Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 17 https://internationalpubls.com Enhancing Network Security in Distributed Systems Using Middle Roman Dominating Functions R. Vinodhini1, T. N. M. Malini Mai2 1,2 SIMATS School of Engineering, Chennai – 600 097, Tamil, India vinodhinir41034.sse@saveetha.com1, malinimait.sse@saveetha.com2 Article History: Received: 05-08-2024 Revised: 25-09-2024 Accepted: 07-10-2024 Abstract: A Middle Roman dominating function (MRDN) on a graph G = (V,E) is a function f:v→{0,1,2,3} satisfying the condition that every vertex u with f(u)=0 is adjacent to at most one vertex v with f(v)=2 or 3. Further if a vertex is assigned 2, then at most two of its vertices can be assigned 0 and if a vertex is assigned 3, then all its neighbours can be assigned 0. The weight of a MRDF is the value f(V(G))=∑_uϵV▒〖f(u〗). The Middle Roman domination number γ_MR (G) is the minimum weight of a MRDF on G. In this paper, we introduce Middle Roman Domination number denoted as γ_MR (G), study the properties of the function, present some characterization and determine γ_MR (G)-value for some graphs. Middle Roman Dominating Functions (MRDFs) present a unique approach within graph theory, with significant implications for various fields in computer science. By assigning values to vertices under specific constraints, MRDFs enable the optimization of resources, enhancing network security, load balancing, and energy efficiency in distributed systems. This paper explores the application of MRDFs in scenarios such as intrusion detection, resilient network design, task scheduling, and sensor activation. By minimizing the overall weight while maintaining functional requirements, MRDFs provide an effective strategy for addressing challenges in network topology, resource allocation, and fault tolerance. The versatility of MRDFs makes them a valuable tool in the development of robust and efficient computing systems. Keywords: Dominating function, Roman dominating function, Weak Roman dominating function, Middle Roman dominating function, Resource Allocation, Optimization, Network Security. Mathematical Subject Classification: 05C69. 1. INTRODUCTION A set 𝐷 of vertices in a graph G is a dominating set if every vertex in 𝑉 − 𝐷 is adjacent to some vertex in 𝐷. The domination number 𝛾(𝐺) is the minimum cardinality of the dominating set of 𝐺 . In 2004, Cockayne et al. published Roman Domination in graphs; as a result, numerous Roman domination parameters were introduced [2, 16, 17]. From then enormous work has been done in Roman domination. Consider a 𝐺 = (𝑉, 𝐸) and define a function 𝑓: 𝑉 → {0, 1, 2}. Unguarded with regard to f is defined as a vertex u with 𝑓(𝑢) = 0 that is not next to a vertex with 1 or 2. The function 𝑓: 𝑉 → {0, 1, 2} satisfying the condition that each vertex u for which 𝑓(𝑢) = 0 is adjacent to at least one vertex v for which 𝑓(𝑣) = 2, is referred to be a Roman dominant function, known as RDF of a graph 𝐺 = (𝑉, 𝐸). Roman domination number (RDN) of G, which is represented by 𝛾𝑅(𝐺), is the bare minimum number of guards that must be employed in any RDF. Let 𝐺 = (𝑉, 𝐸) be a graph and 𝑓 be a function 𝑓: 𝑉 → {0,1,2} . The function 𝑓 is a weak Roman dominating function (WRDF) if each vertex mailto:vinodhinir41034.sse@saveetha.com mailto:malinimait.sse@saveetha.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 18 https://internationalpubls.com 𝑢 with 𝑓(𝑢) = 0 is adjacent to a vertex 𝑉 with 𝑓(𝑣) > 0 such that 𝑓 ′: 𝑉 → {0,1,2} defined by 𝑓′(𝑢) = 1, 𝑓′(𝑣) = 𝑓(𝑣) − 1 and 𝑓′(𝑤) = 𝑓(𝑤) if 𝑤𝜖 𝑉 − 𝑢, 𝑣 has no undefined vertex. The weight of 𝑓 is w(𝑓) = 𝞢𝑣𝜖𝑉𝑓(𝑣) the weak Roman domination number denoted by 𝛾𝑟(𝐺) is the minimum weight of WRDF in G [6, 16, 17, 18, 28]. A Middle Roman dominating function (MRDN) on a graph 𝐺 = (𝑉, 𝐸) is a function 𝑓: 𝑉(𝐺) → {0, 1, 2, 3} satisfying the condition for every vertex 𝑢 ∈ 𝑉, 𝑓(𝑢) = 0, then vertex 𝑢 has at least one neighbor 𝑣 with 𝑓(𝑢) ≥ 2, If 𝑓(𝑢) = 1, then |𝑁(𝑢) ∩ 𝑉0| = 0, 𝑓(𝑢) = 2, then |𝑁(𝑢) ∩ 𝑉0| ≤ 2, 𝑓(𝑢) = 3, then |𝑁(𝑢) ∩ 𝑉0| ≥ 3. Where 𝑉𝑖, 𝑖 = 1,2,3. . . , 𝑛. The weight of a MRDF is the value 𝑓(𝑉(𝐺)) = ∑ 𝑓(𝑢𝑢𝜖𝑉(𝐺) ). The Middle Roman domination number 𝛾𝑀𝑅(𝐺) is the minimum weight of a MRDF on 𝐺. We define the Middle Roman domination number based on the military strategy that if any region is un secured then it must be adjacent to a region with either 2 or 3 legions. If 2 legions are placed at a region then at most two neighboring regions can be undefended. If 3 legions are placed at a region then any number of neighboring legions can be undefended. The idea is that during an attack at any undefended region, the legions placed at the neighboring regions will move and protect this undefended region. Which provides a fruitful level of defence at a cheaper cost. In some highly populated areas of the city where emergency calls for police, ambulance, fire men service etc. are common. three units of servers can be placed, whereas at areas 𝑉𝑖 , 𝑖 = 1,2,3. . . , 𝑛 where only two neighbouring regions need service, exactly two units of servers can be placed. Similarly, in trust worthy areas, one unit of server can be placed. Such type of arrangements can give protection with minimum number of servers which also optimize the cost. Server placements will maximize the service of the servers with optimal cost [11,12,13,14]. Middle Roman Dominating Functions (MRDFs) offer a fascinating intersection of graph theory and computer science, providing powerful tools for addressing complex problems in network security, resource allocation, and optimization. These functions, defined by specific conditions on vertex assignments within a graph, allow for strategic placement and allocation of resources, ensuring that critical nodes are efficiently protected or empowered. By minimizing the overall cost or weight while satisfying the constraints of MRDFs, computer scientists can design robust systems that balance efficiency with security. This introduction explores the diverse applications of MRDFs in various domains, highlighting their significance in optimizing network structures, managing distributed systems, and enhancing the resilience of communication networks. Here we stated some of the characterization theorems of MRDFs followed by its applications. Preposition 1: If 𝐺 a graph of order 𝑛 ≥ 4 with a vertex of degree 𝑛 − 1 then 𝛾(𝐺) = 1 and 𝛾𝑀𝑅(𝐺) = 3 Proposition 2: Let 𝑓 = (𝑉0, 𝑉1, 𝑉2, 𝑉3) be a 𝛾𝑀𝑅-function then, a. 𝐺[𝑉1] the subgraph induced by 𝑉1 has maximum degree 1 b. No edge of 𝐺 joins 𝑉1 and 𝑉3. c. ∀ 𝑣 ∈ 𝑉0, |𝑁(𝑣) ∩ 𝑉1| ≤ 1 = {𝑢, 𝑣2} ∩ {𝑣1, 𝑣2, 𝑣3} = {𝑣2} d. 𝑉2 ∪ 𝑉3 is a 𝛾- set of 𝑉0. 𝑉2 ∪ 𝑉3 ≻ 𝑉0 Theorem 1: 𝛾𝑀𝑅(𝑃𝑛) = 𝛾𝑀𝑅(𝐶𝑛) = ⌈ 2𝑛 3 ⌉ Proof is obvious for the above theorem. Theorem 2: For any graph 𝐺, 𝛾𝑀𝑅(𝐺) = 3𝛾(𝐺) if and only if 𝑉1 = ∅ and 𝑉2 = ∅ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 19 https://internationalpubls.com Proof: Let 𝐺 be a middle roman graph and let 𝑓 = (𝑉0, 𝑉1, 𝑉2, 𝑉3) be a 𝛾𝑀𝑅-function of 𝐺. From the proposition 3(d) 𝑉2, 𝑉3 ≻ 𝑉0 and 𝑉1 ∪ 𝑉2 ∪ 𝑉3 ≻ 𝑉0 and hence, 𝛾(𝐺) ≤ |𝑉1 ∪ 𝑉2 ∪ 𝑉3| = |𝑉1| + |𝑉2| + |𝑉3| ≤ |𝑉1| + 2|𝑉2| + 3|𝑉3| = 𝛾𝑀𝑅(𝐺). But since, 𝐺 is Middle Roman, we know that, 3𝛾(𝐺) = 3|𝑉1| + 3|𝑉2| + 3|𝑉3| = 𝛾𝑀𝑅(𝐺) = |𝑉1| + |𝑉2| + 3|𝑉3|. Hence 𝑛1 = |𝑉1| = |𝑉2| = 0. Conversely, let 𝑓 = (𝑉0, 𝑉1, 𝑉2, 𝑉3) be a 𝛾𝑀𝑅-function of 𝐺 𝑛1 = |𝑉1| = |𝑉2| = 0. Therefore 𝛾𝑀𝑅(𝐺) = 3|𝑉3| and since by the definition 𝑉1 ∪ 𝑉2 ∪ 𝑉3 ≻ 𝑉0 it follows that 𝑉3 is a dominating set of 𝐺. We know that 𝑉3 is a 𝛾- set of 𝐺(𝑉2 ∪ 𝑉3), i.e., |𝑉3| = 𝛾(𝐺) and 𝛾𝑀𝑅(𝐺) = 3𝛾(𝐺) i.e., is a Middle Roman graph. The theorem states that for any graph 𝐺, the Middle Roman domination number 𝛾𝑀𝑅(𝐺) equals three times the domination number 𝛾(𝐺) if and only if two specific vertex sets, 𝑉1 and 𝑉2, are empty. These vertex sets likely represent particular configurations or substructures within the graph that affect the domination and Middle Roman domination properties and the applications of the theorem are as follows Network Design and Optimization-Simplified Network Structures: In the design and optimization of networks (such as communication or transportation networks), this theorem implies that achieving a specific domination-related property (where 𝛾𝑀𝑅(𝐺) = 3𝛾(𝐺)) requires the absence of certain substructures (𝑉1 and 𝑉2). This can guide the design of networks to ensure they meet specific criteria, such as minimal redundancy or efficient coverage, by avoiding these substructures [20, 21]. Fault-Tolerant Systems-Design of Fault-Tolerant Architectures: In systems that require fault tolerance, ensuring that 𝛾𝑀𝑅(𝐺) = 3𝛾(𝐺) can lead to designs where the system's resilience is maximized under specific conditions. The absence of the vertex sets 𝑉1 and 𝑉2 may correspond to configurations that prevent certain types of failures or ensure that every component is adequately backed up. Graph-Based Modelling in Biological Networks-Stability in Ecological or Biological Systems: In ecological networks or biological interaction networks, this theorem can be used to ensure that the network is stable and robust. For example, if 𝛾𝑀𝑅(𝐺) = 3𝛾(𝐺) holds, it might indicate that the system is free from certain destabilizing interactions or species (represented by 𝑉1 and 𝑉2), leading to a more stable and resilient ecosystem or biological network [25]. Algorithm Design in Graph Theory-Simplified Algorithms for Specific Graph Classes: In algorithmic graph theory, this theorem provides a characterization that can be used to design simplified algorithms for specific classes of graphs. If 𝛾𝑀𝑅(𝐺) = 3𝛾(𝐺) and 𝑉1 = ∅ and 𝑉2 = ∅, algorithms can be tailored to efficiently handle these graphs, taking advantage of their simplified structure. This theorem provides a valuable tool for understanding when certain domination-related properties hold in a graph, with wide-ranging applications in network design, optimization, security, biological modelling, and more. The conditions 𝑉1 = ∅ and 𝑉2 = ∅ highlight specific configurations that need to be avoided to achieve these properties, offering practical guidance in various fields. Characterization of 𝜸(𝑮) = 𝜸𝑴𝑹(𝑮) Theorem 3: For any graph 𝐺 of order 𝑛, 𝛾(𝐺) = 𝛾𝑀𝑅(𝐺) if and only if 𝐺 = 𝐾𝑛 ̅̅̅̅ Proof: It is obvious that if 𝐺 = 𝐾𝑛 ̅̅̅̅ then 𝛾(𝐺) = 𝛾𝑀𝑅(𝐺). Let 𝑓 = (𝑉0, 𝑉1, 𝑉2, 𝑉3) be a 𝛾𝑀𝑅(𝐺)- function. The equality 𝛾(𝐺) = 𝛾𝑀𝑅(𝐺) implies that we have equality in 𝛾(𝐺) ≤ |𝑉1| + |𝑉2| + |𝑉3| = |𝑉1| + |𝑉2| + 3|𝑉3| = 𝛾𝑀𝑅(𝐺). Hence |𝑉3| = 0, which implies that 𝑉0 = ∅. Therefore 𝛾𝑀𝑅(𝐺) = |𝑉1| + |𝑉2| = |𝑉| = 𝑛. This implies that 𝐺 = 𝐾𝑛 ̅̅̅̅ The theorem states that for any graph 𝐺 of order 𝑛, the equality 𝛾𝑀𝑅(𝐺) holds if and only if 𝐺 is the complement of the complete graph 𝐾𝑛. This result has specific implications in areas where graph Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 20 https://internationalpubls.com structure plays a crucial role and the applications are as follows Network Security and Vulnerability Analysis-Detection of Isolated Nodes: In network security, this theorem can be used to identify networks (or parts of networks) that are completely disconnected. The complement of a complete graph has no edges, meaning each node is isolated. If 𝛾(𝐺) = 𝛾𝑀𝑅(𝐺), it implies the network's vulnerability to isolation, which could be critical in detecting potential points of failure or attack. Social Network Analysis-Analysis of Isolated Communities: In social networks, where nodes represent individuals and edges represent interactions, this theorem helps identify when a community (represented by a graph 𝐺) has no internal interactions (as indicated by 𝐺 = 𝐾𝑛 ̅̅̅̅ ). This could be useful in sociological studies to identify isolated or inactive groups within a larger social structure. Biological Networks- Study of Non-Interacting Species or Genes: In biological networks, such as gene interaction or ecological networks, this theorem can be applied to understand when a set of species or genes are completely non-interacting. If a biological network is structured as 𝐾𝑛 ̅̅̅̅ , this suggests no direct interaction between the components, which could have implications for understanding certain biological phenomena or for the design of experiments. Quantum Computing and Information Theory-Graph-Based Quantum State Analysis: In quantum computing, where graph structures can represent quantum states or information flow, this theorem helps identify scenarios where quantum states or bits are completely independent, corresponding to the complement of a complete graph. This could be relevant in designing quantum algorithms or analyzing quantum entanglement properties. Optimization of Isolated Systems-Design of Independent Subsystems [33]: In systems design, particularly where subsystems must operate independently (such as in modular robotics or independent software components), this theorem can guide the design to ensure that no interaction occurs between subsystems, modelled by 𝐺 = 𝐾𝑛 ̅̅̅̅ . This theorem is significant in identifying when a graph structure corresponds to a completely independent set of vertices (no edges), which has applications across various fields where isolation or independence of components is a key consideration. Theorem 4: For any complete bipartite graph 𝛾𝑀𝑅(𝐺) = { 4, min(|𝑋|, |𝑌|) = 2 5, min(|𝑋|, |𝑌|) = 3 6, min(|𝑋|, |𝑌|) = 4 The theorem describes the Middle Roman domination number 𝛾𝑀𝑅(𝐺) for complete bipartite graphs 𝐺 = 𝐾|𝑋||𝑌|, where the vertex set is partitioned into two independent sets 𝑋 and 𝑌. The theorem states that 𝛾𝑀𝑅(𝐺) takes the following values based on the sizes of the partitions: • 𝛾𝑀𝑅(𝐺) = 4, min(|𝑋|, |𝑌|) = 2 • 𝛾𝑀𝑅(𝐺) = 5, min(|𝑋|, |𝑌|) = 3, • 𝛾𝑀𝑅(𝐺) = 6, min(|𝑋|, |𝑌|) = 4. Which can provide the following applications Bipartite Network Design-Optimal Resource Allocation: In designing bipartite networks, such as communication networks or supply chains, this theorem provides a clear guideline on the minimal resources required to dominate the entire network. For example, in a communication network modelled as a bipartite graph, knowing the Middle Roman domination number helps in optimizing the placement of communication nodes to ensure minimal resource usage while maintaining network coverage. Bi-partitioned Social Networks-Influence and Control: In social networks with two distinct groups (e.g., buyers and sellers in a marketplace), this theorem provides insights into the minimal influence needed to control or monitor the interactions between these groups. It can be useful in marketing strategies, where the goal is to dominate a bipartite network of consumers and products. Sensor Networks-Energy-Efficient Sensor Placement: In wireless sensor networks where sensors are deployed to monitor two different types of areas (e.g., Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 21 https://internationalpubls.com indoor vs. outdoor), the theorem can be used to determine the minimal number of sensor nodes required to ensure full coverage and minimal energy consumption [31, 32]. Market Research and Analysis: Consumer-Product Matching: In market analysis, where consumer preferences and products are modelled as a bipartite graph, the theorem assists in understanding the minimal marketing effort needed to ensure that all product categories are effectively promoted to consumers. This theorem is particularly useful in applications where bipartite graphs are prevalent, providing a direct method to evaluate the minimal resources or influence needed to dominate such graphs efficiently. Theorem 5: For any double star 𝑇, 𝛾𝑀𝑅(𝐺) = { 4, if deg(𝑢1) = deg(𝑢2) = 2 5, if deg(𝑢1) = 3, deg(𝑢2) = 2 6, if deg(𝑢1) = deg(𝑢2) = 3 The theorem specifies the Middle Roman domination number 𝛾𝑀𝑅(𝐺) for a double star graph T, where 𝑇 consists of two central vertices 𝑢1 and 𝑢2 with specified degrees. The values of 𝛾𝑀𝑅(𝐺) depend on the degrees of these central vertices: • 𝛾𝑀𝑅(𝐺) = 4, if deg(𝑢1) = deg(𝑢2) = 2, • 𝛾𝑀𝑅(𝐺) = 5, if deg(𝑢1) = 3, deg(𝑢2) = 2, • 𝛾𝑀𝑅(𝐺) = 6, if deg(𝑢1) = deg(𝑢2) = 3. The above has various applications as follows Network Design and Reliability-Optimal Node Placement: In network design, where nodes represent key components or hubs, understanding the Middle Roman domination number of a double star graph helps in determining the minimum number of resources required to ensure network reliability and coverage, particularly in hub-and-spoke models where two main hubs are critical. Telecommunications-Minimal Resource Deployment: In telecommunications, where double star structures may represent network topologies with two central communication nodes, the theorem helps in deciding the minimal number of backup resources (e.g., routers, servers) needed to maintain network functionality in case of failures. Biological Networks- Gene Interaction Networks: In gene interaction networks where two key genes regulate other genes, modelled as a double star graph, the theorem provides insights into the minimum interventions required (e.g., genetic modifications or treatments) to influence or control the network effectively. Data Centre Management-Redundancy Planning: In data centres organized with two main server clusters (central vertices), the theorem aids in determining the minimal redundancy needed to ensure data security and availability across all connected storage units or servers. Transportation and Logistics-Airport and Seaport Operations: In transportation logistics, particularly for airports or seaports that function as central hubs, this theorem helps in planning the minimal yet effective allocation of resources like security, maintenance, or logistics personnel to ensure smooth operations across the network [4, 5]. This theorem is applicable in various domains where the network or structure resembles a double star graph, providing practical insights into the minimal resource allocation needed to maintain effective operations. Proof is obvious for the above theorems. Theorem 6: For any graph 𝐺 with 𝑛 vertices if there exists a vertex 𝑛 with degree 𝑛 − 1 then 𝛾𝑀𝑅(𝐺) = 2 𝑜𝑟 3. Proof: Let 𝑣 ∈ 𝑉(𝐺) with deg(𝑣) = 𝑛 − 1. We define a function 𝑓 = (𝑉0, 𝑉1, 𝑉2, 𝑉3) as follows with 𝑛 = 3, 𝑉1 = ∅, 𝑉2 = {𝑣}, 𝑉3 = ∅ and 𝑉0 = 𝑉 − 𝑉2. Then clearly 𝑓 is a MRDF and 𝛾𝑀𝑅(𝐺) = 2. when 𝑛 ≥ 4, 𝑉1 = ∅, 𝑉2 = ∅, 𝑉3 = 3 and 𝑉0 = 𝑉 − 𝑉3. Then 𝑓 is a MRDF with 𝛾𝑀𝑅(𝐺) = 3|𝑉3| = 3. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 22 https://internationalpubls.com The theorem states that for any graph 𝐺 with 𝑛 vertices, if there exists a vertex 𝑣 with degree 𝑛 − 1(i.e., 𝑣 is adjacent to every other vertex in the graph), then the Middle Roman domination number 𝛾𝑀𝑅(𝐺) is either 2 or 3 [7, 9]. This situation typically arises in graphs where a single vertex has a dominating influence over the rest of the graph, such as in complete graphs or star graphs. Here are some applications of this theorem are as follows Network Hub Design-Optimal Placement of Central Nodes: In network design, especially in star or hub-and-spoke topologies, the theorem can be used to determine the minimal effort needed to dominate the entire network. If a central hub node is connected to all other nodes, this configuration ensures that the Middle Roman domination number is minimized (either 2 or 3), leading to efficient network design and resource allocation. Communication Networks-Designing Efficient Broadcast Networks: In communication networks, where one central node needs to broadcast to all other nodes, this theorem implies that such a network is optimally dominated with minimal additional resources. This can be applied in the design of broadcasting stations, Wi-Fi networks, or cellular networks, where a central node serves as a primary transmitter. Graph Theory Simplification-Characterization of Simple Graphs: This theorem provides a straightforward criterion to identify graphs with low Middle Roman domination numbers. Graphs meeting the condition can be easily classified and analysed, simplifying the study of certain classes of graphs, such as complete or nearly complete graphs. Social Network Analysis-Influential Nodes Identification: In social networks, a vertex with degree 𝑛 − 1 represents an individual connected to every other person in the network. The theorem suggests that such a highly influential individual can dominate the network with minimal additional influence 𝛾𝑀𝑅(𝐺) being 2 or 3) [44, 45]. This insight can be used in identifying key influencers or leaders in social networks for marketing or information dissemination. Security and Defence Applications: Strategic Positioning: In scenarios where a central control or command node must oversee or secure an entire network, this theorem indicates that minimal resources are needed to ensure complete coverage and control. This can apply to military networks, cybersecurity frameworks, or surveillance systems, where a central node needs to ensure the security of the entire system [42, 43]. Biological Networks-Central Nodes in Biological Systems: In biological networks, such as protein interaction networks, a central protein (node) that interacts with nearly all others can ensure network stability or functionality with minimal intervention. The theorem highlights the significance of such central nodes in maintaining biological processes. Urban Planning and Infrastructure-Centralized Service Locations: In urban planning, where a central service location (e.g., a hospital or fire station) serves an entire community, this theorem can help in determining the minimal additional infrastructure required to ensure full coverage. This leads to cost- effective and efficient service provision in cities. Data Centre and Cloud Computing-Efficient Resource Allocation: In data centres or cloud computing environments, where one node acts as a central coordinator, the theorem indicates that minimal resources are needed to maintain overall system performance and redundancy. This can optimize the design and operation of cloud services. Strategic Game Design-Central Control in Games: In strategic games modelled by graphs, where one player or element controls all others (degree 𝑛 − 1), the theorem provides insights into minimal strategies needed to dominate the game, making it useful in game theory and AI. This theorem is particularly valuable in scenarios where a single, highly connected node plays a critical role in the structure and function of a graph. It provides a clear understanding of how such a node can influence the entire system with minimal additional resources or effort. Theorem 7: For any connected graph 𝐺, 𝛾𝑀𝑅(𝐺) = 𝛾 + 1 if and only if the following conditions holds (i) deg(𝑣) ≤ 2 ∀ 𝑣 ∈ 𝑉 (ii) 𝑛 ≤ 4 In order for a Middle Roman dominating function 𝑓 = (𝑉0, 𝑉1, 𝑉2, 𝑉3) to have 𝛾𝑀𝑅(𝐺) = 𝛾 + 1 weight 𝛾(𝐺) + 1, either (i) |𝑉1| = 𝛾(𝐺) + 1, |𝑉2| = 0 and |𝑉3| = 0 or (ii) |𝑉1| = 𝛾(𝐺) − 1, |𝑉2| = 1 and |𝑉3| = 0. Any other arrangement of weight 𝛾(𝐺) + 1 would have |𝑉1| + |𝑉2| < 𝛾(𝐺). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 23 https://internationalpubls.com In case (i): since, |𝑉2| = 0 and |𝑉3| = 0, |𝑉1| = 𝑉. Then by Ore’s theorem, for any connected graph with 𝑛 vertices, 𝛾(𝐺) ≤ 𝑛 2 . Thus 𝑛 = 𝛾(𝐺) + 1 ≤ 𝑛 2 + 1. Hence 𝑛 = 2. Let 𝑉1, 𝑉2 ∈ 𝑉. If 𝑉1 and 𝑉2 are not adjacent, then 𝛾(𝐺) = 𝛾𝑀𝑅(𝐺). Hence 𝑉1 and 𝑉2 are adjacent and hence 𝐺 is a 𝑃2. In case (ii) 𝑓 = (𝑉0, 𝑉1, 𝑉2, 𝑉3) be a 𝛾𝑀𝑅(𝐺) function for 𝐺 of weight 𝛾(𝐺) + 1 with |𝑉1| = 𝛾(𝐺) − 1, |𝑉2| = 1 and |𝑉3| = 0. Let 𝑉2 = {𝑢}. Let 𝑣 ∈ 𝑉1 be adjacent to 𝑢. Then {𝑢} is the 𝛾-set of 𝐺 and 𝛾𝑀𝑅(𝐺) = |𝑉1| + |𝑉2| = 𝛾 + 2 which is a contradiction. Hence no edge of 𝐺 joins {𝑢} and 𝑉1 {𝑢} ≻ 𝑉0, deg(𝑢) = 2. Let 𝑢1 and 𝑢2 be the vertices adjacent to 𝑢. Suppose deg(𝑢1) = deg(𝑢2) = 2. Then {𝑢1, 𝑢2} is the 𝛾-set of 𝐺, 𝑉2 = {𝑢1, 𝑢2}, 𝑉3 = ∅ and 𝑉1 = ∅, which implies that 𝛾𝑀𝑅(𝐺) = 2|𝑉2| = 0, 𝛾 + 2 a contradiction. Suppose deg(𝑢1) = 2 and deg(𝑢2) = 1 and 𝑢1, 𝑢2 are not adjacent. Let 𝑤 be the vertex adjacent to 𝑢1. Then {𝑢1, 𝑢2} is the -set of 𝐺 and 𝛾𝑀𝑅(𝐺) = |𝑉1| + |𝑉2| = 𝛾(𝐺) + 1. Suppose deg(𝑢1) = 3, deg(𝑢2) = 1 and 𝑢1, 𝑢2 are adjacent. Then 𝑉3 = {𝑢1}, i.e., |𝑉3| = 1 which is a contradiction. Hence 𝑛 = 4 and deg(𝑣) ≤ 2 ∀ 𝑣 ∈ 𝑉. Conversely, assume that for every vertex of 𝐺 deg (𝑣) ≤ 2 and 𝑛 ≤ 4. If 𝑉2 = {𝑢}, 𝑉1 = 𝑉 − 𝑁[𝑢] and 𝑉0 = 𝑉 − 𝑉1 − 𝑉2, suppose deg (𝑣) ≤ 2. Case (ii) 𝑛 = 2, then since, 𝐺 is a connected deg(𝑣) = 1 ∀ 𝑣 ∈ 𝑉. Hence 𝐺 is a 𝑃2, 𝑉1 = {𝑣1, 𝑣2} and {𝑣1} is a 𝛾- set. Hence 𝛾𝑀𝑅(𝐺) = 𝛾 + 1. Case (ii) 𝑛 = 2 Let 𝑉1, 𝑉2, 𝑉3 be the vertices of 𝐺. If deg(𝑣𝑖) = 2 ∀ 𝑖 = 1, 2, 3…, then 𝐺 ≅ 𝐶3 and if deg(𝑣𝑖) = 2 for some 𝑖 = 1 say, then 𝐺 ≅ 𝑃3. Hence 𝑉2 = {𝑣1}, 𝑉1 = ∅ and {𝑣1} is the 𝛾- set. Therefore 𝛾𝑀𝑅(𝐺) = 𝛾 + 1. The theorem provides a specific condition under which the Middle Roman domination number 𝛾𝑀𝑅(𝐺) of a connected graph 𝐺 is equal to 𝛾 + 1, where 𝛾 is the domination number of the graph. The condition given is: 1. deg(𝑣) ≤ 2 ∀ 𝑣 ∈ 𝑉 (i.e., each vertex has a degree of at most 2), and 2. 𝑛 ≤ 4 (the number of vertices in the graph is at most 4). This characterization has several practical applications in various field such as, Small-Scale Network Design-Design of Simple Networks: In small networks (with up to 4 vertices), this theorem can simplify the process of determining optimal placements of resources or nodes to ensure full coverage. Since all nodes have a maximum degree of 2, the network structures are simple (such as paths or cycles), which can be used to design and analyse very small and specific network configurations. Algorithmic Efficiency-Exact Computation for Small Graphs: For graphs with up to 4 vertices, the theorem provides an exact and straightforward way to compute the Middle Roman domination number [34, 35]. This can be useful in algorithm design for small-scale problems where brute-force methods or exhaustive search can be feasible and efficient [41]. Graph Theory Education-Teaching and Learning: The theorem can serve as a pedagogical tool to illustrate the concepts of domination numbers and Middle Roman domination numbers in graph theory. Its simplicity makes it an excellent example for teaching these concepts to students in an introductory graph theory course. Optimizing Small-Scale Graphs in Research-Specialized Applications: In research areas that involve small, specialized networks (e.g., certain types of molecular networks, small-scale computational models), the theorem can be used to determine optimal configurations or understand the properties of these networks with precision. Resource Allocation in Simple Systems-Efficient Resource Placement: For systems or models with up to 4 nodes, where resources need to be allocated or placed optimally, the theorem provides a clear understanding of the relationship between the domination number and the Middle Roman domination number. Hence, the theorem provides a clear and specific framework for understanding and solving problems related to Middle Roman domination in very simple or small Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 24 https://internationalpubls.com graphs, making it valuable in theoretical studies, educational contexts, and practical applications involving small-scale systems. Theorem 8: For any connected graph 𝐺, 𝛾𝑀𝑅(𝐺) = 𝛾 + 2 if and only if there exist a minimal dominating set 𝑆 satisfying one of the following conditions: (i) there exist a vertex 𝑣𝜖𝑆 such that 𝑣 is a support and 𝑣 ⊆ 𝑁[𝑃𝑛𝑠(𝑣, 𝑠)] (ii) there exist two vertices 𝑢 and 𝑣 in 𝑆 such that 𝑣 ⊆ 𝑁[𝑃𝑛𝑠(𝑢, 𝑆)] ∪ 𝑁[𝑃𝑛𝑠(𝑣, 𝑆)] 𝐺 has a vertex 𝑣 of degree 𝑛 − 𝛾 or 𝐺 has two vertices 𝑣 and 𝑤 such that |𝑉[𝑣] ∪ 𝑁[𝑤]| = 𝑛 − 𝛾(𝐺) + 2 Proof: If 𝐺 has a vertex 𝑣 of degree 𝑛 − 𝛾 ≥ 3, we define 𝑉0 = 𝑁(𝑣), 𝑉1 = 𝑉 − 𝑁[𝑣], 𝑉2 = ∅ and 𝑉3 = {𝑣}, then 𝑓 = (𝑉0, 𝑉1, 𝑉2, 𝑉3) is a MRDF with 𝑓(𝑣) = 𝛾(𝐺) + 2 and hence is a 𝛾𝑀𝑅- function with 𝑓(𝑣) = 𝛾(𝐺) + 2 . If there are two vertices 𝑢 and 𝑣 such that 𝑁[𝑢] ∪ 𝑁[𝑣] = 𝑛 − 𝛾(𝐺) + 1, 𝑉3 = ∅ we define 𝑉2 = {𝑢, 𝑣}, 𝑉0 = 𝑁[𝑢] ∪ 𝑁[𝑣] − {𝑢, 𝑣}, 𝑉1 = 𝑉 − 𝑁[𝑢] ∪ 𝑁[𝑣]. Then 𝑓 = (𝑉0, 𝑉1, 𝑉2, 𝑉3) is a MRDF with 𝑓(𝑣) = 𝛾(𝐺) + 2 and hence is a 𝛾𝑀𝑅- function. In order for a MRDF 𝑓 = (𝑉0, 𝑉1, 𝑉2, 𝑉3) to have weight 𝛾(𝐺) + 2 either (i) |𝑉3| = 1, |𝑉2| = 0 and |𝑉1| = 𝛾(𝐺) − 1 or (ii) |𝑉3| = 0, |𝑉2| = 2 and |𝑉1| = 𝛾(𝐺) − 2. In case (i) let 𝑓 = (𝑉0, 𝑉1, 𝑉2, 𝑉3) be a 𝛾𝑀𝑅- function for 𝐺 of weight 𝛾(𝐺) + 2 with |𝑉3| = 1, |𝑉1| = 𝛾(𝐺) − 1. Let 𝑉3 = {𝑣}. Since, no edge of 𝐺 join 𝑉1 and 𝑣, 𝑣 ≻ 𝑉0, it follows thatdeg(𝑣) = |𝑉0| = 𝑛 − |𝑉1| − |𝑉3| = 𝑛 − (𝛾(𝐺) − 1) − 1 = 𝑛 − 𝛾(𝐺). In case (ii) let 𝑓 = (𝑉0, 𝑉1, 𝑉2, 𝑉3) be a 𝛾𝑀𝑅- function for 𝐺 of weight 𝛾(𝐺) + 2 with |𝑉3| = 0, |𝑉2| = 2 and |𝑉1| = 𝛾(𝐺) − 2. Let 𝑉2 = {𝑢, 𝑣}. Since, no edge of 𝐺 join 𝑉1 and to 𝑢 or 𝑣, {𝑢, 𝑣} ≻ 𝑉0. It follows that |𝑁[𝑢] ∪ 𝑁[𝑣]| = 𝑛 − |𝑉1| = 𝑛[𝛾(𝐺) − 2] = 𝑛 − 𝛾(𝐺) + 2. The theorem provides a characterization of the Middle Roman domination number 𝛾𝑀𝑅(𝐺) for a connected graph 𝐺 in terms of its minimal dominating sets and specific conditions involving vertex degrees and neighborhoods. This result has several applications in various domains, particularly where network design, analysis, and optimization are critical. Here are some potential applications, Network Design and Optimization-Optimal Resource Placement: The theorem helps identify minimal dominating sets with specific properties that can be used to optimize the placement of resources or infrastructure within a network [23, 24]. For example, if a network (graph) needs to ensure full coverage with minimal redundancy, the theorem can guide the placement of resources such that the network is efficiently dominated. Robustness and Fault Tolerance-Identifying Critical Nodes: The conditions specified in the theorem help in identifying critical nodes that can be used to enhance the robustness of a network [14, 24, 42]. If a minimal dominating set satisfies one of the conditions, it suggests that certain nodes are crucial for maintaining network coverage, which can be useful for designing fault-tolerant networks. Wireless Sensor Networks-Coverage and Connectivity: In wireless sensor networks, ensuring coverage and connectivity with minimal nodes is crucial. The theorem can be applied to identify configurations of sensor nodes that provide optimal coverage and redundancy, which is important for maintaining network performance and reliability [39, 40]. Biological Network Analysis-Understanding Biological Interactions: In biological networks, such as protein-protein interaction networks, identifying minimal sets of proteins (nodes) that dominate the network can help in understanding key interactions and functional modules [26, 27, 29]. The theorem provides a framework for analyzing and optimizing these networks. Social Network Analysis- Influence and Control: In social networks, identifying key individuals or groups that can effectively influence or control the network is crucial [1, 3]. The theorem can be used to find minimal sets of influential nodes that meet certain criteria, aiding in strategic decision-making and intervention planning. Security and Surveillance-Optimizing Coverage: In security and surveillance systems, the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 25 https://internationalpubls.com theorem can be used to determine the optimal placement of cameras or sensors to ensure full coverage of an area with minimal equipment. The conditions help in selecting strategic locations for surveillance that provide comprehensive coverage [36,37]. Overall, the theorem provides a theoretical foundation for understanding and optimizing various network structures and configurations. It helps in solving practical problems related to network coverage, resource allocation, and system robustness across different applications. Determination of 𝜸𝑴𝑹 value of some graphs: In this section we first determine the value of 𝛾𝑀𝑅 for a caterpillar 𝑇. For this purpose, we discuss as follows: Let 𝑣1, 𝑣2, … , 𝑣𝑛 be the consecutive super strong support of 𝑇. We call the graph induced by 𝑁[𝑣𝑖] as super strong support chain [SSS chain] for convenience. Next, we consider a moderate support 𝑢𝑟 as an artificial SSS of the following conditions hold. Let 𝑢𝑟−1 and 𝑢𝑟+1 be the preceding and succeeding vertices of a moderate support 𝑢𝑟. Let 𝑛𝑖 and 𝑛𝑗 be the number of internal vertices of the (𝑢𝑟−1, 𝑢𝑟) path and (𝑢𝑟 , 𝑢𝑟+1) respectively. Then 𝑢𝑟 is said to be an artificial super strong support if one of the following conditions hold. (i) If both 𝑢𝑟−1 and 𝑢𝑟+1 are weak support then 𝑛𝑖 ≡ 0, 2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≡ 0, 2(𝑚𝑜𝑑 3). (ii) If 𝑢𝑟−1 is a moderate support and 𝑢𝑟+1 is a weak support then 𝑛𝑖 ≡ 1, 2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≡ 0, 2(𝑚𝑜𝑑 3) and vice versa. (iii) If both 𝑢𝑟−1 and 𝑢𝑟+1 are moderate supports then 𝑛𝑖 ≡ 1, 2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≡ 1, 2(𝑚𝑜𝑑 3). Let 𝑢𝑠 be the weak support on the spine of 𝑇. Then it will be considered as an artificial super strong support if one of the following conditions hold. (i) If both 𝑢𝑟−1 and 𝑢𝑟+1 are weak support then 𝑛𝑖 ≡ 0, 2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≡ 0, 2(𝑚𝑜𝑑 3). (ii) If 𝑢𝑟−1 is a moderate support and 𝑢𝑟+1 is a weak support then 𝑛𝑖 ≡ 1, 2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≡ 0, 2(𝑚𝑜𝑑 3). (iii) If both 𝑢𝑟−1 and 𝑢𝑟+1 are moderate supports then 𝑛𝑖 ≡ 1, 2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≡ 1, 2(𝑚𝑜𝑑 3). Next, identify the following (i) All consecutive SSS say 𝑠𝑛1, 𝑠𝑛2, 𝑠𝑛3, … , 𝑠𝑛𝑘 where 𝑠𝑛𝑖 = {𝑠1, 𝑠2, 𝑠3, … , 𝑠𝑛𝑖} such that each vertex of 𝑠𝑛𝑖 is a SSS. (ii) consecutive weak supports 𝑤𝑛1, 𝑤𝑛2, 𝑤𝑛3, … , 𝑤𝑛𝑟 where each 𝑤𝑛𝑖 = {𝑤1, 𝑤2, 𝑤3, … , 𝑤𝑛𝑖} such that each vertex of 𝑤𝑛𝑖 is a weak support. Now we determine the value of 𝛾𝑀𝑅 for a caterpillar in the next theorem Theorem 9: Let 𝑇 be any caterpillar. let 𝑆 = {𝑆𝑛𝑖 𝑖 = 1, 2, … , 𝑘 𝑆𝑖 𝑖𝑠 𝑎 𝑆𝑆𝑆 𝑐ℎ𝑎𝑖𝑛}, 𝑆1 = {𝑠/ 𝑒𝑎𝑐ℎ 𝑣𝑒𝑟𝑡𝑒𝑥 𝑠 𝑖𝑠 𝑒𝑖𝑡ℎ𝑒𝑟 𝑎 𝑆𝑆𝑆 𝑜𝑟 𝐴𝑆𝑆𝑆}, 𝑆2 = {𝑚/ 𝑒𝑎𝑐ℎ 𝑚 𝑖𝑠 𝑒𝑖𝑡ℎ𝑒𝑟 𝑎 𝑀𝑆 𝑜𝑟 𝐴𝑀𝑆}, 𝑊 = {𝑊𝑛𝑗, 𝐽 = 1, 2, … , 𝑙 / 𝑒𝑎𝑐ℎ 𝑊𝑛𝑗 𝑖𝑠 𝑎 𝑐𝑜𝑚𝑏 𝑔𝑟𝑎𝑝ℎ 𝑤𝑖𝑡ℎ 𝑛𝑗 𝑤𝑒𝑎𝑘 𝑠𝑢𝑝𝑝𝑜𝑟𝑡𝑠}. Let 𝑇1 = [𝑁[𝑆] ∪ 𝑁[𝑆1] ∪ 𝑁[𝑆2] ∪ 𝑁[𝑊]]. Let 𝑇2 = 𝑇 − 𝑇1 and 𝑟 = 1, 2, … , 𝑡. 𝑅1, 𝑅2, … , 𝑅𝑡 be the components of 𝑇2. Then 𝛾𝑀𝑅 = 3[(𝑥 + 𝑎)𝑛] + [∑ 𝑛𝑖 +𝑘 𝑖=1 |𝑆1|] + 2|𝑆2| + 3 4 ∑ 𝑇1 𝑙 𝑗=1 + 2|𝑇2| Proof: Let 𝑇 be any caterpillar. identify the SSS chain, SSS, ASSS, MSS, AMSS and comb graphs using the above procedure. Let 𝑆, 𝑆1, 𝑆2, 𝑊 be as defined in the theorem. Let 𝑣 be an artificial super strong support. Let 𝑣1 and 𝑣2 be the supports that precede and succeed 𝑣 on the spine. Let 𝑃 be the (𝑣1, 𝑣2) path. Let 𝑢1, 𝑢2, … , 𝑢𝑥 be the internal vertices of (𝑣1, 𝑣)- path and 𝑤1, 𝑤2, … , 𝑤𝑦 be the internal vertices of (𝑣, 𝑣2)- path. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 26 https://internationalpubls.com Case (i) 𝑣1 and 𝑣2are the weak supports. If two legions are posted at 𝑣 then ⌈ 2(𝑥−1) 3 ⌉ + ⌈ 2(𝑦−1) 3 ⌉ + 2 = 𝐾1 legions are needed to safeguard the vertices on the path 𝑃. But, on the other hand if three legions are posted at 𝑣1 then ⌈ 2(𝑥−2) 3 ⌉ + ⌈ 2(𝑦−2) 3 ⌉ + 3 legions are required to safeguard the path 𝑃, which is less than or equal to 𝑀1. Hence, we assign three legions at 𝑣 to safeguard 𝑁[𝑣]. Case (iii) 𝑣1 is a weak support and 𝑣2 is a moderate support. If the legions are posted at 𝑣 then ⌈ 2(𝑥−1) 3 ⌉ + ⌈ 2𝑦 3 ⌉ + 2 = 𝐾2 legions are required to safeguard the vertices on the path 𝑃. But, on the other hand, if three legions are posted at 𝑣, then ⌈ 2(𝑥−2) 3 ⌉ + ⌈ 2(𝑦−1) 3 ⌉ + 3 legions are required to safeguard the path 𝑃 which is less than 𝐾2. Hence, we assign three legions at 𝑣. Case (iv) 𝑣1 and 𝑣2 are moderate supports. If legions are posted at 𝑣1 then ⌈ 2𝑥 3 ⌉ + ⌈ 2𝑦 3 ⌉ + 2 = 𝐾3 legions are required to safeguard the legions on the path. But, on the other hand if three legions are posted at 𝑣, then ⌈ 2(𝑥−1) 3 ⌉ + ⌈ 2𝑦−1 3 ⌉ + 3 legions are required to safeguard the path 𝑃, which is less than or equal to 𝐾3. Hence, we assign three legions at 𝑣 to safeguard 𝑁[𝑣]. Let 𝑤 be an artificial strong support. Let 𝑤1 and 𝑤2 be the supports that precede and succeed 𝑤 on the spine. Let 𝑃 be the (𝑤1, 𝑤2) path. Let 𝑥1, 𝑥2, … , 𝑥𝑝 and 𝑦1, 𝑦2, … , 𝑦𝑞 be the internal vertices of (𝑤1, 𝑤) path and (𝑤, 𝑤2) path respectively. Case (v) 𝑤1 and 𝑤2 are weak supports. If two legions are posted at 𝑤 then, ⌈ 2(𝑝−1) 3 ⌉ + 2 = 𝐿1 legions are required to safeguard the internal vertices on the path (𝑤1, 𝑤). But on the other hand, if three legions are posted at 𝑤, then ⌈ 2(𝑝−2) 3 ⌉ + ⌈ 2𝑞−2 3 ⌉ + 3 legions are required to safeguard the path 𝑃, which is less than 𝐿1. Hence, we assign three legions at 𝑤 to safeguard 𝑁[𝑤]. Case (vi) 𝑤1 is a moderate support and 𝑤2 is a weak support. If two legions are posted at 𝑤 then ⌈ 2𝑝 3 ⌉ + ⌈ 2(𝑞−1) 3 ⌉ + 2 = 𝐿2 legions are required to safeguard the vertices on the path 𝑃. But on the other hand, if three legions are posted at 𝑤, then ⌈ 2(𝑝−1) 3 ⌉ + ⌈ 2(𝑞−2) 3 ⌉ + 3 legions are required which is less than 𝐿2. Hence, we assign three at 𝑤 to safeguard 𝑁[𝑤]. Case (ii) Both, 𝑤𝑟−1 and 𝑤𝑟+1 are moderate supports. If two legions are posted at 𝑤, then ⌈ 2𝑝 3 ⌉ + ⌈ 2𝑞 3 ⌉ + 2 = 𝐿3 legions are required to safeguard the path 𝑃. But, on the other hand if three legions are placed at 𝑤 then ⌈ 2(𝑝−1) 3 ⌉ + ⌈ 2(𝑞−1) 3 ⌉ + 3 legions are required to safeguard the vertices on the path (𝑤1, 𝑤2). Which is less than 𝐿3. Hence, we assign three legions at 𝑤 to safeguard 𝑁[𝑤]. Hence, in all the cases we see that three legions are required to safeguard 𝑁[𝑤]. Next, we consider a weak support 𝑢𝑟 as an artificial moderate support, if any one of the following conditions hold. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 27 https://internationalpubls.com (i) If both 𝑢𝑟−1 and 𝑢𝑟+1 are weak supports then 𝑛𝑖 ≡ 0, 2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≡ 1(𝑚𝑜𝑑 3) or vice versa. (ii) If both 𝑢𝑟−1 and 𝑢𝑟+1 are moderate supports then 𝑛𝑖 ≡ 1, 2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≡ 0(𝑚𝑜𝑑 3) or vice versa. (iii) It either 𝑢𝑟−1 is moderate support and 𝑢𝑟+1 is a weak support then 𝑛𝑖 ≡ 1, 2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≡ 0(𝑚𝑜𝑑 3) Let 𝑢 be an artificial moderate support. Let 𝑢1 and 𝑢2 be the supports that precede and succeed 𝑢 on the spine. let 𝑃 be the (𝑢1, 𝑢2)- path. Let 𝑡1, 𝑡2, … , 𝑡𝑚 and 𝑙1, 𝑙2, … , 𝑙𝑠 be the internal vertices of (𝑢1, 𝑢)- path and (𝑢, 𝑢2)-path respectively. Case (i) Both, 𝑢𝑟−1 and 𝑢𝑟+1 are weak supports. Claim: 𝑛𝑖 ≡ 0, 2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≡ 1(𝑚𝑜𝑑 3) Sub case (i): 𝑛𝑖 ≡ 0, 2(𝑚𝑜𝑑 3) Suppose, 𝑛𝑗 ≢ 1(𝑚𝑜𝑑 3). Then either 𝑛𝑗 ≡ 0, (𝑚𝑜𝑑 3) or 𝑛𝑗 ≡ 2, (𝑚𝑜𝑑 3). In both cases 𝑢 will become an artificial super strong support which is a contradiction to the artificial moderate support 𝑢𝑟. Sub case (ii): 𝑛𝑗 ≡ 1(𝑚𝑜𝑑 3) Suppose 𝑛𝑖 ≢ 0, 2(𝑚𝑜𝑑 3). Then 𝑛𝑖 ≡ 1(𝑚𝑜𝑑 3) legions are posted at 𝑢, it can safeguard only two vertices 𝑢 and the corresponding leaf vertices, where as by definition two legions placed at a region can safeguard at most two adjacent regions. Hence the middle Roman domination number in this case may not be minimum. Hence, 𝑛𝑖 ≡ 0,2(𝑚𝑜𝑑 3). Case (ii) Both, 𝑢𝑟−1 and 𝑢𝑟+1 are moderate supports Claim: 𝑛𝑖 ≡ 1,2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≡ 0(𝑚𝑜𝑑 3) or vice versa. Subcase (a): suppose 𝑛𝑖 ≡ 1,2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≢ 0(𝑚𝑜𝑑 3) then 𝑛𝑗 ≡ 1, 2(𝑚𝑜𝑑 3). Then in both cases 𝑢 will become artificial SSS which is a contradiction to the artificial moderate support 𝑢𝑟. Hence, the claim. Subcase (iii): 𝑢𝑟−1 is a moderate support and 𝑢𝑟+1 is a weak support. Claim: 𝑛𝑖 ≡ 1,2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≢ 1(𝑚𝑜𝑑 3). Then 𝑛𝑗 ≡ 0(𝑚𝑜𝑑 3) or 𝑛𝑗 ≡ 2(𝑚𝑜𝑑 3). In both the cases, 𝑢𝑟 will become artificial SSS a contradiction to the artificial moderate support 𝑢𝑟. 𝑛𝑗 ≡ 2(𝑚𝑜𝑑 3), Subcase (b): 𝑛𝑗 ≡ 1(𝑚𝑜𝑑 3) and 𝑛𝑖 ≢ 1,2(𝑚𝑜𝑑 3) suppose not. Let 𝑛𝑗 ≡ 1(𝑚𝑜𝑑 3) and 𝑛𝑖 ≡ 0(𝑚𝑜𝑑 3). Then the two legions placed at 𝑢𝑟 protect only two regions whereas, by definition two legions can protect at most three regions which may or may not minimize the middle Roman domination number. Hence, 𝑛𝑖 ≡ 0(𝑚𝑜𝑑 3). Similarly, we can bring 𝑛𝑖 ≡ 1, 2(𝑚𝑜𝑑 3) and 𝑛𝑗 ≢ 1 (𝑚𝑜𝑑 3). Hence the claim. Let 𝐶1, 𝐶2, … 𝐶𝑛𝑖 be the support vertices of Ƈ𝑛𝑖. If 𝑛𝑖 is odd, then omit 𝐶1 or 𝐶𝑗 according to the following (i) If 𝑢𝑟−1 is a moderate support, then 𝑛𝑖 ≡ 1,2(𝑚𝑜𝑑 3) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 28 https://internationalpubls.com (ii) If 𝑢𝑟−1 is a weak support, then 𝑛𝑖 ≡ 0,2(𝑚𝑜𝑑 3) (iii) If 𝑢𝑟+1 is a moderate support, then omit 𝐶𝑛 if 𝑛𝑗 ≡ 1,2(𝑚𝑜𝑑 3) (iv) If 𝑢𝑟+1 is a weak support, then omit 𝐶𝑛 𝑛𝑗 ≡ 0,2(𝑚𝑜𝑑 3) let ʂ = {𝑆𝑚1, 𝑆𝑚1, … , 𝑆𝑚𝑟} be the SSS chain in 𝑇 and with each chain 𝑆𝑚𝑖, 𝑖 = 1, 2, 3, … , 𝑚𝑟 has 𝑚𝑖 SSS. Let 𝑥 ∈ 𝑁(𝑆𝑚1) and 𝑦 ∈ 𝑁(𝑆𝑚𝑟), 𝑆 = {𝑠1, 𝑠2, 𝑠3, … , 𝑠𝑛} be the SSS or ASSS of 𝑇, Ƈ = {Ƈ𝑛1,Ƈ𝑛2,Ƈ𝑛3, … ,Ƈ𝑛𝑝} be the comb graph in 𝑇, where 𝑛𝑗 , 𝑗 = 1, 2, 3, … , 𝑝 are the even number of supports in Ƈ𝑗. Let 𝑇1 = 𝑇 − [𝑁[𝑆] ∪ 𝑁[𝑠] ∪ 𝑁[Ƈ] − {𝑥, 𝑦 𝑥 ∈ 𝑁(𝑐1), 𝑦 ∈ 𝑁(𝑐𝑛)}] ∪ 𝐿 ∪ 𝐴. Where, 𝐿 = { 𝑙 𝑙 𝑖𝑠 𝑎 𝑙𝑒𝑎𝑓 𝑣𝑒𝑟𝑡𝑒𝑥 𝑖𝑛 𝑁(𝑚𝑘), 𝑘 = 1, 2, 3, … , 𝑝}, 𝐴 = {𝑁[𝑎] − {ℎ}, where a is an artificial SSS or h is the vertex on the internal path (𝑢𝑟−1, 𝑎) 𝑜𝑟 (𝑎, 𝑢𝑟−1) 𝑤𝑖𝑡ℎ {𝑛𝑗 ≢ 0, 2 𝑚𝑜𝑑 3} Hence, 𝛾𝑀𝑅 = 3[(𝑥 + 𝑎)𝑛] + [∑ 𝑛𝑖 +𝑘 𝑖=1 |𝑆1|] + 2|𝑆2| + 3 4 ∑ 𝑇1 𝑙 𝑗=1 + 2|𝑇2|. The above theorem has various applications in various applications including Network Design and Optimization: In telecommunications and computer networks, designing efficient network topologies and optimizing resource allocation often involves understanding the domination properties of the network. The theorem can be used to determine the optimal placement of network nodes or resources to achieve desired network performance or reliability. VLSI Circuit Design: In Very-Large- Scale Integration (VLSI) circuit design, the problem of minimizing the number of components (e.g., transistors, gates) while ensuring that all necessary connections and functionalities are present can be modelled using graph theory. The Middle Roman domination number can help in optimizing circuit layouts to minimize power consumption or improve performance. Bioinformatics: In bioinformatics, especially in the study of protein-protein interaction networks or gene regulatory networks, graph- theoretic models are used to understand the structure and function of biological networks. The theorem can be used to analyse and optimize these networks by providing insights into their dominating sets and efficient network configurations. Urban Planning and Infrastructure: For urban planning and the design of infrastructure networks (such as roads, utilities, and public services), the theorem can assist in optimizing the placement of facilities and services to ensure maximum coverage and efficiency. This can help in minimizing costs while ensuring adequate service provision. Social Network Analysis: In social network analysis, the Middle Roman domination number can be used to identify key individuals or groups that influence the entire network. This can be useful in marketing, spreading information, or managing social dynamics [46, 47]. Algorithm Design: The theorem provides a formula that can be used in the development of algorithms for graph-based problems. It can be applied to design algorithms for computing the Middle Roman domination number in practical scenarios where such graphs are encountered [38]. Conclusion: The concept of a Middle Roman Dominating Function (MRDF) in graph theory has wide-ranging applications in computer science, particularly in optimizing resource allocation, enhancing network security, and improving efficiency in distributed systems. By strategically assigning values to nodes, MRDF enables efficient task scheduling, load balancing, and energy conservation in sensor networks. 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