Abstract
Let be the connected graph. For any vertex and a subset , the distance between and is . The ordered -partition of is . The representation of vertex with respect to is the -vector, that is, . The partition is called the resolving (distinguishing) partition if , for all distinct . The minimum cardinality of the resolving partition is called the partition dimension, denoted as . In this paper, we consider the upper bound for the partition dimension of the generalized Petersen graph in terms of the cardinalities of its partite sets.
1. Introduction
The graphs considered are simple, undirected, and without loops. The classical distance between the two vertices is the length of the shortest path between them. For an ordered set , the ordered -tuple,is known as metric representation of with respect to . When the vertices of the graph have distinct representation, the set is termed as the resolving (distinguishing) set which contains minimum number of vertices. The minimum cardinality of such a resolving set is the metric dimension of the , denoted as . Slater introduced this concept in 1975 [1], after which Melter and Harary independently renamed it the resolving set [2]. In the graph’s theoretical study, this concept is called a metric basis or basis set. The concept of metric dimension does find applications in chemistry [3], else problem concerning pattern recognition and image processing, some involves the utilization of hierarchical data structures [4].
For an ordered-partition , the partition representation of the vertex with respect to iswhere denotes the distance between the vertex and , that is,
We can affirm that is the resolving partition of the graph if different vertices have unique partition representation, that is, , where . The partition dimension of the graph is than the minimum number of resolving partition set in . It is denoted as . This concept was introduced by Chartrand et al. in 2000 [5].
The concept of resolving partition sets and partition dimensions extensively appeared in the literature. For example, the graph with partition dimension is discussed [6] and the graph obtained by graph operations and its corresponding partition dimension is studied in [7]. The bounds on the partition dimension for convex polytopes are studied in [8–10] and bounds of partition on the circulant and multipartite discussed in [11, 12]. The partition dimension of chemical structure fullerenes graphs is studied in [13], and on the bounded partition, dimension of the Cartesian product of graphs are studied in [14]. Furthermore, in [15], yielded bounds for the subdivision of different graphs, than in [16] presented the bounds on tree graphs. The bounds of unicyclic graphs in the form of subgraphs are considered in [17]. For more recent literature and results, we refer to see [18–24]. The graph obtained by graph operations and its corresponding partition dimension is considered in [7]. The relation between the metric dimension and partition dimension in a connected graph is represented as follows.
Theorem 1 (see [3]). For a nontrivial connected graph ,
Some of the known results are related to the parameter of partition dimension of the graphs.
Theorem 2 (see [5]). Let be a partition resolving set of and . If , for all vertices , then belongs to different subsets of .
Theorem 3 (see [5]). Let be a simple and connected graph; then,(i) is two iff is the only path graph(ii) is iff is a complete graph
2. Generalized Petersen Graph
Generalized Petersen graph , where and , is a graph having the vertex setand the edge set
Petersen graph is a cubic graph, and it has three types of edges according to the definition of generalized Petersen graphs; the edges between the vertices and are called outer edges, the edges produced by the vertices and are called inner edges, and the edges created by and are called spokes. The vertices are called outer vertices and are known as the inner vertices. The generalized Petersen graph is shown in Figure 1. The study on the resolving set and metric dimension of Petersen and generalized Petersen graphs, and we refer the readers to [25–27]. Motivated by these results on the metric dimension of the generalized Petersen graphs, we study the problem of partition dimension and have shown the sharp upper bound.
Let be the positive integer called the modulus. We can say that two integers and are congruent modulo m if is divisible by m. It can be written as follows.
, for some positive integer . The symbol denotes the smallest positive integer x.
, where is the remainder when is divided by as many times as possible.
The difference between and is the first one is the equality relation, while the second one is the equivalence relation.
In this paper, we employed the equivalence relation because it can have many solutions for . The partition dimension of the generalized Petersen graph is solved while taking the (mod 4), that is, for .
3. Results on the Generalized Petersen Graph
In this section, we provide sharp bounds on the partition dimension of the generalized Petersen graph .
In the following section, , , and is 1 if the representation of the vertices belongs to the or , otherwise 0.
Theorem 4. Let is a generalized Petersen graph of order , when . Then, .
Proof. Let , where . The proof is exhibited by considering two cases.
Case 1. When is odd.
Assume the partition resolving set , where , , , and . Now, we exhibit the representations of vertices of with respect to .
The representations of the outer vertices; if ; if ; if ; if :Further representation of outer vertices are if , if , if , if , and if .
The above representations indicate that there exist two no vertices having the exact representation in the outer cycle.
Now, the representations of the inner vertices are shown:From the representation of inner vertices, it is evident that no two vertices have the exact representation in the inner cycle.
Case 2. When is even.
Let the partition resolving set be , where , , , and ; the following are the representations of the entire vertex set of with respect to .
The representation of the outer vertices:The representation of outer vertices within themselves is unique, with no two vertices consisting of the exact representation.
Now, we exhibit the representations of the inner vertices:The representations indicate that no two inner vertices among them have the exact representation.
The outer and inner vertices in the graph of have distinct representations. Thus, , where , , , and is the partition resolving set for odd . Moreover, , where , , , and is the partition resolving for even . Hence,
From the following theorem is 1, if the representation of the vertices belongs to the or , otherwise .
Theorem 5. Let is a generalized Petersen graph with , . Then, pd .
Proof. Let , .
Case 3. Let the partition resolving set be , where , , , , and ; the following are the representations of the entire vertex set of with respect to .
The representation of the outer vertices: The representation of outer vertices between them is distinct concerning .
The representation of the inner vertices: The inner vertices have unique representation with respect to . Moreover, the outer and inner vertices have distinct representation among them. Thus, , where , , , , and , is the partition resolving set for . Hence,
Theorem 6. Let be a generalized Petersen graph with , when . Then, pd .
Proof. Let , . The proof is split into two cases.
Case 4. When is odd.
Let the partition resolving set be , where , , , , and .
The representation of the outer vertices:The representation of the outer cycle vertices is distinct concerning .
Now, the representation of the inner vertices:The representations of inner cycle vertices with respect to are distinct. Both the outer and inner cycle vertices show distinct representations. Thus, , where , , , , and , is the partition resolving set when is odd.
Case 5. When is even.
Let the partition resolving set be , where , , , , and .
The representation of the outer vertices:The representations of the outer cycle vertices are distinct concerning .
Now, the representations of the inner vertices:The representations of the inner cycle vertices are distinct. Also, outer and inner vertices are distinct as well with respect to . Thus, , where , , , , and , is the partition resolving set, when is even. From above facts, we have
Theorem 7. Let be a generalized Petersen graph with and . Then, pd .
Proof. When , , and . The proof is divided in two cases.
Case 6. When is odd.
Let the partition resolving set be , where , , , , and . The representation of vertices with respect to is presented.
The representations of the outer vertices:The representation of the outer cycle vertices with respect to is distinct.
Now, the representation of the inner vertices:The representations of the inner cycle vertices are distinct with respect to . Also, the outer and inner vertices have distinct representations as well. Thus, , where , , , , and , is the partition resolving set when is odd.
Case 7. When is even.
Let the partition resolving set be , where , , , , and .
The representation of the outer vertices:The representation of the outer cycle vertices with respect to is distinct.
Now, the representation of the inner vertices:The representation of the inner cycle vertices is distinct. Moreover, the outer and inner vertices are distinct as well. Thus, , where , , , , and , is the partition resolving set when is even. From all these facts, we conclude that
Example 1. Let be a generalized Petersen graph; then, .
Proof. Let the partition resolving set be , where , , , , and . The representation of the entire vertex set of with respect to is presented in Table 1.
4. Conclusion
In this paper, authors provide the sharp bounds of the partition dimension for the generalized Petersen network , and by using Theorem 1.3 for lower bound, we concluded that
Theorem 4 discusses the partitioning of generalized Petersen network when , and Theorems 5–7 contain the partitioning when , respectively.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
The authors contributed equally to this paper.
Acknowledgments
The work of Hassan Raza was supported by the post-doctoral funding of University of Shanghai for Science and Technology, under Grant no. 10-20-303-302.