Abstract
In any interconnection network, fault tolerance is the most desirable property to achieve reliability. Toeplitz networks are used as interconnection networks due their smaller diameter, symmetry, simpler routing, high connectivity, and reliability. The partition dimension of a network is presented as an extension of metric dimension of networks. Its applications can be seen in several areas including robot navigation, network designing, image processing, and chemistry. In this article, the fault tolerant partition dimension, , of Toeplitz networks, is shown to be bounded below by 4 for , whereas it is bounded above by 5 for . Further, it is shown that the exact value of equals 4 for ; ; and .
1. Introduction
Computer networking gives a way of communication among several processors of a computer and numerous computers connected to a network. The objective of the interconnection networks is to reduce the time when it becomes impossible to manage large amount of data by a single processor, so the job is divided among the several processors working at the same time. Effective interconnection networks play vital rule in transferring data briskly among the different components of parallel processing. It is always preferable for a computer network to have smaller diameter, lesser degree, alternate paths among the nodes, higher level of symmetry, simpler routing, and high level of connectivity. Fault tolerance of a network ensures that the system will continue to work even if some of its components fail to work. This also guarantees the lower maintenance cost and longer durability of the network. Toeplitz networks are used as interconnection networks due their smaller diameter, symmetry, simpler routing, high connectivity, and reliability [1]. In this article, we have computed the fault tolerant partition dimension of some particular classes of Toeplitz networks.
1.1. Background and Related Work
The metric basis is a set that contains least number of nodes of a network such that all nodes are exclusively identified by the distances from nodes in the basis. Motivated by the need of such a set of nodes, Slater [2] and Harary and Melter [3] independently introduced the concept of metric basis in 1975 and 1976. Slater [2] used metric basis to locate the invader in the given network. Khuller et al. [4] used it for robot navigation in 1996. In 2000, Chartrand et al. [5] used it to locate position of functional groups in chemical compounds. In 2008, Hernando et al. [6] presented the notion of fault tolerant metric dimension of graphs. In 2015, Estrado-Moreno et al. [7] presented the concept of metric. For , the metric dimension is known as fault tolerant metric dimension denoted by . Further study on metric dimension can be seen in [8, 9].
An important variant of metric basis is the partition basis which is the partition of the nodes of a network with minimum size such that all nodes are exclusively identified by the distances from sets in the partition. The minimum size of a resolving partition is called partition dimension of the network. In 2000, Chartrand et al. [10] presented the concept of partition dimension which is an extension of metric dimension. In 2009, Javaid et al. [11] introduced the concept of fault tolerant partition dimension. In 2014, the partition dimension of unicyclic graphs was discussed in [12]. The partition dimension of some families of trees was studied by Fredlina and Baskaro [13] in 2015. In 2020, Estrado-Moreno [14] presented the concept of partition dimension of the network. For , the partition dimension is known as fault tolerant partition dimension denoted by . The was computed for some important networks in [15, 16].
Harary and Melter in [3] concluded that computation of metric basis of a network is an NP-hard problem; it further implies that computation of partition-related metric parameters for general graphs is also NP-hard. This lead the current research trends to focus on the computation of these metric parameters for specific classes of graphs. Recently, Liu et al. in [17] calculated the metric dimension of Toeplitz networks. The present research on Toeplitz networks and their merits in the interconnection networks [1] motivated us to study fault tolerant partition dimension of Toeplitz networks.
1.2. Preliminaries
Considering a connected network , the node and edge sets of are denoted by and , respectively. The length of the shortest path, connecting two nodes and , is known as the distance between and denoted by . The distance of a node from a subset of the node set is , whereas the set is called the neighbourhood of node . Let be an ordered set of nodes, the representation of a node with respect to is a vector such that . If the vectors are different, for every node of the network, then is termed as a resolving set of . The minimal cardinality of is termed as the metric dimension of , represented by . If the vectors are different in at least places for every node of the graph, then is termed as a metric generator for . metric basis is a generator with least number of nodes. The minimal cardinality of is termed as the metric dimension of , represented by .
Let be an ordered partition of a connected network . The partition representation of node with respect to is a vector such that . If the vectors are different for every node of the graph, then partition is termed as a resolving partition of . The minimal cardinality of for which there is a resolving partition is termed as the partition dimension of , represented by . If the vectors are different in at least places for every node of the network, then is termed as a partition generator of . partition basis is a generator with least number of sets. The minimal cardinality of is termed as a the partition dimension of , represented by .
1.3. Main Results
The study conducted in this paper leads to the following results.
Theorem 1. Letbe the Toeplitz network, then(1)For,(2)For, (3)For, (4)For, (5)Forand, In Section 2, Toeplitz networks are defined and the fault tolerant partition dimension of Toeplitz networks, , for different values of and have been computed. In Section 3, the results are summarized and two open problems are proposed.
2. Fault Tolerant Partition Dimension of Toeplitz Networks
For a sequence of positive integers, with , the Toeplitz network is a network on the vertex set with two vertices and which are adjacent if and only if . Figure 1 shows .
The following lemma gives the lower bound on the fault tolerant partition dimension of Toeplitz networks .
Lemma 1. Letbe the Toeplitz network with, then.
Proof. Assume that for . Let be a fault tolerant partition basis of . Clearly, one of the sets , and contains at least two vertices. Suppose that . Here, at least one node of , say , has a coordinate of . Otherwise, each node in will have the same representation. So, without loss of generality, we can assume that the third coordinate of is greater than 1. i.e., . Let be a node in such that . Let be the neighbourhood of . So, as for all . Since in the Toeplitz network, so we have the following three cases. Case 1: when , let . Case 1.1: suppose all the nodes of are in the same set or . Assume that , then and for , , for . Hence, . Now, pigeonhole principle implies that at least two of the nodes and will be equidistant from and , which is a contradiction, whereas generates a trivial case of contradiction. Case 1.2: assume that or contains three vertices from . Let , and , then , , and . In this case, we have . Again, pigeonhole principle implies that at least two of the nodes , and will be equidistant from and , leading to a contradiction, whereas three nodes of in generate a trivial case of contradiction. Case 1.3: assume that each of and contains two nodes of . Let and , then and . In this case, we have . Again by pigeonhole principle, at least two of the nodes and will have the same distances from and . This leads to a contradiction. Now, if we consider nodes in , then and , which is again a contradiction. Case 2: when , let . The subcases in this case are similar to Case 1.2 and Case 1.3. Case 3: when , let . Case 3.1: if both of the nodes of are in the same set or , assume that , then and . Hence, . Now, if any of and is equal to , then we have a contradiction. Since in Toeplitz networks the only nodes with two neighbours are and , so in all other cases, we will have and which are similar to Case 1 and Case 2, whereas generates a trivial case of contradiction. Case 3.2: assume that each of and contains a node from the set . Let and , then we can consider the and to get contradiction as in the previous cases.Hence, for .
In the following theorems, we calculate the fault tolerant partition dimension of Toeplitz networks for different values of .
Theorem 2. Letbe the Toeplitz network with, then.
Proof. Let be a partition of . The proof is divided into three cases, and in each case, we will show that is the fault tolerant partition generator of . Let . Case 1: for , let , , and . When , the are tabulated in Table 1. Case 2: for +1, let , , and When , the are tabulated in Table 2. Case 3: for +2, let , , and . When , the are tabulated in Table 3.Tables 1–3 clearly show that is a resolving generator of for , therefore for . Lemma 1 implies that for . This establishes our claim.
Theorem 3. Letbe the Toeplitz network, thenwhenandfor.
Proof. The proof is divided into two parts. Case 1: the partition for is tabulated in Table 4 which clearly shows that is the fault tolerant partition generator; therefore, for . Lemma 1implies that for . This establishes our claim. Case 2: suppose be a partition of for . Let , and . For , the are tabulated in Tables 5–7, respectively.Tables 5–7 clearly show that is a resolving generator of for , so for . Lemma 1 implies that for . This establishes that .
Theorem 4. Letbe the Toeplitz network with oddand, then.
Proof. Let be a partition of . The proof consists of three parts. Case 1: for , let , and . When , the are tabulated in Table 8. Case 2: for , let , and . When , the are tabulated in Table 9. Case 3: for , let , and . When , the are tabulated in Table 10.Tables 8–10 clearly show that is a resolving generator of for , therefore for . Lemma 1 implies that for . This establishes our claim.
Theorem 5. Considerwith even, then, for.
Proof. Let be a partition of . The proof consists of three parts. Case 1: for , let , and . When , the are tabulated in Table 11. Case 2: for , let . Case 2a: when (mod 4), let , and . When , the are tabulated in Table 12. Case 2b: when (mod 4), let , and . When , the are tabulated in Table 13. Case 3: for , let , and . When , the are tabulated in Table 14.It is clear from Tables 11–14 that is a resolving generator of for , therefore for . Lemma 1 implies that for . This establishes our claim.
3. Conclusion
In this research article, we conclude that the fault tolerant partition dimension of is 4 when and . When and , the is 4 and its value is less than or equal to 5 for . It is also inferred that the fault partition dimension of Toeplitz networks is 4 when for even and odd [18].
Open Problem 1. Calculate the for odd and .
Open Problem 2. Calculate the for .
Data Availability
All data are included within this article. However, the reader may contact the corresponding author for more details of the data.
Conflicts of Interest
The authors declare that they have no conflicts of interest.