Abstract
Convex polytopes are special types of polytopes having an additional property that they are also convex sets in the n-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.
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 graph , 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 p and denoted by . For any node of and a nonempty subset of V(W), the sets and are called the distance of a node from subset and the neighborhood of node , respectively. Let be the set of ordered nodes; the representation of a node with reference to is a vector such that . If the vectors are different, for every node of the graph, then is termed as a resolving set of . The minimal cardinality of is termed as the metric dimension of , represented by . Let be an ordered partition of a connected graph . The representations of nodes with reference to are vectors such that . If vectors are different for every node of the graph, then partition is termed as a resolving partition of . The minimal cardinality of a resolving partition is termed as the partition dimension of , represented by . 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 metric. If vectors are different in at least places for every node of the graph, then is termed as a metric generator for . The metric basis is a generator with least number of nodes. The minimal cardinality of is termed as the metric dimension of , represented by . The 2-metric dimension was first presented in [14]. Further study on metric dimension can be seen in [15, 16].
In 2020, Moreno et al. [17] presented the concept of partition dimension of the graph. If vectors are different in at least places for every node of the graph, then is termed as a partition generator of . The partition basis is a generator with least number of sets. The minimal cardinality of is termed as the partition dimension of , represented by . For , the partition dimension is known as fault tolerant partition dimension denoted by . 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 graph of order and having a vertex of degree at least 4, (2)For , (3)For ,(a)(b)(c)(d)The rest of the article is organised in the following manner. In Section 2, the convex polytopes 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 polytopes have been computed. In the following theorem, the graphs with fault tolerant partition dimension bounded below by 4 has been characterized.
Lemma 1. Let be a graph of order . If has a node of degree at least 4, then .
Proof. Assume that for . Let be a partition basis of . Let be the node of degree 4. Assume that . Let . Case 1: When , the subcases are as follows: Case 1.1: If each node of is in either set or , assuming that , then, for , , for , which results in a contradiction. Now if , then, for , , for , which again contradicts our assumption. Case 1.2: Assume that or contains three nodes of . Let and ; then , , and , which clearly leads to a contradictory situation. Meanwhile three nodes of in generate a similar case of contradiction. Case 1.3: Assume that each of and contains two nodes from set . Let ; then and , which again results in a contradiction. Case 2: When , the subcases are as follows: Case 2.1: When , this implies that , and then and, for , , for . Hence, . Pigeonhole principle implies that at least two of nodes and are equidistant from and , which results in a contradiction. Case 2.2: When . Let and , and then , , and . In this situation, we get . Again Pigeonhole principle implies that at least two nodes among , and are equidistant from and , which results in a contradiction. Meanwhile generates a similar case of contradiction. Case 2.3: When . Case 2.3.1: Let and , and then , and . In this situation, we get . Now the Pigeonhole principle implies that minimum two nodes among , and are at the same distances from and . This results in a contradiction. Meanwhile generates a similar case of contradiction. Case 2.3.2: Let , , and , and then , and . In this situation, we get . Now the Pigeonhole principle implies that the minimum two nodes among , and are at the same distances from and . This results in a contradiction. Case 2.4: When . Case 2.4.1: Let and , and then , , , , and , which is a clear case of contradiction. Meanwhile generates a similar case of contradiction. Case 2.4.2: Let , , and , and then , , , , and , which is again a clear case of contradiction. Meanwhile and generates a similar case of contradiction. Case 2.5: When . This case is similar to Case 1.Hence, for with (HTML containing a node of degree at least 4.
2.1. Fault Tolerant Partition Dimension of
In [21], Baca defined convex polytope for with and .
Conventionally, we assume that , and . The graph of convex polytope is shown in Figure 1.
In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytope .
Theorem 2. Consider convex polytope with ; then .
Proof. Let be a partition of the node set . We divide the proof in four parts and, in each part, it will be shown that is the fault tolerant partition generator for . Let . Case 1: For . Let , , , and . are tabulated in Table 1. Case 2: For . Let , , , and . are tabulated in Table 2. Case 3: For . Let , , , and . are tabulated in Table 3. Case 4: For . Let , , , and . are tabulated in Table 4.When is odd, we have , whereas when is even, in the tables. Tables 1–4 clearly show that is the resolving generator for when ; therefore, for . Lemma 1 implies that for . This establishes our claim.
2.2. Fault Tolerant Partition Dimension of
In [22], Baca et al. defined convex polytope for consisting of 3-sided, 4-sided, 5-sided, and -sided faces, respectively. Here and . Conventionally, we assume that , and . The graph of convex polytope is shown in Figure 2.
In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytope .
Theorem 3. Consider convex polytope with ; then .
Proof. Let be a partition of the node set . We divide the proof in four parts and, in each part, it will be shown that is the fault tolerant partition generator for . Let . Case 1: For . Let , , , and . are tabulated in Table 5. Case 2: For . Let , , , and . are tabulated in Table 6. Case 3: For . Let , , , and . are tabulated in Table 7. Case 4: For . Let , , , and . are tabulated in Table 8.When is odd, we have , whereas when is even, in the tables. Tables 5–8 clearly show that is the resolving generator for when ; therefore, for . Lemma 1 implies that for . This establishes our claim.
2.3. Fault Tolerant Partition Dimension of
In [12], Imran et al. defined convex polytope for consisting of -sided faces, -sided faces, and a pair of -sided faces which can be constructed by combining convex polytope with prism graph. Here and . Conventionally, we assume that , and . The graph of convex polytope is shown in Figure 3.
In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytope .
Theorem 4. Consider convex polytope with ; then .
Proof. Let be a partition of the node set . The partition sets , and defined for convex polytope are the same for . Let . The representations of nodes , and are the same in both and . The representations of nodes are shown in the tables below. When is odd, we have , whereas when is even, in the tables. Tables 9–12 clearly show that is the resolving generator for when ; therefore, for . Lemma 1 implies that for . This establishes our claim.
2.4. Fault Tolerant Partition Dimension of
In [12], Imran et al. defined convex polytope for consisting of -sided faces, -sided faces, and a pair of -sided faces which can be constructed by combining convex polytope with antiprism graph. Here and . Conventionally, we assume that , and . The graph of convex polytope is shown in Figure 4.
In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytope .
Theorem 5. Consider convex polytope with ; then .
Proof. Let be a partition of the node set . The partition sets , and defined for convex polytope are the same for . Let . The representations of nodes and are the same in both and . The representations of nodes and are shown in the tables below. When is odd, we have , whereas when is even, in the tables. Tables 13–16 clearly show that is the resolving generator for when ; therefore, for . Lemma 1 implies that for . This establishes our claim.
2.5. Fault Tolerant Partition Dimension of
In [4], Baca et al. defined convex polytope for consisting of -sided faces. Here and . Conventionally, we assume that , and . The graph of convex polytope is shown in Figure 5.
The following results in [6, 8] are important for us.
Proposition 1. Let G be a connected graph of order n; then(1) if and only if , where is a path of order ;(2).In the subsequent theorem, we compute the bounds of the fault tolerant partition dimension of convex polytope .
Theorem 6. Consider convex polytope with ; then .
Proof. Let be a partition of the node set . The partition sets defined for convex polytope are the same for . The representations of all the nodes are the same in both and . Hence, in view of Theorem 3 and Proposition 1, we have for .
3. Conclusion
In this research article, we conclude that the fault tolerant partition dimension of the convex polytopes 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 polytopes is 5.
Open Problem 1. Compute the exact value of fault tolerant partition dimension of the convex polytopes .
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.