⌉ = 3$ sentinels and a maximum of 5 sentinels (where the graph contains only isolated vertices) are necessary to ensure the graph’s protection. With these observations, we establish a theorem to determine the eternal 1-security number of a fuzzy path at level $J$ .

Theorem 5: Let $P_{n}$ be a fuzzy path with $n$ vertices, and let $J = J_{1} ⩽ J_{2} ⩽ \dots ⩽ J_{k}$ represent the edge membership levels of $P_{n}$ . If each alpha cut $G_{J_{i}}$ of $P_{n}$ comprises $m$ components $G_{1}, G_{2}, \dots, G_{m}$ , then the eternal 1-security number at level $J_{i}$ , denoted $σ_{J_{i}, 1} (P_{n})$ , is given by: $σ_{J_{i}, 1} (P_{n}) = ⌈ \frac{| G_{1} |}{2} ⌉ + ⌈ \frac{| G_{2} |}{2} ⌉ + \dots + ⌈ \frac{| G_{m} |}{2} ⌉$

Here, $| G_{1} |$ represents the number of vertices in component $G_{1}$ .

Proof. For a fuzzy path $P_{n}$ , the eternal 1-security number at level $J_{i}$ represents the minimum number of sentinels required to safeguard the graph $G_{J_{i}} (V, E_{J_{i}})$ against a sequence of single attacks with a single sentinel movement. If $G_{J_{i}}$ consists of $m$ components, then the eternal 1-security of $P_{n}$ at level $J_{i}$ is the sum of the eternal 1-security numbers of all the components. Since $P_{n}$ is a fuzzy path, each component will either be a path or an isolated vertex. Since, the eternal 1-security number of a crisp path $P_{n}$ is $⌈ \frac{n}{2} ⌉$ , we can express the eternal 1-security number of $P_{n}$ at level $J_{i}$ as: $σ_{J_{i}, 1} (P_{n}) = ⌈ \frac{| G_{1} |}{2} ⌉ + ⌈ \frac{| G_{2} |}{2} ⌉ + \dots + ⌈ \frac{| G_{m} |}{2} ⌉$ .

Goddard et al. also proved that the eternal 1-seurity number of a cycle $C_{n}$ is $⌈ \frac{n}{2} ⌉$ . Hence we write the following corollary.

Corollary 1: Let $C_{n}$ be a fuzzy cycle, and let $J = J_{1} ⩽ J_{2} ⩽ \dots ⩽ J_{k}$ represent the edge membership levels of $C_{n}$ . If each alpha cut $G_{J_{i}}$ of $C_{n}$ comprises $m$ components $G_{1}, G_{2}, \dots, G_{m}$ , then the eternal 1-security number at level $J_{i}$ , denoted $σ_{J_{i}, 1} (C_{n})$ , is given by: $σ_{J_{i}, 1} (C_{n}) = ⌈ \frac{| G_{1} |}{2} ⌉ + ⌈ \frac{| G_{2} |}{2} ⌉ + \dots + ⌈ \frac{| G_{m} |}{2} ⌉$ when $J_{i} \neq J_{1}$ for $J_{i} = J_{1}, σ_{J_{1}, 1} (C_{n}) = ⌈ \frac{n}{2} ⌉$ .

4.2 Need and limitations

Need: This study is crucial for addressing the practical challenges of deploying ambulances efficiently in urban settings like Chennai, India, where traffic congestion and roadblocks can impede emergency response. The concept of eternal 1-secure sets, introduced by A.P. Burger, provides a framework for maintaining a dominating set in a graph, yet its application in fuzzy graphs remains underexplored. By extending this concept to fuzzy graphs and leveraging real-time traffic data, this research fills a significant gap in the literature, offering flexible, adaptive strategies for ambulance deployment. Consequently, it advances fuzzy graph theory and provides actionable insights to improve emergency response systems, ensuring timely medical assistance in dynamic urban environments.

Limitations: This article does not analyze the eternal 1-security number of various types of fuzzy graphs. Only specific cases are discussed. The case study relies on real-time traffic data that may be incomplete and uses static adjacency matrices that might not reflect rapid traffic changes. Limiting traffic levels and generalizing road conditions can overlook local variations. Focusing on Guindy Chennai limits its applicability to other regions, and the fuzzy graph model may not capture all real-world complexities. The study does not fully address varying emergency response times or consider factors like medical personnel availability and ambulance conditions. Additionally, implementing dynamic deployment strategies poses significant technological and logistical challenges.

4.3 Future work

In future research, exploring the applicability and effectiveness of the eternal 1-secure set concept for fuzzy graphs in various practical domains would be valuable. Some potential directions for future investigation include:

Algorithm Development: Develop efficient algorithms for computing the eternal 1-secure set for fuzzy graphs, considering different types of graph structures and membership level sets. This would involve exploring optimization techniques and heuristics to enhance computational efficiency, particularly for large-scale graphs.

Real-World Applications: Investigate the practical implications of the eternal 1-secure set in real-world scenarios such as transportation networks, urban planning, disaster management, and public safety. Conduct case studies or simulations to assess the effectiveness of the proposed vertex selection strategies in enhancing resilience and efficiency in these domains.

Dynamic Fuzzy Graphs: Extend the concept of the eternal 1-secure set to dynamic fuzzy graphs, where the edge membership levels may change over time. Explore how the notion of security evolves and develop dynamic vertex selection algorithms to adapt to changing network conditions.

Multi-Objective Optimization: Consider multi-objective optimization frameworks for vertex selection in fuzzy graphs, incorporating additional criteria such as cost, resource utilization, and connectivity maintenance. Develop methodologies to balance the trade-offs between different objectives while ensuring security and efficiency.

Graph Theory Extensions: Investigate extensions of the eternal 1-secure set concept to other areas of graph theory, such as directed fuzzy graphs, hypergraphs, or higher-dimensional structures. Explore how security can be generalized and applied in these contexts.

Experimental Validation: Validate the theoretical findings and proposed algorithms through experimental studies on real-world or synthetic datasets. This could involve empirical evaluations using network simulation platforms or deployment in practical systems to assess their performance under various scenarios.

5. Discussion

The analysis of the eternal 1-security number for different traffic conditions in Guindy, Chennai, demonstrates that the number of ambulances required varies significantly with traffic intensity. The findings indicate that more ambulances are needed during heavy traffic and congestion to maintain effective emergency response coverage. The comparison between the security number of the fuzzy graph G with level J and the domination number of the spanning subgraph of G with level J highlights the effectiveness of the proposed method in adapting to real-world traffic conditions. Additionally, the theorems and corollary presented provide a theoretical foundation for understanding the sensitivity of the eternal 1-security number in fuzzy graphs, ensuring robustness against varying traffic scenarios.

6. Conclusions

The study introduces the concept of the eternal 1-security number with level $J$ in fuzzy graph theory for optimizing ambulance deployment in urban traffic conditions, using real-time data to offer valuable insights. The comparison with the existing parameter domination number of the fuzzy graph demonstrates the robustness of the approach. The theorems and corollary provide a theoretical foundation, showing the sensitivity of the eternal 1-security number of traffic variations, essential for adaptive emergency response strategies. These findings enhance both practical and academic understanding. Future research could extend this concept to other areas and incorporate more variables to improve accuracy and reliability, ultimately enhancing emergency service deployment.

Funding

Not applicable.

Institutional review board statement

Not applicable.

Informed consent statement

Not applicable.

Data availability statement

Data generated during research used in the present work.

Footnotes

Conflict of interest

The authors declare no conflict of interest.

References

Xavior

Isido

Chitra

. On domination in fuzzy graphs. International Journal of Computing Algorithm. 2013; 2(2): s81-2.

Mordeson

Mathew

Malik

. Fuzzy graph theory with applications to human trafficking. Berlin: Springer; 2018 Jan 1.

Gani

Subahashini

. Properties of fuzzy labeling graph. Applied Mathematical Sciences. 2012; 6(70): 3461-6.

Nagoorgani

Subahashini

. Fuzzy labeling tree. International Journal of Pure and Applied Mathematics. 2014; 90(2): 131-41.

Nagoorgani

Subahashini

. Fuzzy labeling tree. International Journal of Pure and Applied Mathematics. 2014; 90(2): 131-41.

Zadeh

Tanaka

, editors. Fuzzy sets and their applications to cognitive and decision processes: Proceedings of the us–japan seminar on fuzzy sets and their applications, held at the university of california, berkeley, california, July 1–4, 1974. Academic press; 2014 Jun 28.

Kaufman

, Introduction a la Theorie des Sous-emsembles Flous. Masson et Cie. 1973.

Mordeson

. Fuzzy line graphs. Pattern Recognition Letters. 1993 May 1; 14(5): 381-4.

Zadeh

Klir

Yuan

. Fuzzy sets, fuzzy logic, and fuzzy systems: selected papers. World Scientific. 1996.

10.

Somasundaram

. Domination in fuzzy graphs–I. Pattern Recognition Letters. 1998 Jul 1; 19(9): 787-91.

11.

Ghobadi

Soner

Mahyoub

. Inverse dominating Set in fuzzy graphs. Journal of Proceedings of the Jangjeon Mathematical Society. 2008 Jan; 11(1): 75-81.

12.

Munoz

Ortuno

Ramírez

Yanez

. Coloring fuzzy graphs. Omega. 2005 Jun 1; 33(3): 211-21.

13.

Nagoor Gani

Akram

Subahashini

. Novel properties of fuzzy labeling graphs. Journal of Mathematics. 2014; 2014(1): 375135.

14.

Gani

Devi

. 2-domination in fuzzy graphs. International Journal of Fuzzy Mathematical Archive. 2015; 9(1): 119-24.

15.

Mary Jiny

Shanmugapriya

. Fuzzy resolving number and real basis generating fuzzy graphs. Afrika Matematika. 2023 Jun; 34(2): 25.

16.

Burger

Cockayne

Grundlingh

Mynhardt

Van Vuuren

Winterbach

. Finite order domination in graphs. Journal of Combinatorial Mathematics and Combinatorial Computing. 2004; 49: 159-76.

17.

Burger

Cockayne

Grundlingh

Mynhardt

, van Vuuren

Winterbach

. Infinite order domination in graphs. Journal of Combinatorial Mathematics and Combinatorial Computing. 2004; 50: 179-94.

18.

Klostermeyer

Mynhardt

. Eternal Total Domination in Graphs. Ars Comb. 2012 Sep 27; 107: 473-92.

19.

Goddard

Hedetniemi

. Eternal security in graphs. J Combin Math Combin Comput. 2005; 52(1).

20.

Chakraborty

Mondal

Alam

Dey

. Classification of trapezoidal bipolar neutrosophic number, de-bipolarization technique and its execution in cloud service-based MCGDM problem. Complex and Intelligent Systems. 2021 Feb; 7: 145-62.

21.

Haque

Chakraborty

Mondal

Alam

. New exponential operational law for measuring pollution attributes in mega-cities based on MCGDM problem with trapezoidal neutrosophic data. Journal of Ambient Intelligence and Humanized Computing. 2022 Dec; 13(12): 5591-608.

22.

Bera

Pal

. A novel concept of domination in m-polar interval-valued fuzzy graph and its application. Neural Computing and Applications. 2022 Jan; 34(1): 745-56.

23.

Ahmad

Batool

. Domination in rough fuzzy digraphs with application. Soft Computing. 2023 Mar; 27(5): 2425-42.

24.

Shanmugam

Aishwarya

Shreya

. Bridge domination in fuzzy graphs. Journal of Fuzzy Extension and Applications. 2023; 4(3): 148-54.