Abstract
Graph invariants provide an amazing tool to analyze the abstract structures of networks. The interaction and interconnection between devices, sensors, and service providers have opened the door for an eruption of mobile over the web applications. Structure of web sites containing number of pages can be represented using graph, where web pages are considered to be the vertices, and an edge is a link between two pages. Figuring resolving partition of the graph is an intriguing inquest in graph theory as it has many applications such as sensor design, compound classification in chemistry, robotic navigation, and Internet network. The partition dimension is a graph parameter akin to the concept of metric dimension, and fault-tolerant partition dimension is an advancement in the line of research of partition dimension of the graph. In this paper, we compute fault-tolerant partition dimension of alternate triangular cycle, mirror graph, and tortoise graphs.
1. Introduction and Basic Terminologies
Graph theory is an intense region of arithmetic that has capacious variety of implementations in numerous regions of science, such as chemistry, biology, software engineering, and electrical and hardware engineering. Graphing is a powerful tool for representing and understanding objects and their relationships. Currently, online social networks are considered as an essential element for interpersonal relationships where people and smart objects are connected together in smart environments. Theoretical principles of graph theory are applied to practical fields, by determining graph invariants such as vertices, edges, diameter, and degree and mapping them to real-time problems. These invariants are the supporting tools between science and engineering or computational techniques in the fields of chemical, electrical, computer, and telecommunication engineering. One concept that pervades all the graph theory is that of distance and is used in isomorphism testing, graph operations, maximal and minimal problems on connectivity, and diameter. Metric dimension is one of the distance related parameter in graphs that has attracted the attention of several researchers. The generalized version of metric dimension of the graph is unique and an important parameter of graph theory called partition dimension of the graph. It is considered as an applied topic of graph theory and has applications in structure-activity issues in drug design, network discovery and verification [5], pattern recognition and image processing [16], and modelling of chemical substances [14].
In 2000, the concept of partition dimension of graph was initiated by Chartrand et al. as an extension of metric dimension of the graph [6]. Let be a connected graph of order with vertex set and edge set . If two vertices , then the length of shortest path between and in is distance between these vertices and is denoted by . The distance between a vertex and is defined as and is denoted by . For a vertex , will denote the open neighbourhood of in , , and closed neighbourhood of will be denoted by [18]. Let be an ordered subset of vertex set of . The representation of with respect to is tuple and is denoted by . The subset is called a resolving set of if distinct vertices of have distinct representations with respect to . The metric dimension of is defined as and is denoted by . Ali et al. discussed that path-related graphs have constant metric dimension [1]. Zuo et al. computed constant metric dimension of some generalized convex polytopes [27]. Rehman et al. computed the metric dimension of arithmetic graph of a composite number [20]. In 2021, metric dimension of windmill graph was computed by Singh et al. [24].
In 2008, Hernando et al. initiated the concept of fault-tolerant metric dimension of graphs [10]. If, for every pair of distinct vertices , there exists at least two vertices such that for , then the resolving set of is called fault tolerant. The fault-tolerant metric dimension of is the minimum cardinality of fault-tolerant resolving set and is denoted by . Ahmad et al. computed fault-tolerant metric dimension of graph [3]. Hayat et al. discussed fault-tolerant metric dimension of interconnection networks [9].
Let be a partition with partition classes of vertex set of connected graph . The representation of vertex with respect to partition set is vector denoted by . The partition is called resolving partition of if representation of all the vertices in is different. We define the partition dimension of graph as and is denoted by . Chartrand et al. [6] characterised the graphs having to be 2 or . For various classes of connected graphs, the partition dimension has been obtained. For instance, Ayesha et al. computed the partition dimension of trihexagonal boron nanotube [23]. Mehreen et al. computed the partition dimension of the fullerene graph [15]. Hussain et al. provided the bounds on partition dimension of generalized Mobius ladder [11]. Monica et al. studied the partition dimension problem for certain classes of the series-parallel graph [17]. Chu et al. calculated the sharp bounds for partition dimension of convex polytopes [7]. Wei et al. studied the partition dimension problem for cycle-related graphs [25]. Yero et al. studied the partition dimension of strong product graphs and Cartesian product graphs [26].
Gary et al. and Khuller et al. mentioned the computational complexity of metric dimension of general graphs [8, 13]. Computation of is more complex as it is more harder than computing metric dimension of a graph.
The concept of fault-tolerant partition dimension of the graph was initiated by Salman et al. [22]. Let be a partition with partition classes of the vertex set of connected graph . The partition is called fault-tolerant resolving partition of if for every pair of distinct vertices , and and differ by at least two places. The fault-tolerant partition dimension of is defined as and is denoted by . Imran et al. characterised that of all the graphs of order is [12]. Kamran et al. computed the of homogeneous caterpillar, tadpole, and necklace graphs [2, 4]. Asim et al. computed of circulant graphs with connection set in [19]. In this paper, we extend this study by considering alternate triangular cycle, mirror graph, and tortoise graphs and show that they have constant fault-tolerant partition dimension.
Chartrand et al. revealed the following basic results on .
Proposition 1 (see [6]). Let be a graph; then,(1)(2) if , where is a path(3) if , where is the complete graph
Salman et al. revealed the following basic results on .
Proposition 2 (see [21]). For ,(1)(2) if or
Proposition 3 (see [22]). (1)For , (2)For ,
The remaining part of the paper is structured in the following manner. In Section 2, we are concerned with the computation of , , and , where , , and are alternate triangular cycle, mirror graph, and tortoise graphs, respectively. Finally, we conclude the paper in Section 3, by giving future research direction.
2. Fault-Tolerant Partition Dimension of Alternate Triangular Cycle
In this section, we compute , where is alternate triangular cycle. Vertices of , for , are divided into three sets, , , and . An alternate triangular cycle is obtained by replacing alternate edge of an even cycle by . The set and are vertex set and edge set of alternate triangular cycle, respectively. Order of is . The alternate triangular cycle is shown in Figure 1.
We compute and in the following theorems.
Theorem 1. The partition dimension of alternate triangular cycle , for , is 3.
Proof. Let be a partition with 3 partition classes of vertex set of for . For , we consider , , and . It can easily be observed that is resolving partition. Case 1: for , , , where , , and are shown as follows: Case 2: for , , , where , , and are shown as follows: Case 3: for , , , where , , and are shown as follows:It is obvious from the above distinct representations that is resolving partition of ; therefore, . It follows from Proposition 1(b) that . This completes the proof.
Theorem 2. The fault-tolerant partition dimension of alternate triangular cycle , for , is 4.
Proof. In order to prove that , first, we show that . In this regard, consider be a partition with 4 partition classes of vertex set of , for . For , consider , , , and . It is easy to observe that is fault-tolerant resolving partition. (1)Case 1: for , , , where , , , and are shown as follows:(2)Case 2: for , , , where , , , and are shown as follows:(3)Case 3: for , , , where , , , and are shown as follows:It is obvious from the above distinct representations that is fault-tolerant resolving partition of , so .
Now, we prove that . For this, we show that . For a contradiction, suppose be a fault-tolerant partition basis of . There will be at least one vertex of degree 3 in one of the partition set , or . Without loss of generality, we assume that is a vertex of degree 3 that belongs to , and . Suppose that , and , , or . Without loss of generality, we assume that, at least two vertices, . As and are identical at two places, hence, it is a contradiction.
We consider the following cases when and .(1)Case 1: if , then , , , and . As , so using Pigeonhole principle, there will be similarity in the representation of two vertices at two places; hence, it is a contradiction.(2) Case 2: if two neighbours of and one neighbour of belongs to , then following cases will arise:(1) Case 2(a): if and one vertex , then and . As representation of two vertices has two identical coordinates, hence, it is a contradiction.(2) Case 2(b): if and one vertex , then, , , and . Since , so representation of two vertices will have two identical coordinates, hence, it is a contradiction.(3) Case 2(c): if and one vertex , then , , and . Since , so representation of two vertices have two identical coordinates; hence, it is a contradiction.(3) Case 3: if and two vertices , then, , , , and . Since representation of two vertices has two identical coordinates, thus, it is a contradiction.(4) Case 4: when each of , and contains one neighbour of , then we have following cases:(1)Case 4(a): if , , and , then and , which leads to a contradiction.(2) Case 4(b): if , and , then and . Since , so let . Now, , where , and , where . It is obvious that representation of two vertices is identical at two places, so a contradiction.(5) Case 5: if , at least two vertices from belong to . Without loss of generality, we suppose that ; then, , , and . It is again a contradiction.This discussion concludes the proof; hence, .
Example 1. Consider the alternate triangular cycle , as shown in Figure 1. If , where , , , and is a partition of , then the representations of vertices of are as follows: , , , , , , , , , , , , , , , , , and .
It can be seen from the above representations that is a fault-tolerant resolving partition of .
2.1. Fault-Tolerant Partition Dimension of Mirror Graph
Mirror graph is defined as the disjoint union of graph and its copy with additional edges joining each vertex of to its corresponding vertex in . The set and are vertex set and edge set of mirror graph, respectively. Mirror graph is shown in Figure 2.
Yero et al. computed the partition dimension of the mirror graph in the following lemma.
Lemma 1 (see [26]). Let be the mirror graph ; then, .
The following theorem allows us to compute .
Theorem 3. For every ,
Proof. Let be a partition set of vertices of for . Considering , , and , it is easy to observe that is fault-tolerant resolving partition of .
Now, for , consider be a partition with 4 partition classes of vertices of . , considering , , , and , are as follows:As all the above representations are different, so is fault-tolerant resolving partition of ; therefore, .
Now, we prove that ; for this, we show that . For contradiction, let be a fault-tolerant partition basis of . One of the partition sets contains at least one vertex of degree 3. Without loss of generality, we assume that is a vertex of degree 3 that belongs to , and . Suppose that and , , or . Without loss of generality, we assume that at least two vertices . As and have two identical coordinates, so it is a contradiction. Now, we discuss the following cases when and .(1)Case 1: if , then , , , and . As , so using Pigeonhole principle, representation of two vertices have two identical coordinates; therefore, it is a contradiction.(2)Case 2: if and one vertex , then , , , and . Since , so there will be similarity in the representation of two vertices at two places from and ; hence, it is a contradiction.(3) Case 3: if and two vertices , then , , , and . Since and have two identical coordinates, therefore, it is a contradiction.(4) Case 4: if , , and , then , , and . Let , and , then and , which leads to a contradiction. Now, let , , and ; then, and , which leads to a contradiction.(5) Case 5: if and at least two vertices from belong to , without loss of generality, we suppose that ; then, , , and . Again and have two identical coordinates, a contradiction.The above discussions show that , which completes the proof.
Example 2. Consider the mirror graph , as shown in Figure 2. If , where , , , and is a partition of , then the representations of vertices of are as follows: , , , , , , , and . It is obvious from the above representations that is a fault-tolerant resolving partition of .
2.2. Fault-Tolerant Partition Dimension of Tortoise Graph
The tortoise graph, denoted by , has vertex set and edge set . Tortoise graph is shown in Figure 3.
The following theorems allow us to compute and .
Theorem 4. For every , where and , .
Proof. Let be a partition set of for . with respect to , , and are as follows:It is obvious from the above distinct representations that is resolving partition of ; therefore, . It follows by Proposition 1(b) that . Hence, .
Theorem 5. For every , where and ,
Proof. Let be a partition set of for . Considering , , and , it is easy to see that is fault-tolerant resolving partition of . Now, consider be a partition with 4 partition classes of vertex set of for , where . considering , , and are as follows:Distinct representations given above show that is fault-tolerant resolving partition of ; therefore, .
Now, we prove that . For this, we show that . For contradiction, suppose be a fault-tolerant partition basis of . One of the partition sets contains at least one vertex of degree 3. Without loss of generality, we assume that is a vertex of degree 3 that belongs to , and . Suppose that , and , , or . Without loss of generality, we assume at least two vertices . As and has two identical coordinates, therefore, it is a contradiction. Now, we consider the following cases when and .(1) Case 1: if , then , , , and . As , so using Pigeonhole principle, there will be similarity in the representation of two vertices at two places, hence a contradiction.(2) Case 2: if and one vertex , then , , , and . Since , so representation of two vertices will have two identical coordinates, hence a contradiction.(3) Case 3: if and two vertices , then , , , and . Since, representation of two vertices has two identical coordinates, thus, it is a contradiction.(4) Case 4: if , and .(1) For , , , and . Let ; then , , and , which leads to a contradiction.(2) For , , , and . Let, and ; then, , , , and , which leads to a contradiction. Now, let and ; then, , , , and , which leads to a contradiction.(5) Case 5: if , contains at least two neighbours of . Without loss of generality, we suppose that ; then, , , and . Again and have two identical coordinates; hence, it is a contradiction.The above discussion follows that . Thus, .
Example 3. Consider the tortoise graph , as shown in Figure 3. If , where , , , and is a partition of , then are as follows: , , , , , , and . The above distinct representations verify that is a fault-tolerant resolving partition of .
3. Conclusion
In this paper, authors conclude that of alternate triangular cycle and tortoise graph for is 3. of alternate triangular cycle, mirror graph, and tortoise graph for is between 3 and 4. The obtained results led us to the conclusion that the discussed cyclic networks have constant partition and fault-tolerant partition dimension. and of these graphs are independent of the number of vertices of the graph. Future research can focus on computing the fault-tolerant partition dimension of alternate square and alternate pentagonal cycle.
The significance of the current work can be seen in optimization related to resource management, routing problem, and supply chain problem. These require to group certain nodes in the network to optimally approach the whole network. The least number of grouping required in such scenario can be realised as partition dimension problem. Also, the least number of grouping required to approach the whole network in case of inaccessibility of one of the group relates to fault-tolerant partition dimension of the graph.
Data Availability
All data used to support the findings of the study are included within this article and can be obtained from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.