# Fault Tolerant Partition Resolvability in Convex Polytopes

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

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## Abstract

Convex polytopes are special types of polytopes having an additional property that they are also convex sets in then-dimensional Euclidean space. The convex polytope topologies are being used in the antitracking networks due to their stability, resilience, and destroy-resistance. The metric related parameters have been extensively studied in the recent times due to their applications in several areas including robot navigation, network designing, image processing, and chemistry. In this article, the sharp bounds for the fault tolerant partition dimension of certain well-known families of convex polytopes Rn, Qn, Sn, Tn, and Dn have been computed. Furthermore, we have studied the graphs having fault tolerant partition dimension bounded below by 4.


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## Body

## 1. Introduction

The ability of a network to maintain its working capacity and functionality if one of its components fails to work is called fault tolerance. This guarantees the lower maintenance cost and longer durability of the network. Antitracking networks are used to protect users’ identities and communication confidentiality. A robust and resilient network topology is always required to ensure the strength and protection of antitracking network. Tian et al. [1], in their recent research, proposed the convex polytope topology in the antitracking network due to the stability, resilience, and destroy-resistance. The convex polytope topology can preserve the network stability, avoid the threat to key nodes and cut vertices, and achieve the self-optimization of network topology. Convex polytopes have been studied to a great extent for metric dimension and partition dimension. Imran et al. [2] computed the metric dimension of convex polytopes, whereas the bounds of fault tolerant metric dimension for some convex polytopes have been discussed in [3, 4]. Furthermore, fractional versions of metric related parameters for certain convex polytopes have been calculated in [5, 6].Garey and Johnson in [7] concluded that computation of metric basis of a network is an NP-hard problem; it was further implied that computation of partition related metric parameters for general graphs is also NP-hard. This leads the current research trends to focus on the computation of these metric parameters for specific classes of graphs. In this regard, Liu et al. [8] and Chu et al. [9] recently computed sharp bounds of partition dimension for some families of convex polytopes. Following the same spirit, the fault tolerant partition dimension of certain convex polytopes is computed in the current work.In a connected graphW, the node and edge sets of W are denoted by VW and EW, respectively. The length of the shortest path connecting two nodes p and q is known as the distance between p and q denoted by dp,q. For any node p of W and a nonempty subset A of V(W), the sets HTML translation failed and Np=q∈VG|pq∈EW are called the distance of a node p from subset A and the neighborhood of node p, respectively. Let Ω=x1,x2,…,xl be the set of ordered nodes; the representation of a node p with reference 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 graph, 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. Let Ψ=B1,B2,…,Bl be an ordered l−partition of a connected graph W. The representations of nodes with reference to Ψ are vectors rp|ψ such that rp|ψ=dp,B1,dp,B2,…,dp,Bl. If 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 a resolving partition Ψ is termed as the partition dimension of W, represented by pdW. The basic relationship between metric and partition dimension of graphs is given in [10].Further study on partition dimension can be seen in [11, 12]. In 2015, Moreno et al. [13] presented the concept of κ− metric. If 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. The 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. The 2-metric dimension was first presented in [14]. Further study on κ− metric dimension can be seen in [15, 16].In 2020, Moreno et al. [17] presented the concept of κ− partition dimension of the graph. If vectors rp|ψ are different in at least κ places for every node of the graph, then Ψ is termed as a κ− partition generator of W. The κ− partition basis is a generator with least number of sets. The minimal cardinality of Ψ is termed as the κ− partition dimension of W, represented by pdκW. For κ=2, the κ− partition dimension is known as fault tolerant partition dimension denoted by pd2W. pd2W was computed for some important graphs in [18, 19] and further for circulant graphs in [20].


### 1.1. Main Results

The research conducted in this article leads to the subsequent results.Theorem 1.
(1)
For a graphW of order m≥5 and having a vertex of degree at least 4, pd2W≥4(2)
Forn≥5, 4≤pd2Rn≤5(3)
Forn≥6,(a)
4≤pd2Qn≤5(b)
4≤pd2Sn≤5(c)
4≤pd2Tn≤5(d)
3≤pd2Dn≤5
The rest of the article is organised in the following manner. In Section2, the convex polytopes Rn,Qn,Sn,Tn,Dn are defined and lower and upper bounds of fault tolerant partition dimension for these infinite families of graphs are also computed. In Section 3, the article is concluded with a conjecture and an open problem.


## 1.1. Main Results

The research conducted in this article leads to the subsequent results.Theorem 1.
(1)
For a graphW of order m≥5 and having a vertex of degree at least 4, pd2W≥4(2)
Forn≥5, 4≤pd2Rn≤5(3)
Forn≥6,(a)
4≤pd2Qn≤5(b)
4≤pd2Sn≤5(c)
4≤pd2Tn≤5(d)
3≤pd2Dn≤5
The rest of the article is organised in the following manner. In Section2, the convex polytopes Rn,Qn,Sn,Tn,Dn are defined and lower and upper bounds of fault tolerant partition dimension for these infinite families of graphs are also computed. In Section 3, the article is concluded with a conjecture and an open problem.


## 2. Fault Tolerant Partition Dimension of Convex Polytopes

In this section, the bounds of fault partition dimensions of certain classes of convex polytopesRn,Qn,Sn,Tn,Dn have been computed. In the following theorem, the graphs with fault tolerant partition dimension bounded below by 4 has been characterized.Lemma 1.
LetW be a graph of order m≥5. If W has a node of degree at least 4, then pd2W≥4.Proof.
Assume thatpd2W=3 for m≥5. Let Λ=B1,B2,B3 be a 2− partition basis of VW. Let vk be the node of degree 4. Assume that vk∈B1. Let Nvk=x1,x2,x3,x4.Case 1: WhenB1=1, the subcases are as follows:Case 1.1: If each node ofNvk is in either set B2 or B3, assuming that Nvk⊆B2, then, for x∈Nvk, rxi|Λ=1,0,bi, for 1≤i≤4, which results in a contradiction.Now ifNvk⊆B3, then, for x∈Nvk, rxi|Λ=1,ai,0, for 1≤i≤4, which again contradicts our assumption.Case 1.2: Assume thatB2 or B3 contains three nodes of Nvk.Letxp,xq,xr∈Nvk∩B2 and xt∈Nvk∩B3; then rvk|Λ=0,1,1, rxq|Λ=1,0,b1,rxq|Λ=1,0,b2, and rxr|Λ=1,0,b3, which clearly leads to a contradictory situation. Meanwhile three nodes of Nvk in B3 generate a similar case of contradiction.Case 1.3: Assume that each ofB2 and B3 contains two nodes from set Nvk. Let xp,xq∈Nvk∩B2; then rxp|Λ=1,0,b1 and rxq|Λ=1,0,b2, which again results in a contradiction.Case 2: WhenB1≥2, the subcases are as follows:Case 2.1: WhenNvk∩B1=4, this implies that Nvk⊆B1, and then rvk|Λ=0,a0,b0 and, for x∈Nvk, rxi|Λ=0,ai,bi, for 1≤i≤4. Hence, a0−1≤a1,a2,a3,a4≤a0+1. Pigeonhole principle implies that at least two of nodes vk and xi are equidistant from B1 and B2, which results in a contradiction.Case 2.2: WhenNvk∩B1=3.Letxp,xq,xr∈Nvk∩B1 and xt∈Nvk∩B2, and then rvk|Λ=0,1,b0, rxp|Λ=0,a1,b1,rxq|Λ=0,a2,b2, and rxr|Λ=0,a3,b3. In this situation, we get 1≤a1,a2,a3≤2. Again Pigeonhole principle implies that at least two nodes among vk,xp,xq, and xr are equidistant from B1 and B2, which results in a contradiction. Meanwhile xt∈Nvk∩B3 generates a similar case of contradiction.Case 2.3: WhenNvk∩B1=2.Case 2.3.1: Letxp,xq∈Nvk∩B1 and xr,xt∈Nvk∩B2, and then rvk|Λ=0,1,b0,rxp|Λ=0,a1,b1, and rxq|Λ=0,a2,b2. In this situation, we get 1≤a1,a2≤2. Now the Pigeonhole principle implies that minimum two nodes among vk,xp, and xq are at the same distances from B1 and B2. This results in a contradiction. Meanwhile xr,xt∈Nvk∩B3 generates a similar case of contradiction.Case 2.3.2: Letxp,xq∈Nvk∩B1, xr∈Nvk∩B2, and xt∈Nvk∩B3, and then rvk|Λ=0,1,1,rxp|Λ=0,a1,b1, and rxq|Λ=0,a2,b2. In this situation, we get 1≤a1,a2≤2. Now the Pigeonhole principle implies that the minimum two nodes among vk,xp, and xq are at the same distances from B1 and B2. This results in a contradiction.Case 2.4: WhenNvk∩B1=1.Case 2.4.1: Letxp∈Nvk∩B1 and xq,xr,xt∈Nvk∩B2, and then rvk|Λ=0,1,b0, rxp|Λ=0,a1,b1, rxq|Λ=1,0,b2, rxr|Λ=1,0,b3, and rxt|Λ=1,0,b4, which is a clear case of contradiction. Meanwhile xq,xr,xt∈Nvk∩B3 generates a similar case of contradiction.Case 2.4.2: Letxp∈Nvk∩B1, xq,xr∈Nvk∩B2, and xt∈Nvk∩B3, and then rvk|Λ=0,1,1, rxp|Λ=0,a1,b1, rxq|Λ=1,0,b2, rxr|Λ=1,0,b3, and rxt|Λ=1,a4,0, which is again a clear case of contradiction. Meanwhile xq,xr∈Nvk∩B3 and xt∈Nvk∩B2 generates a similar case of contradiction.Case 2.5: WhenNvk∩B1=0. This case is similar to Case 1.
Hence,pd2W≥4 for m≥5 with (HTML translation failed containing a node of degree at least 4.


### 2.1. Fault Tolerant Partition Dimension ofRn

In [21], Baca defined convex polytope Rn for n≥5 with VRn=ui,vi,xi:1≤i≤n and ERn=uiui+1;ui,vi;ui+1vi;vivi+1;vixi;xixi+1:1≤i≤n.Conventionally, we assume thatun+1=u1,vn+1=v1, and xn+1=x1. The graph of convex polytope Rn is shown in Figure 1.Figure 1
Convex polytopeRn.In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytopeRn.Theorem 2.
Consider convex polytopeRn with n≥5; then 4≤pd2Rn≤5.Proof.
LetΨ=B1,B2,B3,B4,B5 be a partition of the node set VRn. We divide the proof in four parts and, in each part, it will be shown that Ψ is the fault tolerant partition generator for Rn. Let a=⌊p/2⌋.Case 1: Forn=4p.LetB1=ui,vi:1≤i≤n,B2=xi:1≤i≤p, B3=xi:p+1≤i≤2p, B4=xi:2p+1≤i≤3p, and B5=xi:3p+1≤i≤3n.rq|Ψ are tabulated in Table 1.Case 2: Forn=4p+1.LetB1=ui,vi:1≤i≤n,B2=xi:1≤i≤p+1, B3=xi:p+2≤i≤2p+1, B4=xi:2p+2≤i≤3p+1, and B5=xi:3p+2≤i≤3n.rq|Ψ are tabulated in Table 2.Case 3: Forn=4p+2.LetB1=ui,vi:1≤i≤n,B2=xi:1≤i≤p+1, B3=xi:p+2≤i≤2p+2, B4=xi:2p+3≤i≤3p+2, and B5=xi:3p+3≤i≤3n.rq|Ψ are tabulated in Table 3.Case 4: Forn=4p+3.LetB1=ui,vi:1≤i≤n,B2=xi:1≤i≤p+1, B3=xi:p+2≤i≤2p+2, B4=xi:2p+3≤i≤3p+3, and B5=xi:3p+4≤i≤3n.rq|Ψ are tabulated in Table 4.
Whenp is odd, we have t=2, whereas when p is even, t=1 in the tables. Tables 1–4 clearly show that Ψ is the 2− resolving generator for Rn when n≥5; therefore, pd2Rn≤5 for n≥5. Lemma 1 implies that pd2Rn≥4 for n≥5. This establishes our claim.Table 1
rq|Ψ for Rn when n=4p and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+102p−i+3p+i+1i+1uia+2≤i≤p+102p−i+32p−i+3i+1uip+2≤i≤p+a+10i−p+122p−i+3i+1uip+a+2≤i≤2p+10i−p+122p−i+33p−i+3ui2p+2≤i≤2p+a+10i−p+1i−2p+123p−i+3ui2p+a+2≤i≤3p+10n−i+3i−2p+123p−i+3ui3p+2≤i≤n−2t+10n−i+3i−2p+1i−n+p+12uin−2t+2≤i≤n0n−i+35p−i+3i−n+p+12vi1≤i≤a+t−101p−i+2p+i+1i+1via+t≤i≤p01p−i+22p−i+2i+1vip+1≤i≤p+a+t−10i−p+112p−i+2i+1vip+a+t≤i≤2p0i−p+112p−i+23p−i+2vi2p+1≤i≤2p+a+t−10i−p+1i−2p+113p−i+2vi2p+a+t≤i≤3p0n−i+2i−2p+113p−i+2vi3p+1≤i≤n−a0n−i+2i−2p+1i−3p+11vin−a+1≤i≤n0n−i+25p−i+2i−3p+11xi1≤i≤a+t−110p−i+1i+pixia+t≤i≤p10p−i+12p−i+1ixip+1≤i≤p+a+t−11i−p02p−i+1ixip+a+t≤i≤2p1i−p02p−i+13p−i+1xi2p+1≤i≤2p+a+t−11i−pi−2p03p−i+1xi2p+a+t≤i≤3p1n−i+1i−2p03p−i+1xi3p+1≤i≤n−a1n−i+1i−2pi−3p0xin−a+1≤i≤n1n−i+15p−i+1i−3p0Table 2
Representations of nodes forRn when n=4p+1 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+102p−i+4p+i+1i+1uia+t+1≤i≤p+202p−i+42p−i+4i+1uip+3≤i≤p+a+t0i−p22p−i+4i+1uip+a+t+1≤i≤2p+20i−p22p−i+43p−i+4ui2p+3≤i≤2p+a+20i−pi−2p23p−i+4ui2p+a+3≤i≤3p+20n−i+3i−2p23p−i+4ui3p+3≤i≤n−a+10n−i+3i−2pi−3p2uin−a+2≤i≤n0n−i+35p−i+5i−n+p+12vi1≤i≤a+101p−i+3p+i+1i+1via+2≤i≤2a+t01p−i+32p−i+3i+1vi2a+t+1≤i≤p+a+10i−p12p−i+3i+1vip+a+2≤i≤2p+10i−p12p−i+33p−i+3vi2p+2≤i≤2p+a+t0i−pi−2p13p−i+3vi2p+a+t+1≤i≤3p+10n−i+2i−2p13p−i+3vi3p+2≤i≤n−a−t+20n−i+2i−2pi−3p1vin−a−t+3≤i≤n0n−i+25p−i+4i−3p1xi1≤i≤a+110p−i+2i+pixia+2≤i≤p+110p−i+22p−i+2ixip+2≤i≤p+a+11i−p−102p−i+2ixip+a+2≤i≤2p+11i−p−102p−i+23p−i+2xi2p+2≤i≤2p+a+11i−p−1i−2p−103p−i+2xi2p+a+2≤i≤3p+11n−i+1i−2p−103p−i+2xi3p+2≤i≤n−a−t+21n−i+1i−2p−1i−3p−10xin−a−t+3≤i≤n1n−i+15p−i+3i−3p−10Table 3
Representations of nodes forRn when n=4p+2 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+202p−i+4p+i+1i+1uia+3≤i≤p+202p−i+42p−i+5i+1uip+3≤i≤p+a+20i−p22p−i+5i+1uip+a+3≤i≤2p+30i−p22p−i+53p−i+5ui2p+4≤i≤2p+a+t+10i−pi−2p−123p−i+5ui2p+a+t+2≤i≤3p+30n−i+3i−2p−123p−i+5ui3p+4≤i≤n−a+10n−i+3i−2p−1i−n+p+12uin−a+2≤i≤n0n−i+35p−i+6i−n+p+12vi1≤i≤a+t01p−i+3p+i+1i+1via+t+1≤i≤p+101p−i+32p−i+4i+1vip+2≤i≤p+a+t0i−p12p−i+4i+1vip+a+t+1≤i≤2p+20i−p12p−i+43p−i+4vi2p+3≤i≤2p+a+20i−pi−2p−113p−i+4vi2p+a+3≤i≤3p+20n−i+2i−2p−113p−i+4vi3p+3≤i≤n−a−t+20n−i+2i−2p−1i−3p−11vin−a−t+3≤i≤n0n−i+25p−i+5i−3p−11xi1≤i≤a+t10p−i+2i+pixia+t+1≤i≤p+110p−i+22p−i+3ixip+2≤i≤p+a+t1i−p−102p−i+3ixip+a+t+1≤i≤2p+21i−p−102p−i+33p−i+3xi2p+3≤i≤2p+a+21i−p−1i−2p−203p−i+3xi2p+a+3≤i≤3p+214p−i+3i−2p−203p−i+3xi3p+3≤i≤n−a−t+214p−i+3i−2p−2i−3p−20xin−a−t+3≤i≤n14p−i+35p−i+4i−3p−20Table 4
Representations of nodes forRn when n=4p+3 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+202p−i+4p+i+1i+1uia+3≤i≤p+202p−i+42p−i+5i+1uip+3≤i≤p+a+t+10i−p22p−i+5i+1uip+a+t+2≤i≤2p+30i−p22p−i+53p−i+6ui2p+4≤i≤2p+a+30i−pi−2p−123p−i+6ui2p+a+4≤i≤3p+40n−i+3i−2p−123p−i+6ui3p+5≤i≤n−a−t+20n−i+3i−2p−1i−3p−22uin−a−t+3≤i≤n0n−i+35p−i+7i−n+p+12vi1≤i≤a+t01p−i+3p+i+1i+1via+t+1≤i≤p+101p−i+32p−i+4i+1vip+2≤i≤p+a+20i−p12p−i+4i+1vip+a+3≤i≤2p+20i−p12p−i+43p−i+5vi2p+3≤i≤2p+a+t+10i−pi−2p−113p−i+5vi2p+a+t+2≤i≤3p+30n−i+2i−2p−113p−i+5vi3p+4≤i≤n−a0n−i+2i−2p−1i−3p−21vin−a+1≤i≤n0n−i+25p−i+6i−3p−21xi1≤i≤a+t10p−i+2i+pixia+t+1≤i≤p+110p−i+22p−i+3ixip+2≤i≤p+a+21i−p−102p−i+3ixip+a+3≤i≤2p+21i−p−102p−i+33p−i+4xi2p+3≤i≤2p+a+t+11i−p−1i−2p−203p−i+4xi2p+a+t+2≤i≤3p+31n−i+1i−2p−203p−i+4xi3p+4≤i≤n−a1n−i+1i−2p−2i−3p−30xin−a+1≤i≤n1n−i+15p−i+5i−3p−30


### 2.2. Fault Tolerant Partition Dimension ofQn

In [22], Baca et al. defined convex polytope Qn for n≥6 consisting of 3-sided, 4-sided, 5-sided, and n-sided faces, respectively. Here VQn=ui,vi,xi,yi:1≤i≤n and EQn=uiui+1;vivi+1;yiyi+1;uivi;vixi;vi+1xi;xiyi:1≤i≤n. Conventionally, we assume that un+1=u1,vn+1=v1,xn+1=x1, and yn+1=y1. The graph of convex polytope Qn is shown in Figure 2.Figure 2
Convex polytopeQn.In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytopeQn.Theorem 3.
Consider convex polytopeQn with n≥6; then 4≤pd2Qn≤5.Proof.
LetΨ=B1,B2,B3,B4,B5 be a partition of the node set VQn. We divide the proof in four parts and, in each part, it will be shown that Ψ is the fault tolerant partition generator for Qn. Let a=⌊p/2⌋.Case 1: Forn=4p.LetB1=ui,vi,xi:1≤i≤n,B2=yi:1≤i≤p, B3=yi:p+1≤i≤2p, B4=yi:2p+1≤i≤3p, and B5=yi:3p+1≤i≤4n.rq|Ψ are tabulated in Table 5.Case 2: Forn=4p+1.LetB1=ui,vi,xi:1≤i≤n,B2=yi:1≤i≤p+1, B3=yi:p+2≤i≤2p+1, B4=yi:2p+2≤i≤3p+1, and B5=yi:3p+2≤i≤4n.rq|Ψ are tabulated in Table 6.Case 3: Forn=4p+2.LetB1=ui,vi,xi:1≤i≤n,B2=yi:1≤i≤p+1, B3=yi:p+2≤i≤2p+2, B4=yi:2p+3≤i≤3p+2, and B5=yi:3p+3≤i≤4n.rq|Ψ are tabulated in Table 7.Case 4: Forn=4p+3.LetB1=ui,vi,xi:1≤i≤n,B2=yi:1≤i≤p+1, B3=yi:p+2≤i≤2p+2, B4=yi:2p+3≤i≤3p+3, and B5=yi:3p+4≤i≤4n.rq|Ψ are tabulated in Table 8.
Whenp is odd, we have t=2, whereas when p is even, t=1 in the tables. Tables 5–8 clearly show that Ψ is the 2− resolving generator for Qn when n≥6; therefore, pd2Qn≤5 for n≥6. Lemma 1 implies that pd2Qn≥4 for n≥6. This establishes our claim.Table 5
rq|Ψ for Qn when n=4p and a=⌊p/2⌋.
Nodesdq,B1dq,B2ui1≤i≤a+103uia+2≤i≤p+103uip+2≤i≤p+a+10i−p+2uip+a+2≤i≤2p+10i−p+2ui2p+2≤i≤2p+a+10i−p+2ui2p+a+2≤i≤3p+105p−i−3t+3ui3p+2≤i≤n−2t+105p−i−3t+3uin−2t+2≤i≤n05p−i−3t+3vi1≤i≤a+102via+2≤i≤p+102vip+2≤i≤p+a+10i−p+1vip+a+2≤i≤2p+10i−p+1vi2p+2≤i≤2p+a+10i−p+1vi2p+a+2≤i≤3p+105p−i−3t+2vi3p+2≤i≤n−2t+105p−i−3t+2vin−2t+2≤i≤n05p−i−3t+2xi1≤i≤a+t−101xia+t≤i≤p01xip+1≤i≤p+a+t−10i−p+1xip+a+t≤i≤2p0i−p+1xi2p+1≤i≤2p+a+t−10i−p+1xi2p+a+t≤i≤3p05p−i−3t+1xi3p+1≤i≤n−t−105p−i−3t+1xin−t≤i≤n05p−i−3t+1yi1≤i≤a+t−110yia+t≤i≤p10yip+1≤i≤p+a+t−11i−pyip+a+t≤i≤2p1i−pyi2p+1≤i≤2p+a+t−11i−pyi2p+a+t≤i≤3p15p−i−3tyi3p+1≤i≤n−t−115p−i−3tyin−t≤i≤n15p−i−3tTable 6
rq|Ψ for Qn when n=4p+1 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+t03p−i+5p+i+2i+2uia+t+1≤i≤p+203p−i+52p−i+5i+2uip+3≤i≤p+a+t0i−p+132p−i+5i+2uip+a+t+1≤i≤2p+20i−p+132p−i+53p−i+5ui2p+3≤i≤2p+a+20i−p+1i−2p+133p−i+5ui2p+a+3≤i≤3p+205p−i−3t+4i−2p+133p−i+5ui3p+3≤i≤n−t05p−i−3t+4i−2p+1i−3p+13uin−t+1≤i≤n05p−i−3t+46p−i−3t+5i−3p+13vi1≤i≤a+t02p−i+4p+i+1i+1via+t+1≤i≤p+202p−i+42p−i+4i+1vip+3≤i≤p+a+t0i−p22p−i+4i+1vip+a+t+1≤i≤2p+20i−p22p−i+43p−i+4vi2p+3≤i≤2p+a+20i−pi−2p23p−i+4vi2p+a+3≤i≤3p+205p−i−3t+3i−2p23p−i+4vi3p+3≤i≤n−t05p−i−3t+3i−2pi−3p2vin−t+1≤i≤n05p−i−3t+36p−i−3t+4i−3p2xi1≤i≤a+101p−i+3p+i+1i+1xia+2≤i≤p+101p−i+32p−i+3i+1xip+2≤i≤p+a+10i−p12p−i+3i+1xip+a+2≤i≤2p+10i−p12p−i+33p−i+3xi2p+2≤i≤2p+a+t0i−pi−2p13p−i+3xi2p+a+t+1≤i≤3p+105p−i−3t+2i−2p13p−i+3xi3p+2≤i≤n−2t+105p−i−3t+2i−2pi−3p1xin−2t+2≤i≤n05p−i−3t+26p−i−3t+3i−3p1yi1≤i≤a+110p−i+2p+iiyia+2≤i≤p+110p−i+22p−i+2iyip+2≤i≤p+a+11i−p−102p−i+2iyip+a+2≤i≤2p+11i−p−102p−i+23p−i+2yi2p+2≤i≤2p+a+t1i−p−1i−2p−103p−i+2yi2p+a+t+1≤i≤3p+115p−i−3t+1i−2p−103p−i+2yi3p+2≤i≤n−2t+115p−i−3t+1i−2p−1i−3p−10yin−2t+2≤i≤n15p−i−3t+16p−i−3t+2i−3p−10Table 7
rq|Ψ for Qn when n=4p+2 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+203p−i+5p+i+2i+2uia+3≤i≤p+203p−i+52p−i+6i+2uip+3≤i≤p+a+20i−p+132p−i+6i+2uip+a+3≤i≤2p+a+10i−p+132p−i+63p−i+6ui2p+a+2≤i≤3p0i−p+1i−2p33p−i+6ui3p+1≤i≤n−a−t05p−i+2i−2p33p−i+6uin−a−t+1≤i≤n−105p−i−t+3i−2pi−3p3uii=n05p−i−t+36p−i−t+4i−3p3vi1≤i≤a+202p−i+4p+i+1i+1via+3≤i≤p+202p−i+42p−i+5i+1vip+3≤i≤p+a+20i−p22p−i+5i+1vip+a+3≤i≤3p−t0i−p22p−i+53p−i+5vi3p−t+1≤i≤3p0i−pi−2p−123p−i+5vi3p+1≤i≤n−a−t05p−i−t+2i−2p−123p−i+5vin−a−t+1≤i≤n−105p−i−t+2i−2p−1i−3p−12vii=n05p−i−t+26p−i−t+3i−3p−12xi1≤i≤a+t01p−i+3p+i+1i+1xia+t+1≤i≤p+101p−i+32p−i+4i+1xip+2≤i≤p+a+t0i−p12p−i+4i+1xip+a+t+1≤i≤3p−t−10i−p12p−i+43p−i+4xi3p−t≤i≤3p−t+10i−pi−2p−113p−i+4xi3p−t+2≤i≤n−a−t−105p−i−t+1i−2p−113p−i+4xin−a−t≤i≤n−t05p−i−t+1i−2p−1i−3p−11xin−t+1≤i≤n05p−i−t+16p−i−t+2i−3p−11yi1≤i≤a+t10p−i+2p+iiyia+t+1≤i≤p+110p−i+22p−i+3iyip+2≤i≤p+a+t1i−p−102p−i+3iyip+a+t+1≤i≤3p−t−11i−p−102p−i+33p−i+3yi3p−t≤i≤3p−t+11i−p−1i−2p−203p−i+3yi3p−t+2≤i≤n−a−t−115p−i−ti−2p−203p−i+3yin−a−t≤i≤n−t15p−i−ti−2p−2i−3p−20yin−t+1≤i≤n15p−i−t6p−i−t+1i−3p−20Table 8
rq|Ψ for Qn when n=4p+3 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+203p−i+5p+i+2i+2uia+3≤i≤p+203p−i+52p−i+6i+2uip+3≤i≤p+a+t+10i−p+132p−i+6i+2uip+a+t+2≤i≤2p+a−t+20i−p+132p−i+63p−i+7ui2p+a−t+3≤i≤3p−2t+30i−p+1i−2p33p−i+7ui3p−2t+4≤i≤n−3t05p−i−3t+6i−2p33p−i+7uin−3t+1≤i≤n−2t+105p−i−3t+6i−2pi−3p−13uin−2t+2≤i≤nZuin−2t+2≤i≤n05p−i−3t+66p−i−3t+7i−3p−13vi1≤i≤a+202p−i+4p+i+1i+1via+3≤i≤p+202p−i+42p−i+5i+1vip+3≤i≤p+a+t+10i−p22p−i+5i+1vip+a+t+2≤i≤2p+30i−p22p−i+53p−i+6vi2p+a+4≤i≤3p+40i−pi−2p−123p−i+6vi2p+a+4≤i≤3p+405p−i−3t+5i−2p−123p−i+6vi3p+5≤i≤n−2t+105p−i−3t+5i−2p−1i−3p−22vin−2t+2≤i≤n05p−i−3t+56p−i−3t+6i−3p−22xi1≤i≤a+t01p−i+3p+i+1i+1xia+t+1≤i≤p+101p−i+32p−i+4i+1xip+2≤i≤p+a+20i−p12p−i+4i+1xip+a+3≤i≤2p+20i−p12p−i+43p−i+5xi2p+3≤i≤2p+a+t+10i−pi−2p−113p−i+5xi2p+a+t+2≤i≤3p+305p−i−3t+4i−2p−113p−i+5xi3p+4≤i≤n−t−105p−i−3t+4i−2p−1i−3p−21xin−t≤i≤n05p−i−3t+46p−i−3t+5i−3p−21yi1≤i≤a+t10p−i+2p+iiyia+t+1≤i≤p+110p−i+22p−i+3iyip+2≤i≤p+a+21i−p−102p−i+3iyip+a+3≤i≤2p+21i−p−102p−i+33p−i+4yi2p+3≤i≤2p+a+t+11i−p−1i−2p−203p−i+4yi2p+a+t+2≤i≤3p+315p−i−3t+3i−2p−203p−i+4yi3p+4≤i≤n−t−115p−i−3t+3i−2p−2i−3p−30yin−t≤i≤n15p−i−3t+36p−i−3t+4i−3p−30


### 2.3. Fault Tolerant Partition Dimension ofSn

In [12], Imran et al. defined convex polytope Sn for n≥6 consisting of 2n3-sided faces, 2n4-sided faces, and a pair of n-sided faces which can be constructed by combining convex polytope Rn with prism graph. Here VSn=ui,vi,xi,yi:1≤i≤n and ESn=uiui+1;vivi+1;xixi+1;yiyi+1;uj+1vi;ui,vi;vi,xi;xi,yi:1≤i≤n. Conventionally, we assume that un+1=u1,vn+1=v1,xn+1=x1, and yn+1=y1. The graph of convex polytope Sn is shown in Figure 3.Figure 3
Convex polytopeSn.In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytopeSn.Theorem 4.
Consider convex polytopeSn with n≥6; then 4≤pd2Sn≤5.Proof.
LetΨ=B1,B2,B3,B4,B5 be a partition of the node set VSn. The partition sets B1,B2,B3,B4, and B5 defined for convex polytope Qn are the same for Sn. Let a=⌊p/2⌋. The representations of nodes ui,xi, and yi are the same in both Qn and Sn. The representations of nodes vi are shown in the tables below. When p is odd, we have t=2 , whereas when p is even, t=1 in the tables. Tables 9–12 clearly show that Ψ is the 2− resolving generator for Sn when n≥6; therefore, pd2Sn≤5 for n≥5. Lemma 1 implies that pd2Sn≥4 for n≥6. This establishes our claim.Table 9
rq|Ψ for Sn when n=4p and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5vi1≤i≤a+t−102p−i+3p+i+2i+2via+t≤i≤p02p−i+32p−i+3i+2vip+1≤i≤p+a+t−10i−p+222p−i+3i+2vip+a+t≤i≤2p0i−p+222p−i+3vip+a+t≤i≤2pvi2p+1≤i≤2p+a+t−10i−p+2i−2p+223p−i+3vi2p+a+t≤i≤3p05p−i−3t+2i−2p+223p−i+3vi3p+1≤i≤n−t−105p−i−3t+2i−2p+2i−3p+22vin−t≤i≤n05p−i−3t+26p−i−3t+2i−3p+22Table 10
rq|Ψ for Sn when n=4p+1 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5vi1≤i≤a+102p−i+4p+i+2i+2via+2≤i≤p+102p−i+42p−i+4i+2vip+2≤i≤p+a+10i−p+122p−i+4i+2vip+a+2≤i≤2p+10i−p+122p−i+43p−i+4vi2p+2≤i≤2p+a+t0i−p+1i−2p+123p−i+4vi2p+a+t+1≤i≤3p+105p−i−3t+3i−2p+123p−i+4vi3p+2≤i≤n−2t+105p−i−3t+3i−2p+1i−3p+12vin−2t+2≤i≤n05p−i−3t+36p−i−3t+4i−3p+12Table 11
rq|Ψ for Sn when n=4p+2 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5vi1≤i≤a+t02p−i+4p+i+2i+2via+t+1≤i≤p+102p−i+42p−i+5i+2vip+2≤i≤p+a+t0i−p+122p−i+5i+2vip+a+t+1≤i≤2p+20i−p+122p−i+53p−i+5vi2p+3≤i≤3p−2t+20i−p+1i−2p23p−i+5vi3p−2t+3≤i≤n−a−2t05p−i−3t+4i−2p23p−i+5vin−a−2t+1≤i≤n−2t+105p−i−3t+4i−2pi−3p2vin−2t+2≤i≤n05p−i−3t+46p−i−3t+5i−3p2Table 12
rq|Ψ for Sn when n=4p+3 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5vi1≤i≤a+t02p−i+4p+i+2i+2via+t+1≤i≤p+102p−i+42p−i+5i+2vip+2≤i≤p+a+20i−p+122p−i+5i+2vip+a+3≤i≤2p+20i−p+122p−i+53p−i+6vi2p+3≤i≤3p−t+10i−p+1i−2p23p−i+6vi3p−t+2≤i≤n−a−2t05p−i−3t+5i−2p23p−i+6vin−a−2t+1≤i≤n−t−105p−i−3t+5i−2pi−3p−12vin−t≤i≤n05p−i−3t+56p−i−3t+5i−3p−12


### 2.4. Fault Tolerant Partition Dimension ofTn

In [12], Imran et al. defined convex polytope Tn for n≥6 consisting of 4n3-sided faces, n4-sided faces, and a pair of n-sided faces which can be constructed by combining convex polytope Rn with antiprism graph. Here VTn=ui,vi,xi,yi,1≤i≤n and ETn=uiui+1;uivi;ui+1vi;vivi+1;vi,xi;xi,xi+1;xiyi;xi+1yi;yiyi+1:1≤i≤n. Conventionally, we assume that un+1=u1,vn+1=v1,xn+1=x1, and yn+1=y1. The graph of convex polytope Tn is shown in Figure 4.Figure 4
Convex polytopeTn.In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytopeTn.Theorem 5.
Consider convex polytopeTn with n≥6; then 4≤pd2Tn≤5.Proof.
LetΨ=B1,B2,B3,B4,B5 be a partition of the node set VTn. The partition sets B1,B2,B3,B4, and B5 defined for convex polytope Qn are the same for Tn. Let a=⌊p/2⌋. The representations of nodes vi and yi are the same in both Qn and Tn. The representations of nodes ui and xi are shown in the tables below. When p is odd, we have t=2, whereas when p is even, t=1 in the tables. Tables 13–16 clearly show that Ψ is the 2− resolving generator for Tn when n≥6; therefore, pd2Tn≤5 for n≥6. Lemma 1 implies that pd2Tn≥4 for n≥6. This establishes our claim.Table 13
rq|Ψ for Tn when n=4p and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5uii=103p−i+4p+i+1i+2ui2≤i≤p−t03p−i+4p+i+1i+1uip−t+1≤i≤p+103p−i+42p−i+4i+1uip+2≤i≤2p−t0i−p+132p−i+4i+1ui2p−t+1≤i≤2p+10i−p+132p−i+43p−i+4ui2p+2≤i≤3p−t0i−p+1i−2p+133p−i+4ui3p−t+1≤i≤n−3t05p−i−3t+3i−2p+133p−i+4uin−3t+1≤i≤n−t05p−i−3t+3i−2p+1i−3p+13uin−t+1≤i≤n05p−i−3t+36p−i−3t+3i−3p+13xi1≤i≤a+101p−i+2p+iixia+2≤i≤p+101p−i+22p−i+2ixip+2≤i≤p+t+20i−p12p−i+2ixip+t+3≤i≤2p+10i−p12p−i+23p−i+2xi2p+2≤i≤3p−2t+10i−pi−2p13p−i+2xi3p−2t+2≤i≤3p+105p−i−3t+1i−2p13p−i+2xi3p+2≤i≤n−2t+105p−i−3t+1i−2pi−3p1xin−2t+2≤i≤n05p−i−3t+16p−i−3t+1i−3p1Table 14
rq|Ψ for Tn when n=4p+1 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5uii=103p−i+5p+i+1i+2ui2≤i≤p−2t+203p−i+5p+i+1i+1uip−2t+3≤i≤p+203p−i+52p−i+5i+1uip+3≤i≤2p−2t+20i−p32p−i+5i+1ui2p−2t+3≤i≤2p+20i−p32p−i+53p−i+5ui2p+3≤i≤3p−t+10i−pi−2p33p−i+5ui3p−t+2≤i≤n−3t05p−i−3t+4i−2p33p−i+5uin−3t+1≤i≤n−2t+105p−i−3t+4i−2pi−3p3uin−2t+2≤i≤n05p−i−3t+46p−i−3t+5i−3p3xi1≤i≤a+t01p−i+3p+iixia+t+1≤i≤p+201p−i+32p−i+3ixip+3≤i≤p+2t+10i−p−112p−i+3ixip+2t+2≤i≤2p+20i−p−112p−i+33p−i+3xi2p+3≤i≤3p−2t+20i−p−1i−2p−113p−i+3xi3p−2t+3≤i≤3p+205p−i−3t+2i−2p−113p−i+3xi3p+3≤i≤n−t05p−i−3t+2i−2p−1i−3p−11xin−t+1≤i≤n05p−i−3t+26p−i−3t+3i−3p−11Table 15
rq|Ψ for Tn when n=4p+2 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5uii=103p−i+5p+i+1i+2ui2≤i≤p−t+103p−i+5p+i+1i+1uip−t+2≤i≤p+203p−i+52p−i+6i+1uip+3≤i≤2p−t+10i−p32p−i+6i+1ui2p−t+2≤i≤2p+30i−p32p−i+63p−i+6ui2p+4≤i≤3p−2t+30i−pi−2p−133p−i+6ui3p−2t+4≤i≤n−3t05p−i−3t+5i−2p−133p−i+6uin−3t+1≤i≤n−t05p−i−3t+5i−2p−1i−3p−13uin−t+1≤i≤n05p−i−3t+56p−i−3t+6i−3p−13xi1≤i≤a+201p−i+3p+iixia+3≤i≤p+201p−i+32p−i+4ixip+3≤i≤p+t+30i−p−112p−i+4ixip+t+4≤i≤2p+30i−p−112p−i+43p−i+4xi2p+4≤i≤3p−t+10i−p−1i−2p−213p−i+4xi3p−t+2≤i≤3p+305p−i−3t+3i−2p−213p−i+4xi3p+4≤i≤n−t05p−i−3t+3i−2p−2i−3p−21xin−t+1≤i≤n05p−i−3t+36p−i−3t+4i−3p−21Table 16
rq|Ψ for Tn when n=4p+3 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5uii=103p−i+5p+i+1i+2ui2≤i≤p−t+103p−i+5p+i+1i+1uip−t+2≤i≤p+203p−i+52p−i+6i+1uip+3≤i≤2p−2t+30i−p32p−i+6i+1ui2p−2t+4≤i≤2p+30i−p32p−i+63p−i+7ui2p+4≤i≤3p−t+20i−pi−2p−133p−i+7ui3p−t+3≤i≤n−3t05p−i−3t+6i−2p−133p−i+7uin−3t+1≤i≤n−t05p−i−3t+6i−2p−1i−3p−23uin−t+1≤i≤n05p−i−3t+66p−i−3t+7i−3p−23xi1≤i≤a+201p−i+3p+iixia+3≤i≤p+201p−i+32p−i+4ixip+3≤i≤p+t+20i−p−112p−i+4ixip+2t+3≤i≤2p+30i−p−112p−i+43p−i+5xi2p+4≤i≤3p−2t+30i−p−1i−2p−213p−i+5xi3p−2t+4≤i≤3p+405p−i−3t+4i−2p−213p−i+5xi3p+5≤i≤n−2t+105p−i−3t+4i−2p−2i−3p−21xin−2t+2≤i≤n05p−i−3t+46p−i−3t+5i−3p−21


### 2.5. Fault Tolerant Partition Dimension ofDn

In [4], Baca et al. defined convex polytope Dn for n≥4 consisting of 2n5-sided faces. Here VDn=ui,vi,xi,yi:1≤i≤n and EDn=uiui+1;yiyi+1;uivi;vixi;vi+1xi;xiyi:1≤i≤n. Conventionally, we assume that un+1=u1,xn+1=v1,xn+1=x1, and yn+1=y1. The graph of convex polytope Dn is shown in Figure 5.Figure 5
Convex polytopeDn.The following results in [6, 8] are important for us.Proposition 1.
LetG be a connected graph of order n; then(1)
pdG=2 if and only if G=Pn, where Pn is a path of order n≥2;(2)
pdG≤pd2G.
In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytopeDn.Theorem 6.
Consider convex polytopeDn with n≥6; then 3≤pd2Dn≤5.
Proof. Letψ=B1,B2,B3,B4,B5 be a partition of the node set VDn. The partition sets B1,B2,B3,B4,B5 defined for convex polytope Qn are the same for Dn. The representations of all the nodes are the same in both Qn and Dn. Hence, in view of Theorem 3 and Proposition 1, we have 3≤pd2Dn≤5 for n≥6.


## 2.1. Fault Tolerant Partition Dimension ofRn

In [21], Baca defined convex polytope Rn for n≥5 with VRn=ui,vi,xi:1≤i≤n and ERn=uiui+1;ui,vi;ui+1vi;vivi+1;vixi;xixi+1:1≤i≤n.Conventionally, we assume thatun+1=u1,vn+1=v1, and xn+1=x1. The graph of convex polytope Rn is shown in Figure 1.Figure 1
Convex polytopeRn.In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytopeRn.Theorem 2.
Consider convex polytopeRn with n≥5; then 4≤pd2Rn≤5.Proof.
LetΨ=B1,B2,B3,B4,B5 be a partition of the node set VRn. We divide the proof in four parts and, in each part, it will be shown that Ψ is the fault tolerant partition generator for Rn. Let a=⌊p/2⌋.Case 1: Forn=4p.LetB1=ui,vi:1≤i≤n,B2=xi:1≤i≤p, B3=xi:p+1≤i≤2p, B4=xi:2p+1≤i≤3p, and B5=xi:3p+1≤i≤3n.rq|Ψ are tabulated in Table 1.Case 2: Forn=4p+1.LetB1=ui,vi:1≤i≤n,B2=xi:1≤i≤p+1, B3=xi:p+2≤i≤2p+1, B4=xi:2p+2≤i≤3p+1, and B5=xi:3p+2≤i≤3n.rq|Ψ are tabulated in Table 2.Case 3: Forn=4p+2.LetB1=ui,vi:1≤i≤n,B2=xi:1≤i≤p+1, B3=xi:p+2≤i≤2p+2, B4=xi:2p+3≤i≤3p+2, and B5=xi:3p+3≤i≤3n.rq|Ψ are tabulated in Table 3.Case 4: Forn=4p+3.LetB1=ui,vi:1≤i≤n,B2=xi:1≤i≤p+1, B3=xi:p+2≤i≤2p+2, B4=xi:2p+3≤i≤3p+3, and B5=xi:3p+4≤i≤3n.rq|Ψ are tabulated in Table 4.
Whenp is odd, we have t=2, whereas when p is even, t=1 in the tables. Tables 1–4 clearly show that Ψ is the 2− resolving generator for Rn when n≥5; therefore, pd2Rn≤5 for n≥5. Lemma 1 implies that pd2Rn≥4 for n≥5. This establishes our claim.Table 1
rq|Ψ for Rn when n=4p and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+102p−i+3p+i+1i+1uia+2≤i≤p+102p−i+32p−i+3i+1uip+2≤i≤p+a+10i−p+122p−i+3i+1uip+a+2≤i≤2p+10i−p+122p−i+33p−i+3ui2p+2≤i≤2p+a+10i−p+1i−2p+123p−i+3ui2p+a+2≤i≤3p+10n−i+3i−2p+123p−i+3ui3p+2≤i≤n−2t+10n−i+3i−2p+1i−n+p+12uin−2t+2≤i≤n0n−i+35p−i+3i−n+p+12vi1≤i≤a+t−101p−i+2p+i+1i+1via+t≤i≤p01p−i+22p−i+2i+1vip+1≤i≤p+a+t−10i−p+112p−i+2i+1vip+a+t≤i≤2p0i−p+112p−i+23p−i+2vi2p+1≤i≤2p+a+t−10i−p+1i−2p+113p−i+2vi2p+a+t≤i≤3p0n−i+2i−2p+113p−i+2vi3p+1≤i≤n−a0n−i+2i−2p+1i−3p+11vin−a+1≤i≤n0n−i+25p−i+2i−3p+11xi1≤i≤a+t−110p−i+1i+pixia+t≤i≤p10p−i+12p−i+1ixip+1≤i≤p+a+t−11i−p02p−i+1ixip+a+t≤i≤2p1i−p02p−i+13p−i+1xi2p+1≤i≤2p+a+t−11i−pi−2p03p−i+1xi2p+a+t≤i≤3p1n−i+1i−2p03p−i+1xi3p+1≤i≤n−a1n−i+1i−2pi−3p0xin−a+1≤i≤n1n−i+15p−i+1i−3p0Table 2
Representations of nodes forRn when n=4p+1 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+102p−i+4p+i+1i+1uia+t+1≤i≤p+202p−i+42p−i+4i+1uip+3≤i≤p+a+t0i−p22p−i+4i+1uip+a+t+1≤i≤2p+20i−p22p−i+43p−i+4ui2p+3≤i≤2p+a+20i−pi−2p23p−i+4ui2p+a+3≤i≤3p+20n−i+3i−2p23p−i+4ui3p+3≤i≤n−a+10n−i+3i−2pi−3p2uin−a+2≤i≤n0n−i+35p−i+5i−n+p+12vi1≤i≤a+101p−i+3p+i+1i+1via+2≤i≤2a+t01p−i+32p−i+3i+1vi2a+t+1≤i≤p+a+10i−p12p−i+3i+1vip+a+2≤i≤2p+10i−p12p−i+33p−i+3vi2p+2≤i≤2p+a+t0i−pi−2p13p−i+3vi2p+a+t+1≤i≤3p+10n−i+2i−2p13p−i+3vi3p+2≤i≤n−a−t+20n−i+2i−2pi−3p1vin−a−t+3≤i≤n0n−i+25p−i+4i−3p1xi1≤i≤a+110p−i+2i+pixia+2≤i≤p+110p−i+22p−i+2ixip+2≤i≤p+a+11i−p−102p−i+2ixip+a+2≤i≤2p+11i−p−102p−i+23p−i+2xi2p+2≤i≤2p+a+11i−p−1i−2p−103p−i+2xi2p+a+2≤i≤3p+11n−i+1i−2p−103p−i+2xi3p+2≤i≤n−a−t+21n−i+1i−2p−1i−3p−10xin−a−t+3≤i≤n1n−i+15p−i+3i−3p−10Table 3
Representations of nodes forRn when n=4p+2 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+202p−i+4p+i+1i+1uia+3≤i≤p+202p−i+42p−i+5i+1uip+3≤i≤p+a+20i−p22p−i+5i+1uip+a+3≤i≤2p+30i−p22p−i+53p−i+5ui2p+4≤i≤2p+a+t+10i−pi−2p−123p−i+5ui2p+a+t+2≤i≤3p+30n−i+3i−2p−123p−i+5ui3p+4≤i≤n−a+10n−i+3i−2p−1i−n+p+12uin−a+2≤i≤n0n−i+35p−i+6i−n+p+12vi1≤i≤a+t01p−i+3p+i+1i+1via+t+1≤i≤p+101p−i+32p−i+4i+1vip+2≤i≤p+a+t0i−p12p−i+4i+1vip+a+t+1≤i≤2p+20i−p12p−i+43p−i+4vi2p+3≤i≤2p+a+20i−pi−2p−113p−i+4vi2p+a+3≤i≤3p+20n−i+2i−2p−113p−i+4vi3p+3≤i≤n−a−t+20n−i+2i−2p−1i−3p−11vin−a−t+3≤i≤n0n−i+25p−i+5i−3p−11xi1≤i≤a+t10p−i+2i+pixia+t+1≤i≤p+110p−i+22p−i+3ixip+2≤i≤p+a+t1i−p−102p−i+3ixip+a+t+1≤i≤2p+21i−p−102p−i+33p−i+3xi2p+3≤i≤2p+a+21i−p−1i−2p−203p−i+3xi2p+a+3≤i≤3p+214p−i+3i−2p−203p−i+3xi3p+3≤i≤n−a−t+214p−i+3i−2p−2i−3p−20xin−a−t+3≤i≤n14p−i+35p−i+4i−3p−20Table 4
Representations of nodes forRn when n=4p+3 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+202p−i+4p+i+1i+1uia+3≤i≤p+202p−i+42p−i+5i+1uip+3≤i≤p+a+t+10i−p22p−i+5i+1uip+a+t+2≤i≤2p+30i−p22p−i+53p−i+6ui2p+4≤i≤2p+a+30i−pi−2p−123p−i+6ui2p+a+4≤i≤3p+40n−i+3i−2p−123p−i+6ui3p+5≤i≤n−a−t+20n−i+3i−2p−1i−3p−22uin−a−t+3≤i≤n0n−i+35p−i+7i−n+p+12vi1≤i≤a+t01p−i+3p+i+1i+1via+t+1≤i≤p+101p−i+32p−i+4i+1vip+2≤i≤p+a+20i−p12p−i+4i+1vip+a+3≤i≤2p+20i−p12p−i+43p−i+5vi2p+3≤i≤2p+a+t+10i−pi−2p−113p−i+5vi2p+a+t+2≤i≤3p+30n−i+2i−2p−113p−i+5vi3p+4≤i≤n−a0n−i+2i−2p−1i−3p−21vin−a+1≤i≤n0n−i+25p−i+6i−3p−21xi1≤i≤a+t10p−i+2i+pixia+t+1≤i≤p+110p−i+22p−i+3ixip+2≤i≤p+a+21i−p−102p−i+3ixip+a+3≤i≤2p+21i−p−102p−i+33p−i+4xi2p+3≤i≤2p+a+t+11i−p−1i−2p−203p−i+4xi2p+a+t+2≤i≤3p+31n−i+1i−2p−203p−i+4xi3p+4≤i≤n−a1n−i+1i−2p−2i−3p−30xin−a+1≤i≤n1n−i+15p−i+5i−3p−30


## 2.2. Fault Tolerant Partition Dimension ofQn

In [22], Baca et al. defined convex polytope Qn for n≥6 consisting of 3-sided, 4-sided, 5-sided, and n-sided faces, respectively. Here VQn=ui,vi,xi,yi:1≤i≤n and EQn=uiui+1;vivi+1;yiyi+1;uivi;vixi;vi+1xi;xiyi:1≤i≤n. Conventionally, we assume that un+1=u1,vn+1=v1,xn+1=x1, and yn+1=y1. The graph of convex polytope Qn is shown in Figure 2.Figure 2
Convex polytopeQn.In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytopeQn.Theorem 3.
Consider convex polytopeQn with n≥6; then 4≤pd2Qn≤5.Proof.
LetΨ=B1,B2,B3,B4,B5 be a partition of the node set VQn. We divide the proof in four parts and, in each part, it will be shown that Ψ is the fault tolerant partition generator for Qn. Let a=⌊p/2⌋.Case 1: Forn=4p.LetB1=ui,vi,xi:1≤i≤n,B2=yi:1≤i≤p, B3=yi:p+1≤i≤2p, B4=yi:2p+1≤i≤3p, and B5=yi:3p+1≤i≤4n.rq|Ψ are tabulated in Table 5.Case 2: Forn=4p+1.LetB1=ui,vi,xi:1≤i≤n,B2=yi:1≤i≤p+1, B3=yi:p+2≤i≤2p+1, B4=yi:2p+2≤i≤3p+1, and B5=yi:3p+2≤i≤4n.rq|Ψ are tabulated in Table 6.Case 3: Forn=4p+2.LetB1=ui,vi,xi:1≤i≤n,B2=yi:1≤i≤p+1, B3=yi:p+2≤i≤2p+2, B4=yi:2p+3≤i≤3p+2, and B5=yi:3p+3≤i≤4n.rq|Ψ are tabulated in Table 7.Case 4: Forn=4p+3.LetB1=ui,vi,xi:1≤i≤n,B2=yi:1≤i≤p+1, B3=yi:p+2≤i≤2p+2, B4=yi:2p+3≤i≤3p+3, and B5=yi:3p+4≤i≤4n.rq|Ψ are tabulated in Table 8.
Whenp is odd, we have t=2, whereas when p is even, t=1 in the tables. Tables 5–8 clearly show that Ψ is the 2− resolving generator for Qn when n≥6; therefore, pd2Qn≤5 for n≥6. Lemma 1 implies that pd2Qn≥4 for n≥6. This establishes our claim.Table 5
rq|Ψ for Qn when n=4p and a=⌊p/2⌋.
Nodesdq,B1dq,B2ui1≤i≤a+103uia+2≤i≤p+103uip+2≤i≤p+a+10i−p+2uip+a+2≤i≤2p+10i−p+2ui2p+2≤i≤2p+a+10i−p+2ui2p+a+2≤i≤3p+105p−i−3t+3ui3p+2≤i≤n−2t+105p−i−3t+3uin−2t+2≤i≤n05p−i−3t+3vi1≤i≤a+102via+2≤i≤p+102vip+2≤i≤p+a+10i−p+1vip+a+2≤i≤2p+10i−p+1vi2p+2≤i≤2p+a+10i−p+1vi2p+a+2≤i≤3p+105p−i−3t+2vi3p+2≤i≤n−2t+105p−i−3t+2vin−2t+2≤i≤n05p−i−3t+2xi1≤i≤a+t−101xia+t≤i≤p01xip+1≤i≤p+a+t−10i−p+1xip+a+t≤i≤2p0i−p+1xi2p+1≤i≤2p+a+t−10i−p+1xi2p+a+t≤i≤3p05p−i−3t+1xi3p+1≤i≤n−t−105p−i−3t+1xin−t≤i≤n05p−i−3t+1yi1≤i≤a+t−110yia+t≤i≤p10yip+1≤i≤p+a+t−11i−pyip+a+t≤i≤2p1i−pyi2p+1≤i≤2p+a+t−11i−pyi2p+a+t≤i≤3p15p−i−3tyi3p+1≤i≤n−t−115p−i−3tyin−t≤i≤n15p−i−3tTable 6
rq|Ψ for Qn when n=4p+1 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+t03p−i+5p+i+2i+2uia+t+1≤i≤p+203p−i+52p−i+5i+2uip+3≤i≤p+a+t0i−p+132p−i+5i+2uip+a+t+1≤i≤2p+20i−p+132p−i+53p−i+5ui2p+3≤i≤2p+a+20i−p+1i−2p+133p−i+5ui2p+a+3≤i≤3p+205p−i−3t+4i−2p+133p−i+5ui3p+3≤i≤n−t05p−i−3t+4i−2p+1i−3p+13uin−t+1≤i≤n05p−i−3t+46p−i−3t+5i−3p+13vi1≤i≤a+t02p−i+4p+i+1i+1via+t+1≤i≤p+202p−i+42p−i+4i+1vip+3≤i≤p+a+t0i−p22p−i+4i+1vip+a+t+1≤i≤2p+20i−p22p−i+43p−i+4vi2p+3≤i≤2p+a+20i−pi−2p23p−i+4vi2p+a+3≤i≤3p+205p−i−3t+3i−2p23p−i+4vi3p+3≤i≤n−t05p−i−3t+3i−2pi−3p2vin−t+1≤i≤n05p−i−3t+36p−i−3t+4i−3p2xi1≤i≤a+101p−i+3p+i+1i+1xia+2≤i≤p+101p−i+32p−i+3i+1xip+2≤i≤p+a+10i−p12p−i+3i+1xip+a+2≤i≤2p+10i−p12p−i+33p−i+3xi2p+2≤i≤2p+a+t0i−pi−2p13p−i+3xi2p+a+t+1≤i≤3p+105p−i−3t+2i−2p13p−i+3xi3p+2≤i≤n−2t+105p−i−3t+2i−2pi−3p1xin−2t+2≤i≤n05p−i−3t+26p−i−3t+3i−3p1yi1≤i≤a+110p−i+2p+iiyia+2≤i≤p+110p−i+22p−i+2iyip+2≤i≤p+a+11i−p−102p−i+2iyip+a+2≤i≤2p+11i−p−102p−i+23p−i+2yi2p+2≤i≤2p+a+t1i−p−1i−2p−103p−i+2yi2p+a+t+1≤i≤3p+115p−i−3t+1i−2p−103p−i+2yi3p+2≤i≤n−2t+115p−i−3t+1i−2p−1i−3p−10yin−2t+2≤i≤n15p−i−3t+16p−i−3t+2i−3p−10Table 7
rq|Ψ for Qn when n=4p+2 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+203p−i+5p+i+2i+2uia+3≤i≤p+203p−i+52p−i+6i+2uip+3≤i≤p+a+20i−p+132p−i+6i+2uip+a+3≤i≤2p+a+10i−p+132p−i+63p−i+6ui2p+a+2≤i≤3p0i−p+1i−2p33p−i+6ui3p+1≤i≤n−a−t05p−i+2i−2p33p−i+6uin−a−t+1≤i≤n−105p−i−t+3i−2pi−3p3uii=n05p−i−t+36p−i−t+4i−3p3vi1≤i≤a+202p−i+4p+i+1i+1via+3≤i≤p+202p−i+42p−i+5i+1vip+3≤i≤p+a+20i−p22p−i+5i+1vip+a+3≤i≤3p−t0i−p22p−i+53p−i+5vi3p−t+1≤i≤3p0i−pi−2p−123p−i+5vi3p+1≤i≤n−a−t05p−i−t+2i−2p−123p−i+5vin−a−t+1≤i≤n−105p−i−t+2i−2p−1i−3p−12vii=n05p−i−t+26p−i−t+3i−3p−12xi1≤i≤a+t01p−i+3p+i+1i+1xia+t+1≤i≤p+101p−i+32p−i+4i+1xip+2≤i≤p+a+t0i−p12p−i+4i+1xip+a+t+1≤i≤3p−t−10i−p12p−i+43p−i+4xi3p−t≤i≤3p−t+10i−pi−2p−113p−i+4xi3p−t+2≤i≤n−a−t−105p−i−t+1i−2p−113p−i+4xin−a−t≤i≤n−t05p−i−t+1i−2p−1i−3p−11xin−t+1≤i≤n05p−i−t+16p−i−t+2i−3p−11yi1≤i≤a+t10p−i+2p+iiyia+t+1≤i≤p+110p−i+22p−i+3iyip+2≤i≤p+a+t1i−p−102p−i+3iyip+a+t+1≤i≤3p−t−11i−p−102p−i+33p−i+3yi3p−t≤i≤3p−t+11i−p−1i−2p−203p−i+3yi3p−t+2≤i≤n−a−t−115p−i−ti−2p−203p−i+3yin−a−t≤i≤n−t15p−i−ti−2p−2i−3p−20yin−t+1≤i≤n15p−i−t6p−i−t+1i−3p−20Table 8
rq|Ψ for Qn when n=4p+3 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5ui1≤i≤a+203p−i+5p+i+2i+2uia+3≤i≤p+203p−i+52p−i+6i+2uip+3≤i≤p+a+t+10i−p+132p−i+6i+2uip+a+t+2≤i≤2p+a−t+20i−p+132p−i+63p−i+7ui2p+a−t+3≤i≤3p−2t+30i−p+1i−2p33p−i+7ui3p−2t+4≤i≤n−3t05p−i−3t+6i−2p33p−i+7uin−3t+1≤i≤n−2t+105p−i−3t+6i−2pi−3p−13uin−2t+2≤i≤nZuin−2t+2≤i≤n05p−i−3t+66p−i−3t+7i−3p−13vi1≤i≤a+202p−i+4p+i+1i+1via+3≤i≤p+202p−i+42p−i+5i+1vip+3≤i≤p+a+t+10i−p22p−i+5i+1vip+a+t+2≤i≤2p+30i−p22p−i+53p−i+6vi2p+a+4≤i≤3p+40i−pi−2p−123p−i+6vi2p+a+4≤i≤3p+405p−i−3t+5i−2p−123p−i+6vi3p+5≤i≤n−2t+105p−i−3t+5i−2p−1i−3p−22vin−2t+2≤i≤n05p−i−3t+56p−i−3t+6i−3p−22xi1≤i≤a+t01p−i+3p+i+1i+1xia+t+1≤i≤p+101p−i+32p−i+4i+1xip+2≤i≤p+a+20i−p12p−i+4i+1xip+a+3≤i≤2p+20i−p12p−i+43p−i+5xi2p+3≤i≤2p+a+t+10i−pi−2p−113p−i+5xi2p+a+t+2≤i≤3p+305p−i−3t+4i−2p−113p−i+5xi3p+4≤i≤n−t−105p−i−3t+4i−2p−1i−3p−21xin−t≤i≤n05p−i−3t+46p−i−3t+5i−3p−21yi1≤i≤a+t10p−i+2p+iiyia+t+1≤i≤p+110p−i+22p−i+3iyip+2≤i≤p+a+21i−p−102p−i+3iyip+a+3≤i≤2p+21i−p−102p−i+33p−i+4yi2p+3≤i≤2p+a+t+11i−p−1i−2p−203p−i+4yi2p+a+t+2≤i≤3p+315p−i−3t+3i−2p−203p−i+4yi3p+4≤i≤n−t−115p−i−3t+3i−2p−2i−3p−30yin−t≤i≤n15p−i−3t+36p−i−3t+4i−3p−30


## 2.3. Fault Tolerant Partition Dimension ofSn

In [12], Imran et al. defined convex polytope Sn for n≥6 consisting of 2n3-sided faces, 2n4-sided faces, and a pair of n-sided faces which can be constructed by combining convex polytope Rn with prism graph. Here VSn=ui,vi,xi,yi:1≤i≤n and ESn=uiui+1;vivi+1;xixi+1;yiyi+1;uj+1vi;ui,vi;vi,xi;xi,yi:1≤i≤n. Conventionally, we assume that un+1=u1,vn+1=v1,xn+1=x1, and yn+1=y1. The graph of convex polytope Sn is shown in Figure 3.Figure 3
Convex polytopeSn.In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytopeSn.Theorem 4.
Consider convex polytopeSn with n≥6; then 4≤pd2Sn≤5.Proof.
LetΨ=B1,B2,B3,B4,B5 be a partition of the node set VSn. The partition sets B1,B2,B3,B4, and B5 defined for convex polytope Qn are the same for Sn. Let a=⌊p/2⌋. The representations of nodes ui,xi, and yi are the same in both Qn and Sn. The representations of nodes vi are shown in the tables below. When p is odd, we have t=2 , whereas when p is even, t=1 in the tables. Tables 9–12 clearly show that Ψ is the 2− resolving generator for Sn when n≥6; therefore, pd2Sn≤5 for n≥5. Lemma 1 implies that pd2Sn≥4 for n≥6. This establishes our claim.Table 9
rq|Ψ for Sn when n=4p and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5vi1≤i≤a+t−102p−i+3p+i+2i+2via+t≤i≤p02p−i+32p−i+3i+2vip+1≤i≤p+a+t−10i−p+222p−i+3i+2vip+a+t≤i≤2p0i−p+222p−i+3vip+a+t≤i≤2pvi2p+1≤i≤2p+a+t−10i−p+2i−2p+223p−i+3vi2p+a+t≤i≤3p05p−i−3t+2i−2p+223p−i+3vi3p+1≤i≤n−t−105p−i−3t+2i−2p+2i−3p+22vin−t≤i≤n05p−i−3t+26p−i−3t+2i−3p+22Table 10
rq|Ψ for Sn when n=4p+1 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5vi1≤i≤a+102p−i+4p+i+2i+2via+2≤i≤p+102p−i+42p−i+4i+2vip+2≤i≤p+a+10i−p+122p−i+4i+2vip+a+2≤i≤2p+10i−p+122p−i+43p−i+4vi2p+2≤i≤2p+a+t0i−p+1i−2p+123p−i+4vi2p+a+t+1≤i≤3p+105p−i−3t+3i−2p+123p−i+4vi3p+2≤i≤n−2t+105p−i−3t+3i−2p+1i−3p+12vin−2t+2≤i≤n05p−i−3t+36p−i−3t+4i−3p+12Table 11
rq|Ψ for Sn when n=4p+2 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5vi1≤i≤a+t02p−i+4p+i+2i+2via+t+1≤i≤p+102p−i+42p−i+5i+2vip+2≤i≤p+a+t0i−p+122p−i+5i+2vip+a+t+1≤i≤2p+20i−p+122p−i+53p−i+5vi2p+3≤i≤3p−2t+20i−p+1i−2p23p−i+5vi3p−2t+3≤i≤n−a−2t05p−i−3t+4i−2p23p−i+5vin−a−2t+1≤i≤n−2t+105p−i−3t+4i−2pi−3p2vin−2t+2≤i≤n05p−i−3t+46p−i−3t+5i−3p2Table 12
rq|Ψ for Sn when n=4p+3 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5vi1≤i≤a+t02p−i+4p+i+2i+2via+t+1≤i≤p+102p−i+42p−i+5i+2vip+2≤i≤p+a+20i−p+122p−i+5i+2vip+a+3≤i≤2p+20i−p+122p−i+53p−i+6vi2p+3≤i≤3p−t+10i−p+1i−2p23p−i+6vi3p−t+2≤i≤n−a−2t05p−i−3t+5i−2p23p−i+6vin−a−2t+1≤i≤n−t−105p−i−3t+5i−2pi−3p−12vin−t≤i≤n05p−i−3t+56p−i−3t+5i−3p−12


## 2.4. Fault Tolerant Partition Dimension ofTn

In [12], Imran et al. defined convex polytope Tn for n≥6 consisting of 4n3-sided faces, n4-sided faces, and a pair of n-sided faces which can be constructed by combining convex polytope Rn with antiprism graph. Here VTn=ui,vi,xi,yi,1≤i≤n and ETn=uiui+1;uivi;ui+1vi;vivi+1;vi,xi;xi,xi+1;xiyi;xi+1yi;yiyi+1:1≤i≤n. Conventionally, we assume that un+1=u1,vn+1=v1,xn+1=x1, and yn+1=y1. The graph of convex polytope Tn is shown in Figure 4.Figure 4
Convex polytopeTn.In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytopeTn.Theorem 5.
Consider convex polytopeTn with n≥6; then 4≤pd2Tn≤5.Proof.
LetΨ=B1,B2,B3,B4,B5 be a partition of the node set VTn. The partition sets B1,B2,B3,B4, and B5 defined for convex polytope Qn are the same for Tn. Let a=⌊p/2⌋. The representations of nodes vi and yi are the same in both Qn and Tn. The representations of nodes ui and xi are shown in the tables below. When p is odd, we have t=2, whereas when p is even, t=1 in the tables. Tables 13–16 clearly show that Ψ is the 2− resolving generator for Tn when n≥6; therefore, pd2Tn≤5 for n≥6. Lemma 1 implies that pd2Tn≥4 for n≥6. This establishes our claim.Table 13
rq|Ψ for Tn when n=4p and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5uii=103p−i+4p+i+1i+2ui2≤i≤p−t03p−i+4p+i+1i+1uip−t+1≤i≤p+103p−i+42p−i+4i+1uip+2≤i≤2p−t0i−p+132p−i+4i+1ui2p−t+1≤i≤2p+10i−p+132p−i+43p−i+4ui2p+2≤i≤3p−t0i−p+1i−2p+133p−i+4ui3p−t+1≤i≤n−3t05p−i−3t+3i−2p+133p−i+4uin−3t+1≤i≤n−t05p−i−3t+3i−2p+1i−3p+13uin−t+1≤i≤n05p−i−3t+36p−i−3t+3i−3p+13xi1≤i≤a+101p−i+2p+iixia+2≤i≤p+101p−i+22p−i+2ixip+2≤i≤p+t+20i−p12p−i+2ixip+t+3≤i≤2p+10i−p12p−i+23p−i+2xi2p+2≤i≤3p−2t+10i−pi−2p13p−i+2xi3p−2t+2≤i≤3p+105p−i−3t+1i−2p13p−i+2xi3p+2≤i≤n−2t+105p−i−3t+1i−2pi−3p1xin−2t+2≤i≤n05p−i−3t+16p−i−3t+1i−3p1Table 14
rq|Ψ for Tn when n=4p+1 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5uii=103p−i+5p+i+1i+2ui2≤i≤p−2t+203p−i+5p+i+1i+1uip−2t+3≤i≤p+203p−i+52p−i+5i+1uip+3≤i≤2p−2t+20i−p32p−i+5i+1ui2p−2t+3≤i≤2p+20i−p32p−i+53p−i+5ui2p+3≤i≤3p−t+10i−pi−2p33p−i+5ui3p−t+2≤i≤n−3t05p−i−3t+4i−2p33p−i+5uin−3t+1≤i≤n−2t+105p−i−3t+4i−2pi−3p3uin−2t+2≤i≤n05p−i−3t+46p−i−3t+5i−3p3xi1≤i≤a+t01p−i+3p+iixia+t+1≤i≤p+201p−i+32p−i+3ixip+3≤i≤p+2t+10i−p−112p−i+3ixip+2t+2≤i≤2p+20i−p−112p−i+33p−i+3xi2p+3≤i≤3p−2t+20i−p−1i−2p−113p−i+3xi3p−2t+3≤i≤3p+205p−i−3t+2i−2p−113p−i+3xi3p+3≤i≤n−t05p−i−3t+2i−2p−1i−3p−11xin−t+1≤i≤n05p−i−3t+26p−i−3t+3i−3p−11Table 15
rq|Ψ for Tn when n=4p+2 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5uii=103p−i+5p+i+1i+2ui2≤i≤p−t+103p−i+5p+i+1i+1uip−t+2≤i≤p+203p−i+52p−i+6i+1uip+3≤i≤2p−t+10i−p32p−i+6i+1ui2p−t+2≤i≤2p+30i−p32p−i+63p−i+6ui2p+4≤i≤3p−2t+30i−pi−2p−133p−i+6ui3p−2t+4≤i≤n−3t05p−i−3t+5i−2p−133p−i+6uin−3t+1≤i≤n−t05p−i−3t+5i−2p−1i−3p−13uin−t+1≤i≤n05p−i−3t+56p−i−3t+6i−3p−13xi1≤i≤a+201p−i+3p+iixia+3≤i≤p+201p−i+32p−i+4ixip+3≤i≤p+t+30i−p−112p−i+4ixip+t+4≤i≤2p+30i−p−112p−i+43p−i+4xi2p+4≤i≤3p−t+10i−p−1i−2p−213p−i+4xi3p−t+2≤i≤3p+305p−i−3t+3i−2p−213p−i+4xi3p+4≤i≤n−t05p−i−3t+3i−2p−2i−3p−21xin−t+1≤i≤n05p−i−3t+36p−i−3t+4i−3p−21Table 16
rq|Ψ for Tn when n=4p+3 and a=⌊p/2⌋.
Nodesdq,B1dq,B2dq,B3dq,B4dq,B5uii=103p−i+5p+i+1i+2ui2≤i≤p−t+103p−i+5p+i+1i+1uip−t+2≤i≤p+203p−i+52p−i+6i+1uip+3≤i≤2p−2t+30i−p32p−i+6i+1ui2p−2t+4≤i≤2p+30i−p32p−i+63p−i+7ui2p+4≤i≤3p−t+20i−pi−2p−133p−i+7ui3p−t+3≤i≤n−3t05p−i−3t+6i−2p−133p−i+7uin−3t+1≤i≤n−t05p−i−3t+6i−2p−1i−3p−23uin−t+1≤i≤n05p−i−3t+66p−i−3t+7i−3p−23xi1≤i≤a+201p−i+3p+iixia+3≤i≤p+201p−i+32p−i+4ixip+3≤i≤p+t+20i−p−112p−i+4ixip+2t+3≤i≤2p+30i−p−112p−i+43p−i+5xi2p+4≤i≤3p−2t+30i−p−1i−2p−213p−i+5xi3p−2t+4≤i≤3p+405p−i−3t+4i−2p−213p−i+5xi3p+5≤i≤n−2t+105p−i−3t+4i−2p−2i−3p−21xin−2t+2≤i≤n05p−i−3t+46p−i−3t+5i−3p−21


## 2.5. Fault Tolerant Partition Dimension ofDn

In [4], Baca et al. defined convex polytope Dn for n≥4 consisting of 2n5-sided faces. Here VDn=ui,vi,xi,yi:1≤i≤n and EDn=uiui+1;yiyi+1;uivi;vixi;vi+1xi;xiyi:1≤i≤n. Conventionally, we assume that un+1=u1,xn+1=v1,xn+1=x1, and yn+1=y1. The graph of convex polytope Dn is shown in Figure 5.Figure 5
Convex polytopeDn.The following results in [6, 8] are important for us.Proposition 1.
LetG be a connected graph of order n; then(1)
pdG=2 if and only if G=Pn, where Pn is a path of order n≥2;(2)
pdG≤pd2G.
In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytopeDn.Theorem 6.
Consider convex polytopeDn with n≥6; then 3≤pd2Dn≤5.
Proof. Letψ=B1,B2,B3,B4,B5 be a partition of the node set VDn. The partition sets B1,B2,B3,B4,B5 defined for convex polytope Qn are the same for Dn. The representations of all the nodes are the same in both Qn and Dn. Hence, in view of Theorem 3 and Proposition 1, we have 3≤pd2Dn≤5 for n≥6.


## 3. Conclusion

In this research article, we conclude that the fault tolerant partition dimension of the convex polytopesRn,Qn,Sn,Tn,Dn is bounded above by 5. We also proved that the lower bound of fault tolerant partition dimension of all the families of graphs with order greater than or equal to 5 and having a node of degree at least 4 is 4 which includes first four of these convex polytopes. We conclude the article with a conjecture and an open problem.Conjecture 1.
The fault tolerant partition dimension of convex polytopesRn,Qn,Sn,Tn,Dn is 5.Open Problem 1.
Compute the exact value of fault tolerant partition dimension of the convex polytopesRn,Qn,Sn,Tn,Dn.



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