Abstract
Metric-related parameters in graph theory have several applications in robotics, navigation, and chemical strata. An important such parameter is the partition dimension of graphs that plays an important role in engineering, computer science, and chemistry. In the context of chemical and pharmaceutical engineering, these parameters are used for unique representation of chemical compounds and their structural analysis. The structure of benzenoid hydrocarbon molecules is represented in the form of caterpillar trees and studied for various attributes including UV absorption spectrum, molecular susceptibility, anisotropy, and heat of atomization. Several classes of trees have been studied for partition dimension; however, in this regard, the advanced variant, the fault-tolerant partition dimension, remains to be explored. In this paper, we computed fault-tolerant partition dimension for homogeneous caterpillars , , and for , , and , respectively, and it is found to be constant. Further numerical examples and an application are furnished to elaborate the accuracy and significance of the work.
1. Introduction and Basic Terminologies
Graph theory is a widely excelling branch of mathematics that is used to model and simplify the solution of daily-life problems. Richly engaged area of research now-a-days is the application of mathematics in chemistry. Graph theory provides simple rules to obtain many qualitative predictions about the structure and reactivity of various compounds. In the chemical graph, vertices of the graph correspond to the atoms of the molecule and edges between the vertices correspond to the chemical bonds. The caterpillar graph plays an important role in chemical graph theory for studying the combinatorial and physical properties of benzenoid hydrocarbons. They have promising uses in data reduction, modeling of interactions, computational chemistry, and ordering of graphs [1]. The partition dimension for various classes of trees such as stars, caterpillars, and homogeneous firecrackers have been computed; however, the values of partition dimensions for most kind of trees are still to be solved completely [2, 3]. Among different parameters of graph theory, partition dimension of the graph is a unique and important parameter and has applications in network discovery and verification [4], mastermind games [5], and image processing [6].
In 2000, the concept of partition dimension of the graph was initiated by Chartrand et al. [7] as another variant of the metric dimension of graphs. The metric dimension of graph was first presented by Slater [8] and later by Harary et al. [9]. Consider to be a connected graph of order , where and are the set of vertices and edges, respectively. If two vertices , then the length of shortest path between and in is the 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 a closed neighbourhood of will be denoted by [10]. Consider to be an ordered subset of . The representation of with respect to is -tuple , denoted by . The subset is called a resolving set of , if representation of with respect to is distinct for all . The metric dimension of is defined as and is denoted by . In 2000, Chartrand et al. [11] observed the application of in pharmaceutical chemistry. Zehui et al. computed the metric dimension of the families of generalized Petersen graphs in [12]. Hussain et al. studied the metric dimension of 1-pentagonal carbon nanocone networks in [13].
In 2008, Hernando et al. [14] initiated the concept of fault-tolerant metric dimension of graphs. The resolving set of is called fault-tolerant if is also a resolving set for each . The fault-tolerant metric dimension of is the minimum cardinality of fault-tolerant resolving set and is denoted by . Raza et al. presented bounds on the fault-tolerant metric dimension of three infinite families of regular graphs [15] and also computed fault-tolerant metric dimension of convex polytopes [16]. Seyedi et al. discussed fault-tolerant metric dimension of circulant graphs in [17].
Let be a partition with partition classes of the vertex set of connected graph . The representation of vertex with respect to partition set is the -vector , denoted by . The partition is called a resolving partition of if the representation of all vertices in is different. We define the partition dimension of graph as , and it is denoted by . In [7], Chartrand et al. characterised the graphs with partition dimension 2 or . The partition dimensions for various classes of connected graphs have been obtained. For instance, some families of trees were discussed by Fredlina et al. [18] and Rodŕiguez et al. [19]. Amrullah studied the partition dimension problem for a subdivision of a homogeneous firecracker in [2]. Boskoro et al. [20] characterised all graphs of order , having diameter 2 with partition dimension . Mardhaningsih provided partition dimension of a thorn of the fan graph in [21]. Monica et al. discussed the problem for certain classes of series-parallel graphs [22]. Chu et al. provided the sharp bounds for partition dimension of convex polytopes and flower graphs [23]. Wei et al. studied the partition dimension problem for cycle-related graphs in [24]. Circulant graphs were discussed by Maritz et al. [25]. Firstly, Gary et al. and, later, Khuller et al. mentioned the computational complexity of metric dimension of general graphs [26, 27]. The partition dimension is a graph parameter akin to the concept of metric dimension, so its computation is also more complex.
The concept of fault-tolerant version of partition dimension of graphs was initiated by Salman et al. [28]. Let be a partition with partition classes of the vertex set of connected graph . If for each pair of distinct vertices , and differ by at least two places, then the partition is called fault-tolerant resolving partition of . The fault-tolerant partition dimension of is defined as and is denoted by . In [29], Imran et al. characterised that of all the graphs of order is . Recently, Kamran et al. have computed the of tadpole and necklace graphs in [30]. Asim et al. have computed of circulant graphs with a connection set in [31]. In this paper, we extend this study by considering homogeneous caterpillars , , and and show that they have constant fault-tolerant partition dimension. Some basic results on are stated as follows.
Salman et al. revealed the following basic results on .
Proposition 1 (see [32]). For ,(a)(b) iff or
Proposition 2 (see [28]). (a)For , (b)For , The remaining part of the article is structured in the following manner: Section 2 is devoted for the computation of , where is a homogeneous caterpillar. In Section 3, we have concluded the paper by giving future research direction and application showing significance of the current work.
2. Fault-Tolerant Partition Dimension of the Homogeneous Caterpillar Graph
A caterpillar graph is a tree having a central path with vertices . Leaves are pendent vertices those are attached to every vertex of the central path. If an equal number of leaf vertices are attached to each where , then caterpillar is called the homogeneous caterpillar and is denoted by . The set , where and , and are the vertex set and edge set of homogeneous caterpillar , respectively. The graph of homogeneous caterpillar is shown in Figure 1.
Lemma 1 (see [18]). Let be a homogeneous caterpillar with . Then, if and only if ( and ) or ( and ) or ( and ).
Lemma 2 (see [3]). Let be a homogeneous caterpillar with any integers . Then, if and only if ( and ) or ( and ).
The following theorems will allow us to compute .
Theorem 1. Let be a homogeneous caterpillar. If and , then .
Proof. Let be a partition with 3 partition classes of the vertices of . We have the following: Case (i): for The of with respect to , , and is as follows: Case (ii): for . Consider , and . The of is as follows: Case (iii): for Consider , , and . The of is as follows:As all the vertices have distinct representations, is fault-tolerant resolving partition of ; therefore, . It follows from Proposition 1 (a) and Lemma 1 that , which completes the proof.
Theorem 2. Let be a homogeneous caterpillar; if and , or and , then .
Proof. In order to prove that , we show that . Suppose on contrary that is a fault-tolerant partition basis of . One of the partition sets , or 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 and , , or . Without loss of generality, we assume that at least two vertices . As and have two identical coordinates, hence a contradiction. Now, we suppose that . We discuss the following cases: Case 1: if , then , , , and . As , by Pigeonhole principle, it is observed that two vertices have two identical coordinates in their representation, which leads to a contradiction. Case 2: if and one vertex , then , , , and . Since , the representation of two vertices will again be identical at two places, which is a contradiction. Case 3: if and two vertices , then , , , and . As and have two identical coordinates, hence a contradiction. Case 4: if , and . We have . Case 4(a): for and Consider , and . Let and ; then, and , which is a contradiction. Now, if , then and , a contradiction. Case 4(b): for and Consider, , , and . Let ; then, , , and which leads to a contradiction. Now, let and ; then and , which leads to a contradiction. Case 5: Let and at least two vertices from belong to . Without loss of generality, we suppose that ; then, , , and . Again, and have two identical coordinates, which leads to a contradiction.It is obvious from this discussion that , which completes the proof.
Theorem 3. Let be a homogeneous caterpillar; if and , then .
Proof. Let be a partition set of . The of , taking , , , and , is as follows:Distinct representations of vertices of show that is a fault-tolerant resolving partition of ; therefore, . Also, by Theorem 2, , which completes the proof.
Theorem 4. Let be a homogeneous caterpillar; if and , then .
Proof. Let be a partition set of . The of , taking , , and , is as follows:It is obvious from the representations in (5) that is a fault-tolerant resolving partition of ; therefore, . It follows from Proposition 1 (a) and Lemma 1 that , which completes the proof.
Theorem 5. Let be a homogeneous caterpillar; if and , then .
Proof. Let be a partition with 4 partition classes of the vertices of . The of with respect to , , , and is as follows:As all the representations are distinct, is a fault-tolerant resolving partition of ; therefore, . Also, from Theorem 2, . This completes the proof.
Example 1. Consider the homogeneous caterpillar , shown in Figure 2.
Let be a partition with 4 partition classes of the vertices of . The representation of vertices of , considering , , , and , is as follows:The representations in (7) shows that is a fault-tolerant resolving partition of .
The following lemma will be used in computing .
Lemma 3. If is a homogeneous caterpillar, then we develop the relations of distances of .
Proof. The relations of distances of the vertices and of homogeneous caterpillar are given as follows:(a), for , where (b), for , where (c), for , where (d), for , where (e), for (f), for (g), for
Theorem 6. Let be a homogeneous caterpillar; if , then
Proof. Let be a partition with 4 partition classes of vertices of for and 3. For , consider , , , and . For , consider , , , and . It can be observed easily that is a fault-tolerant resolving partition of , for and 3.
Now, let be a partition with 5 partition classes of vertices of , for . Partition representations of the vertices of , taking , , , , and are as follows:Unique representations of (9) show that is a fault-tolerant resolving partition of ; therefore, .
Now, we prove that , for . For this, we show that . Suppose on contrary that . Let be a partition set of for . We discuss the following cases: Case 1: when two partitioning sets of are subsets of either or . At least two vertices from , and belong to the remaining two partition sets; then, by Lemma 3 ((a) and (d)), representation of two vertices will have three identical coordinates, hence a contradiction. Now, when two partitioning sets of are subsets of and at least two vertices from , and belong to the remaining two partition sets, then by Lemma 3 (b), representation of two vertices will have three identical coordinates, hence a contradiction. Case 2: we discuss cases when two partitioning sets are a subset of union of 2 sets of vertices: Case 2(a): when two partitioning sets of are subsets of and at least two vertices from , and belong to the remaining two partition sets, then by Lemma 3 (c), representation of two vertices will have three identical coordinates, which is a contradiction. When two partitioning sets of are subsets of either and at least two vertices from , and belong to the remaining two partition sets, then by Lemma 3 (b), representation of two vertices will have three identical coordinates, which is a contradiction. Case 2(b): when two partitioning sets of are subsets of either and at least two vertices from , and belong to the remaining two partition sets, then by Lemma 3 (a), representation of two vertices will have three identical coordinates, which is a contradiction. Case 2(c): when two partitioning sets of are subsets of and at least two vertices from , and belong to the remaining two partition sets, then by Lemma 3 (e), representation of two vertices will have three identical coordinates, which is a contradiction. Finally, when two partitioning sets of are subsets of either and at least two vertices from , and belong to the remaining two partition sets, then by Lemma 3 (f), representation of two vertices will have three identical coordinates, which is a contradiction. Case 3: now we discuss cases when two partitioning sets are subsets of union of sets of vertices: Case 3(a): when two partitioning sets of are subsets of either , the remaining two partition sets will contain at least two . Without loss of generality, we suppose that belong to ; then, and , which is a contradiction. Case 3(b): when two partitioning sets of are subsets of either , and if , at least two vertices from , and belong to the remaining two partition sets, then by Lemma 3 (g), representation of two vertices will have three identical coordinates, which is a contradiction. If , then let be a vertex set not contained in . At least two vertices from , and , belong to the remaining two partition sets. Representation of these two vertices with respect to will have three identical coordinates, which is a contradiction. Case 4: when two partition sets of are subsets of , without loss of generality, we suppose these subsets belong to and : Case 4(a): at least two of belong to one of or . Without loss of generality, we suppose that and ; then, and , where , hence a contradiction. Case 4(b): one of the remaining two partition sets contain at least two . Without loss of generality, we suppose that belong to ; then, and , which is a contradiction. Case 4(c): neither two of where belong to the same partition set, nor two of belong to or . For , at least two vertices in will have representations of forms , , , , , and , which leads to a contradiction. For , at least three vertices , and belong to or . Without loss of generality, we suppose these vertices belong to ; then, or , or , and or . Representation of at least two of will be the same at three places, hence a contradiction.The above discussion shows that . This completes the proof.
Example 2. Consider the homogeneous caterpillar , shown in Figure 3.
Let be a partition with 5 partition classes of the vertices of . The representations of vertices of , considering , , , , and , are as follows:The representations in (10) shows that is a fault-tolerant resolving partition of .
3. Conclusion
In this paper, we have computed that for , and 3 is between 3 and 5. The obtained results led us to the conclusion that the structures of homogeneous caterpillars , , and have constant fault-tolerant partition dimension for , , and , respectively. Future research can focus on computing the fault-tolerant partition dimension for the classes of homogeneous caterpillar , when , and 6.
Here, we include an application of routing optimization problem that shows the significance of the current work. Consider a company that wants to pick passengers from different locations in a certain area using minimum resources and avoiding repeated visits. If locations are considered as nodes and roads connecting them as edges of a graph, then locations can be grouped together that require a single vehicle to pick the passenger. The minimum number of grouping required to represent each location uniquely can be realised as the partition dimension problem of the graph. Also, the minimum number of grouping required to uniquely represent each location even if one of the groups is inaccessible relates to fault-tolerant partition dimension of the graph.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.