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Fault-tolerance of a system measures its working capability in the presence of faulty components in the system. The fault-tolerant partition dimension of a network computes the least number of subcomponents of network required to distinctively identify each node in the presence of faults, having promising applications in telecommunication, robot na...
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The concept of metric-related parameters permeates all of graph theory and plays an important role in diverse networks, such as social networks, computer networks, biological networks and neural networks. The graph parameters include an incredible tool for analyzing the abstract structures of networks. An important metric-related parameter is the p...
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... In addition, this paper includes an implementation of fault resistant partitioning for region-based routing in networks. Azhar et al. [30] examined the partition and fault-tolerant partition resolvability of specific triangular mesh networks, including triangular ladder, triangular mesh, reflection triangular mesh, tower triangular mesh, and reflection tower triangular mesh networks. This demonstrates that the fault-tolerant partition dimension of these networks is 4, while the partition dimension is 3. Continuing the abovementioned investigation, we shall calculate the metric dimensions of two distinct types of claw-free cubic graphs graphs-a string of diamonds (SOD) and a ring of diamonds (ROD) in this paper. ...
Consider the simple connected graph G with vertex set V(G) and edge set E(G). A graph \(G\) can be resolved by \(R\) if each vertex’s representation of distances to the other vertices in \(R\) uniquely identifies it. The minimum cardinality of the set \(R\) is the metric dimension of \(G\). The length of the shortest path between any two vertices, x, y in V(G), is signified by the distance symbol d(x, y). An ordered k-tuple \(r(x/R)=(d(x,z_1),d(x,\ z_2),…,d(x,z_k))\) represents representation of \(x\) with respect to \(R\) for an ordered subset \(R={\{z}_1,z_2,z_3…,z_k\}\) of vertices and vertex \(x\) in a connected graph. Metric dimension is used in a wide range of contexts where connection, distance, and connectedness are essential factors. It facilitates understanding the structure and dynamics of complex networks and problems relating to robotics network design, navigation, optimization, and facility location. Robots can optimize their localization and navigation methods using a small number of reference sites due to the pertinent idea of metric dimension. As a result, many robotic applications, such as collaborative robotics, autonomous navigation, and environment mapping, are more accurate, efficient, and resilient. A claw-free cubic graph (CCG) is one in which no induced subgraph is a claw. CCG proves helpful in various fields, including optimization, network design, and algorithm development. They offer intriguing structural and algorithmic properties. Developing algorithms and results for claw-free graphs frequently has applications in solving of challenging real-world situations. The metric dimension of a couple of claw-free cubic graphs (CCG), a string of diamonds (SOD), and a ring of diamonds (ROD) will be determined in this work.
... The labeling is illustrated in Figure VI. For more study, we refer the readers to [3]. Figure VI: The triangular mesh network T n Lemma 2.1: For n ≥ 1, ...
The robustness of the networks is their capacity to remain operational in the presence of faults and disruptions, and thereby it is an important tool to provide data transmission in telecommunications networks, enterprise networks, and cloud computing networks. The connected collection of nodes in a network, excluding which results in the decomposition of a network into components such that the cardinality of each component is at most equal to the cardinality of the collection, is referred to as a connected safe set (CSS). The least size of CSS is known as connected safe number (CSN). The identification of the connected collection of nodes in the networks, capable of enduring dual load in case of faults and disruption, can be realized as CSS. Mesh networks (MNs) have become an integral part of a variety of domains, such as smart cities, disaster recovery, and military defense, due to their decentralized nature and ability to reconfigure as conditions change dynamically. In this paper, the CSS and CSN for various types of MNs, such as triangular, triangular circular, double triangular circular, and quadrangular necklace mesh are computed. Finally, an application of CSS in the context of optimal router installation on certain MNs is included.
... The FTPDs of toeplitz networks (see [28]), convex polytopes (see [29]) and circulant graphs having a {1, 2} connection set (see [30]) were discussed by Asim et al. For further studies on FTPDs, we refer to [31,32]. In this paper, we protract these studies by computing the FTPD of a kayak paddle graph and flower graph. ...
The concept of metric-related parameters permeates all of graph theory and plays an important role in diverse networks, such as social networks, computer networks, biological networks and neural networks. The graph parameters include an incredible tool for analyzing the abstract structures of networks. An important metric-related parameter is the partition dimension of a graph holding auspicious applications in telecommunication, robot navigation and geographical routing protocols. A fault-tolerant resolving partition is a preference for the concept of a partition dimension. A system is fault-tolerant if failure of any single unit in the originally used chain is replaced by another chain of units not containing the faulty unit. Due to the optimal fault tolerance, cycle-related graphs have applications in network analysis, periodic scheduling and surface reconstruction. In this paper, it is shown that the partition dimension (PD) and fault-tolerant partition dimension (FTPD) of cycle-related graphs, including kayak paddle and flower graphs, are constant and free from the order of these graphs. More explicitly, the FTPD of kayak paddle and flower graphs is four, whereas the PD of flower graphs is three. Finally, an application of these parameters in a scenario of installing water reservoirs in a locality has also been furnished in order to verify our findings.
... In [5], region-based routing puts the destinations into subnets allowing a significant reduction in the sizes of routing tables. Recently, Azhar et al. [6] has discussed the use of fault tolerant addressing schemes in mesh-related networks. Bossard et al. [7] discussed the occurrence of faults as clusters in the larger networks and focused on the topological properties of the interconnection networks. ...
... The pd 2 (W) is termed as a fault tolerant partition dimension. The pd 2 (W) was computed for some important graphs in [6,[29][30][31][32][33][34]. Furthermore, the following lemma characterizes the graphs with a fault tolerant partition dimension bounded below by 4 and will be used in computing the fault tolerant partition dimension of SOXCN, RHOXN and RTOXN interconnection networks in the forthcoming subsections. ...
... The applications of partition and fault tolerant addressing schemes have recently been discussed for routing optimization in [29], supply chain optimization in [31] and sensors deployment in [6]. In this section, an application of the fault tolerant addressing scheme in the context of optimal data flow is included. ...
The symmetry of an interconnection network plays a key role in defining the functioning of a system involving multiprocessors where thousands of processor-memory pairs known as processing nodes are connected. Addressing the processing nodes helps to create efficient routing and broadcasting algorithms for the multiprocessor interconnection networks. Oxide interconnection networks are extracted from the silicate networks having applications in multiprocessor systems due to their symmetry, smaller diameter, connectivity and simplicity of structure, and a constant number of links per node with the increasing size of the network can avoid overloading of nodes. The fault tolerant partition basis assigns unique addresses to each processing node in terms of distances (hops) from the other subnets in the network which work in the presence of faults. In this manuscript, the partition and fault tolerant partition resolvability of oxide interconnection networks have been studied which include single oxide chain networks (SOXCN), rhombus oxide networks (RHOXN) and regular triangulene oxide networks (RTOXN). Further, an application of fault tolerant partition basis in case of region-based routing in the networks is included.
