# On the Fault Tolerant Partition Resolvability of Toeplitz Networks

Authors: Asim Nadeem; Agha Kashif; Amer Aljaedi; Sohail Zafar
Journal: Mathematical Problems in Engineering
 (2022)

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## 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,pd2Tn1,t, of Toeplitz networks, is shown to be bounded below by 4 for t≥2,n≥4, whereas it is bounded above by 5 for t=3,n≥14. Further, it is shown that the exact value of pd2Tn1,t equals 4 for t=2,n≥4; t=3,n∈5,6,…,13; and t≥4,n∈t+2,t+3,t+4.


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## Body

## 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 κ−metric. For κ=2, the κ−metric dimension is known as fault tolerant metric dimension denoted by dim2W. Further study on κ−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 κ−partition dimension of the network. For κ=2, the κ−partition dimension is known as fault tolerant partition dimension denoted by pd2W. The pd2W 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 networkW, the node and edge sets of W are denoted by VW and EW, respectively. The length of the shortest path, connecting two nodes and q, is known as the distance between p and q denoted by dp,q. The distance of a node p from a subset A of the node set is dp,A=mindp,t|t∈A, whereas the set Np=q∈VG|pq∈EG is called the neighbourhood of node p. Let Ω=x1,x2,…,xl be an ordered set of nodes, the representation of a node p with respect to Ω is a vector rp|Ω such that rp|Ω=dp,x1,dp,x2,…,dp,xl. If the vectors rp|Ω are different, for every node of the network, then Ω is termed as a resolving set of W. The minimal cardinality of Ω is termed as the metric dimension of W, represented by dimW. If the vectors rp|Ω are different in at least κ places for every node of the graph, then Ω is termed as a κ−metric generator for W. κ−metric basis is a generator with least number of nodes. The minimal cardinality of Ω is termed as the κ−metric dimension of W, represented by dimκW.LetΨ=B1,B2,…,Bl be an ordered l−partition of a connected network W. The partition representation of node p with respect to Ψ is a vector rp|Ψ such that rp|Ψ=dp,B1,dp,B2,…,dp,Bl. If the vectors rp|Ψ are different for every node of the graph, then l−partition is termed as a resolving partition of W. The minimal cardinality of Ψ for which there is a resolving l−partition is termed as the partition dimension of W, represented by pdW. If the vectors rp|Ψ are different in at least κ places for every node of the network, then Ψ is termed as a κ−partition generator of W. κ−partition basis is a generator with least number of sets. The minimal cardinality of Ψ is termed as a the κ−partition dimension of W, represented by pdκW.


### 1.3. Main Results

The study conducted in this paper leads to the following results.Theorem 1.
LetTn1,tbe the Toeplitz network, then(1)
Forn≥4,pd2Tn1,t≥4(2)
Forn≥4, pd2Tn1,2=4(3)
Forn∈5,6,…,13, pd2Tn1,3=4(4)
Forn≥14, 4≤pd2Tn1,3≤5(5)
Fort≥5andn=t+2,t+3,t+4, pd2Tn1,t=4
In Section2, Toeplitz networks are defined and the fault tolerant partition dimension of Toeplitz networks, Tn1,t, for different values of n and t have been computed. In Section 3, the results are summarized and two open problems are proposed.


## 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 κ−metric. For κ=2, the κ−metric dimension is known as fault tolerant metric dimension denoted by dim2W. Further study on κ−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 κ−partition dimension of the network. For κ=2, the κ−partition dimension is known as fault tolerant partition dimension denoted by pd2W. The pd2W 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 networkW, the node and edge sets of W are denoted by VW and EW, respectively. The length of the shortest path, connecting two nodes and q, is known as the distance between p and q denoted by dp,q. The distance of a node p from a subset A of the node set is dp,A=mindp,t|t∈A, whereas the set Np=q∈VG|pq∈EG is called the neighbourhood of node p. Let Ω=x1,x2,…,xl be an ordered set of nodes, the representation of a node p with respect to Ω is a vector rp|Ω such that rp|Ω=dp,x1,dp,x2,…,dp,xl. If the vectors rp|Ω are different, for every node of the network, then Ω is termed as a resolving set of W. The minimal cardinality of Ω is termed as the metric dimension of W, represented by dimW. If the vectors rp|Ω are different in at least κ places for every node of the graph, then Ω is termed as a κ−metric generator for W. κ−metric basis is a generator with least number of nodes. The minimal cardinality of Ω is termed as the κ−metric dimension of W, represented by dimκW.LetΨ=B1,B2,…,Bl be an ordered l−partition of a connected network W. The partition representation of node p with respect to Ψ is a vector rp|Ψ such that rp|Ψ=dp,B1,dp,B2,…,dp,Bl. If the vectors rp|Ψ are different for every node of the graph, then l−partition is termed as a resolving partition of W. The minimal cardinality of Ψ for which there is a resolving l−partition is termed as the partition dimension of W, represented by pdW. If the vectors rp|Ψ are different in at least κ places for every node of the network, then Ψ is termed as a κ−partition generator of W. κ−partition basis is a generator with least number of sets. The minimal cardinality of Ψ is termed as a the κ−partition dimension of W, represented by pdκW.


## 1.3. Main Results

The study conducted in this paper leads to the following results.Theorem 1.
LetTn1,tbe the Toeplitz network, then(1)
Forn≥4,pd2Tn1,t≥4(2)
Forn≥4, pd2Tn1,2=4(3)
Forn∈5,6,…,13, pd2Tn1,3=4(4)
Forn≥14, 4≤pd2Tn1,3≤5(5)
Fort≥5andn=t+2,t+3,t+4, pd2Tn1,t=4
In Section2, Toeplitz networks are defined and the fault tolerant partition dimension of Toeplitz networks, Tn1,t, for different values of n and t 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,n,k1,k2,…,kn with k1,k2,…,kn≤n−1, the Toeplitz network Tnk1,k2,…,kn is a network on the vertex set x1,x2,…,xn with two vertices xi and xj which are adjacent if and only if i−j∈k1,k2,…,kn. Figure 1 shows T141,3.Figure 1
Toeplitz networkT141,3.The following lemma gives the lower bound on the fault tolerant partition dimension of Toeplitz networksTn1,t.Lemma 1.
LetTn1,tbe the Toeplitz network withn≥4, thenpd2Tn1,t≥4.Proof.
Assume thatpd2Tn1,t=3 for n≥4. Let Ψ=A1,A2,A3 be a fault tolerant partition basis of VTn1,t. Clearly, one of the sets A1,A2, and A3 contains at least two vertices. Suppose that A1≥2. Here, at least one node of A1, say vk, has a coordinate of rvk|Ψ>1. Otherwise, each node in A1 will have the same representation. So, without loss of generality, we can assume that the third coordinate of rvk|Ψ is greater than 1. i.e., dvk,A3≥2. Let vm be a node in A3 such that dvk,vm=dvk,A3. Let Nvk be the neighbourhood of vk. So, Nvk∩A3=∅ as du,vk<dvk,vm for all u∈Nvk. Since 2≤Nvk≤4 in the Tn1,t Toeplitz network, so we have the following three cases.Case 1: whenNvk=4, let Nvk=x1,x2,x3,x4.Case 1.1: suppose all the nodes ofNvk are in the same set A1 or A2. Assume that Nvk⊆A1, then rvk|Ψ=0,a0,b0 and for x∈Nvk, rxl|Ψ=0,al,bl, for 1≤l≤4. Hence, a0−1≤a1,a2,a3,a4≤a0+1. Now, pigeonhole principle implies that at least two of the nodes vk and xl will be equidistant from A1 and A2, which is a contradiction, whereas Nvk⊆A2 generates a trivial case of contradiction.Case 1.2: assume thatA1 or A2 contains three vertices from Nvk.Letxp,xq,xr∈Nvk∩A1, and xt∈Nvk∩A2, then rvk|Ψ=0,1,b0, rxp|Ψ=0,a1,b1,rxq|Ψ=0,a2,b2, and rxq|Ψ=0,a3,b3. In this case, we have 1≤a1,a2,a3≤2. Again, pigeonhole principle implies that at least two of the nodes vk,xp,xq, and xr will be equidistant from A1 and A2, leading to a contradiction, whereas three nodes of Nvk in A2 generate a trivial case of contradiction.Case 1.3: assume that each ofA1 and A2 contains two nodes of Nvk. Let xp,xq∈Nvk∩A1 and xr,xt∈Nvk∩A2, then rvk|Ψ=0,1,b0,rxp|Ψ=0,a1,b1 and rxq|Ψ=0,a2,b2. In this case, we have 1≤a1,a2≤2. Again by pigeonhole principle, at least two of the nodes vk,xp and xq will have the same distances from A1 and A2. This leads to a contradiction. Now, if we consider nodes in A2, then rxr|Ψ=1,0,b1 and rxt|Ψ=1,0,b2, which is again a contradiction.Case 2: whenNvk=3, let Nvk=x1,x2,x3. The subcases in this case are similar to Case 1.2 and Case 1.3.Case 3: whenNvk=2, let Nvk=x1,x2.Case 3.1: if both of the nodes ofNvk are in the same set A1 or A2, assume that Nvk⊆A1, then rvk|Ψ=0,a0,b0,rxp|Ψ=0,a1,b1 and rxq|Ψ=0,a2,b2. Hence, a0−1≤a1,a2≤a0+1. Now, if any of a1 and a2 is equal to a0, then we have a contradiction. Since in Toeplitz networks the only nodes with two neighbours are v1 and vn, so in all other cases, we will have 3≤Nvp≤4 and 3≤Nvq≤4 which are similar to Case 1 and Case 2, whereas Nvk⊆A2 generates a trivial case of contradiction.Case 3.2: assume that each ofA1 and A2 contains a node from the set Nvk. Let xp∈Nvk∩A1 and xq∈Nvk∩A2, then we can consider the Nvp and Nvq to get contradiction as in the previous cases.
Hence,pd2Tn1,t≥4 for n≥4.
In the following theorems, we calculate the fault tolerant partition dimension of Toeplitz networksTn1,t for different values of t.Theorem 2.
LetTn1,2be the Toeplitz network withn≥4, thenpd2Tn1,2=4.Proof.
LetΨ=A1,A2,A3,A4 be a partition of VTn1,2. The proof is divided into three cases, and in each case, we will show that Ψ is the fault tolerant partition generator of Tn1,2. Let κ=⌈n/2⌉.Case 1: forκ=3〉, let A1=v2,v4,…,v2⌊n/2⌋,A2=v1,v3,…,v2〉−1, A3=v2〉+1,v2〉+3,…,v4〉−1, and A4=v4〉+1,v4〉+3,…,v2κ−1.Whenκ=3〉, the rva|Ψ are tabulated in Table 1.Case 2: forκ=3〉 +1, let A1=v2,v4,…,v2⌊n/2⌋,A2=v1,v3,…,v2⌊κ/3⌋+1, A3=v2⌊κ/3⌋+3,v2⌊κ/3⌋+5,…,v4⌊κ/3⌋+1, and A4=v4⌊κ/3⌋+3,v4⌊κ/3⌋+5,…,v2κ−1Whenκ=3〉+1, the rva|Ψ are tabulated in Table 2.Case 3: forκ=3ϱ +2, let A1=v2,v4,…,v2⌊n/2⌋,A2=v1,v3,…,v2⌈κ/3⌉−1, A3=v2⌈κ/3⌉+1,v2⌈κ/3⌉+3,…,v4⌈κ/3⌉−1, and A4=v4⌈κ/3⌉+1,v4⌈κ/3⌉+3,…,v2κ−1. When κ=3〉+2, the rva|Ψ are tabulated in Table 3.
Tables1–3 clearly show that Ψ is a 2−resolving generator of Tn1,2 for n≥4, therefore pd2Tn1,2≤4 for n≥4. Lemma 1 implies that pd2Tn1,2≥4 for n≥4. This establishes our claim.Table 1
rva|Ψ for Tn1,2 when κ=3〉.
Nodesdva,A1dva,A2dva,A3dva,A4v2a1≤a≤〉01〉−a+12〉−a+1v2a〉+1≤a≤2〉0a−〉+112〉−a+1v2a2〉+1≤a≤⌊n/2⌋0a−〉+1a−2〉+11v2a+10≤a≤〉−110〉−a2〉−av2a+1〉≤a≤2〉−11a−〉+102〉−av2a+12〉≤a≤κ−11a−〉+1a−2〉+10Table 2
rva|Ψ for Tn1,2 when κ=3〉+1.
Nodesdva,A1dva,A2dva,A3dva,A4v2a1≤a≤〉+101〉−a+22〉−a+2v2a〉+2≤a≤2〉+10a−〉12〉−a+2v2a2〉+2≤a≤⌊n/2⌋0a−〉a−2〉1v2a+10≤a≤〉10〉−a+12〉−a+1v2a+1〉+1≤a≤2〉1a−〉02〉−a+1v2a+12〉+1≤a≤κ−11a−〉a−2〉0Table 3
rva|Ψ for Tn1,2 when κ=3〉+2.
Nodesdva,A1dva,A2dva,A3dva,A4v2a1≤a≤〉+101〉−a+22〉−a+3v2a〉+2≤a≤2〉+20a−〉12〉−a+3v2a2〉+3≤a≤⌊n/2⌋0a−〉a−2〉−11v2a+10≤a≤〉10〉−a+12〉−a+2v2a+1〉+1≤a≤2〉+11a−〉02〉−a+2v2a+12〉+2≤a≤κ−11a−〉a−2〉-10Theorem 3.
LetTn1,3be the Toeplitz network, thenpd2Tn1,3=4whenn∈5,6,…,13and4≤pd2Tn1,3≤5forn≥14.Proof.
The proof is divided into two parts.Case 1: the partitionΨ for 5≤n≤13 is tabulated in Table 4 which clearly shows that Ψ is the fault tolerant partition generator; therefore, pd2Tn1,3≤4 for 5≤n≤13. Lemma 1implies that pd2Tn1,3≥4 for 5≤n≤13. This establishes our claim.Case 2: supposeΨ=A1,A2,A3,A4,A5 be a partition of VTn1,3 for n≥14. Let A1=v1,v2,…,vn−6,A2=vn−5,vn−4,A3=vn−3,vn−2,A4=vn−1, and A5=vn. For n=3〉,3〉+1,3〉+2, the rva|Ψ are tabulated in Tables 5–7, respectively.
Tables5–7 clearly show that Ψ is a 2−resolving generator of Tn1,3 for n≥14, so pd2Tn1,3≤5 for n≥14. Lemma 1 implies that pd2Tn1,3≥4 for n≥14. This establishes that 4≤pd2Tn1,3≤5.Table 4
Partitions forn=5,6,…,13.
nA1A2A3A45v1,v2v3v4v56v1,v2v3,v4v5v67v1,v2v3v4,v7v5,v68v1,v2,v5v3,v4,v7v6v89v1,v2,v5v3,v4,v7v5,v8,v9v610v1,v2,v3v4,v8v5,v6,v7v9,v1011v1,v2,v3v4,v8v5,v6,v7v9,v10,v1112v1,v2,v3v4,v8,v12v5,v6,v7v9,v10,v1113v1,v2,v12,v13v3,v7,v11v4,v5,v6v8,v9,v10Table 5
rva|Ψ for Tn1,3 when n=3〉.
Nodesdva,A1dva,A2dva,A3dva,A4dva,A5v3a+10≤a≤〉−30〉−a−2〉−a−1〉−a〉−a+1v3a−11≤a≤〉−20〉−a−1〉−a〉−a〉−a+1v3a1≤a≤〉−20〉−a−1〉−a−1〉−a+1〉−avaa=n−510123vaa=n−410112vaa=n−311021vaa=n−221012vaa=n−121101vaa=n22110Table 6
rva|Ψ for Tn1,3 when n=3〉+1.
Nodesdva,A1dva,A2dva,A3dva,A4dva,A5v3a+10≤a≤〉−20〉−a−1〉−a〉−a〉−a+1v3a−11≤a≤〉−20〉−a−1〉−a〉−a+1〉−a+2v3a1≤a≤〉−20〉−a−1〉−a〉−a〉−a+1vaa=n−510123vaa=n−410112vaa=n−311021vaa=n−221012vaa=n−121101vaa=n22110Table 7
rva|Ψ for Tn1,3 when n=3〉+2.
Nodesdva,A1dva,A2dva,A3dva,A4dva,A5v3a+10≤a≤〉−20〉−a−1〉−a〉−a〉−a+1v3a−11≤a≤〉−10〉−a〉−a〉−a+2〉−a+1v3a1≤a≤〉−20〉−a−1〉−a〉−a+1〉−a+2vaa=n−510123vaa=n−410112vaa=n−311021vaa=n−221012vaa=n−121101vaa=n22110Theorem 4.
LetTn1,tbe the Toeplitz network with oddt≥5andn=t+2,t+3,t+4, thenpd2Tn1,t=4.Proof.
LetΨ=A1,A2,A3,A4 be a partition of VTn1,t. The proof consists of three parts.Case 1: forn=t+2, let A1=v1,v2,v3,A2=v4,v5,…,vn−2,A3=vn−1, and A4=vn. When n=t+2, the rva|Ψ are tabulated in Table 8.Case 2: forn=t+3, let A1=v1,v2,v3,v4,A2=v5,v6,…,vn−2,A3=vn−1, and A4=vn. When n=t+3, the rva|Ψ are tabulated in Table 9.Case 3: forn=t+4, let A1=v1,v2,v3,A2=v4,v5,…,vn−3,A3=vn−2,vn−1, and A4=vn. When n=t+4, the rva|Ψ are tabulated in Table 10.
Tables8–10 clearly show that Ψ is a 2−resolving generator of Tn1,t for n=t+2,t+3,t+4, therefore pd2Tn1,t≤4 for n=t+2,t+3,t+4. Lemma 1 implies that pd2Tn1,t≥4 for n=t+2,t+3,t+4. This establishes our claim.Table 8
rva|Ψ for Tn1,t when n=t+2 with odd t≥5.
Nodesdva,A1dva,A2dva,A3dva,A4vaa=10212vaa=20221vaa=30132va4≤a≤n+1/2−1a−30ga−1vaa=n+1/2n+1/2−30n+1/2−2n+1/2−1van+1/2+1≤a≤n−2n−a0n−a−1n−avaa=n−11101vaa=n1210Table 9
rva|Ψ for Tn1,t when n=t+3 with odd t≥5.
Nodesdva,A1dva,A2dva,A3dva,A4vaa=10123vaa=20212vaa=30221vaa=40132va5≤a≤n/2a−40a−1a−2vaa=n/2+1n/2−30n/2−2n/2−1van/2+2≤a≤n−2n−a−10n−a−1n−avaa=n−11101vaa=n1210Table 10
rva|Ψ for Tn1,t when n=t+4 with odd t≥5.
Nodesdva,A1dva,A2dva,A3dva,A4vaa=10124vaa=20213vaa=30112va4≤a≤n+1/2−1a−30a−2a−3vaa=n+1/2n+1/2−30n+1/2−3n+1/2−3van+1/2+1≤a≤n−3n−a−20n−a−2n−avaa=n−21102vaa=n−11201vaa=n2110Theorem 5.
ConsiderTn1,twith event≥4, thenpd2Tn1,t=4, forn=t+2,t+3,t+4.Proof.
LetΨ=A1,A2,A3,A4 be a partition of VTn1,t. The proof consists of three parts.Case 1: forn=t+2, let A1=v1,v2,…,vn−3,A2=vn−2,A3=vn−1, and A4=vn. When n=t+2, the rva|Ψ are tabulated in Table 11.Case 2: forn=t+3, let τ=⌈n/4⌉.Case 2a: whenn≡1 (mod 4), let A1=va|1≤a≤τ,A2=va|τ+1≤a≤2τ−1,A3=va|2τ≤a≤2τ+1, and A4=va|2τ+2≤a≤n.Whenn=t+3, the rva|Ψ are tabulated in Table 12.Case 2b: whenn≡3 (mod 4), let A1=va|1≤a≤τ,A2=va|τ+1≤a≤2τ,A3=va|2τ+1≤a≤3τ, and A4=va|3τ+1≤a≤n.Whenn=t+3, the rva|Ψ are tabulated in Table 13.Case 3: forn=t+4, let A1=v1,v2,v3,A2=v4,v5,…,vn−3,A3=vn−2,vn−1, and A4=vn. When n=t+4, the rva|Ψ are tabulated in Table 14.
It is clear from Tables11–14 that Ψ is a 2−resolving generator of Tn1,t for n=t+2,t+3,t+4, therefore pd2Tn1,t≤4 for n=t+2,t+3,t+4. Lemma 1 implies that pd2Tn1,t≥4 for n=t+2,t+3,t+4. This establishes our claim.Table 11
rva|Ψ for Tn1,t when n=t+2 with even t≥4.
Nodesdva,A1dva,A2dva,A3dva,A4vaa=10212vaa=t/20t/2t/2t/2−1vaa=t/2+10t/2−1t/2t/2vat/2+2≤a≤t−10t−at−a+1t−a+2vaa=t1012vaa=t+11101vaa=t+21210Table 12
rva|Ψ for Tn1,t when n=t+3 with even t≥4.
Nodesdva,A1dva,A2dva,A3dva,A4vaa=10τt/2−21vaa=20τ−1t/2−11va3≤a≤τ0τ−a+1t/2−a+3a−2vaa=τ+110a−2a−2vaτ+2≤a≤t/2+2a−τ0t/2−a+3t/2−a+5vaa=t/2+3τ102vaa=n/2+4τ+1201vat/2+5≤a≤t/2+8a−τa−t/2−2a−t/2−40vat/2+9≤a≤n−7t−a+2a−t/2−2a−t/2−40van−6≤a≤n−2t−a+2t−a+τ+2a−t/2−40vaa=n−11τt/2−20vaa=n1τ−1t/2−10Table 13
rva|Ψ for Tn1,t when n=t+3 with even t≥4.
Nodesdva,A1dva,A2dva,A3dva,A4vaa=10ττ−21vaa=20τ−1τ−11va3≤a≤τ−50τ−a+1a+τ−3a−2vaτ−4≤a≤τ0τ−a+1t/2−a+3a−2vaτ+1≤a≤t/2−4a−τ0t/2−a+3a−2vat/2−3≤a≤t/2+2a−τ0t/2−a+3a−τ+1vat/2+3≤a≤t/2+8a−τa−t/2−20a−τ−1vat/2+9≤a≤3τt−a+2a−t/2−203τ−a+1va3τ+1≤a≤t−4t−a+2a−t/2−2a−3τ0vat−3≤a≤n−2t−a+23τ+t/2−aa−3τ0vaa=n−11ττ−20vaa=n1τ−1τ−10Table 14
rva|Ψ for Tn1,t when n=t+4 with even t≥4.
Nodesdva,A1dva,A2dva,A3dva,A4vaa=10124vaa=20213vaa=30112va4≤a≤t/2+2a−30a−2a−3vaa=t/2+3t−10t−1tvat/2+4≤a≤n−3t−a+20t−a+2t−a+4vaa=n−21102vaa=n−11201vaa=n2110


## 3. Conclusion

In this research article, we conclude that the fault tolerant partition dimension ofTn1,t is 4 when t=2 and n≥4. When t=3 and n∈5,6,…,13, the pd2 is 4 and its value is less than or equal to 5 for n≥14. It is also inferred that the fault partition dimension of Toeplitz networks Tn1,t is 4 when n=t+2,t+3,t+4 for even t≥4 and odd t≥5 [18].Open Problem 1.
Calculate thepd2Tn1,t for odd t≥5 and n≥t+5.Open Problem 2.
Calculate thepd2Tn1,2,t for t≥3.



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*Source: 3429091-2022-01-17.xml*