Abstract
Let be a simple connected graph. Suppose an -partition of . A partition representation of a vertex is the vector , denoted by . Any partition is referred as resolving partition if such that . The smallest integer is referred as the partition dimension of if the -partition is a resolving partition. In this article, we discuss the partition dimension of kayak paddle graph, cycle graph with chord, and a graph generated by chain of cycles. It has been shown that the partition dimension of the said families of graphs is constant.
1. Introduction and Preliminaries
The concept of partition dimension is a natural generalization of metric dimension. It was proposed in [1]. This concept came from the study of metric dimension which was defined independently in [2, 3]. This parameter of a graph was proved NP-complete problem given [4]. The metric dimension of a connected graph is based on the distance between vertices, while partition dimension is based on the distance between a vertex and a set containing some vertices. Many researchers worked on this topic. A significant number of papers were published and some of them are given in [5–9]. The problem of finding partition dimension of a connected graph is still unsolved. There are only some lower and upper bounds for the partition dimension of general connected graph available in the literature and the exact values are still open. An upper bound for the partition dimension of a tree is given in [10]. The partition dimension of certain classes of graphs is given in [11–13].
Graph theory is very vast field of applied and computational mathematics. That is why most of the applied sciences field extensively uses the graph theory. The partition dimension is also considered an applied topic of graph theory, and some of them are Djokovic–Winkler relation [14], strategies for the mastermind game [15], network discovery and verification [16], and in chemistry to represent the chemical compounds [17, 18]. In digital world, it is used to recognize the pattern, in robotics, it used for image processing, and it also plays a key role for the management of hierarchical data structures [19]. However, the applications of partition are still limited because the computational cost to compute the partition dimension is very complex. Some other applications of this concept is the navigation of robots in networks, and for interest, few areas appear in these literature [2, 20].
The partition dimension of a graph can be constant, but some graphs have bounded. Here, are some literature works. Ahmed et al. [21] computed the metric dimension of kayak paddle graph and cycle graph with chords and proved that adding an edge in a cycle graph will not affect the metric dimension of a cycle graph. Mehreen et al. [5] computed the partition dimension of fullerene graph is 3. Rajan et al. [7] described the constant partition dimension of hexagonal and honeycomb networks. Partition dimension of certain honeycomb-derived networks is computed in [6]. The partition dimension of some wheel-related graphs is computed in [22]. Rodrìguez-Velàzquez et al. [8, 9] worked on the trees and unicyclic graphs and computed the bounds on the partition dimension. For more details, we refer to [23–26].
The distance between two vertices is a shortest path between them and is denoted by . Let be an -partition of the vertex set and be an -tuple representing a unique code of w.r.t to . If all representation codes of the vertex set of the graph are unique w.r.t to , then is a resolving partition. This partition with the minimum value of is referred as partition dimension of and is denoted by .
This paper deals with the partition dimension of the kayak paddle graph, a cycle graph with a chord, and a graph generated by a chain of cycles. More precisely, we computed the partition dimension of said families of graphs and it turns out to be constant, i.e., 3.
Following theorems are very helpful in finding the partition dimension of a connected graph.
Theorem 1. (see [1]). Let be a resolving partition of of a graph and . If for all vertices , then belong to different classes of .
Theorem 2. (see [1]). Let be a simple and connected graph of order ; then,(i) is 2 iff is a path graph(ii) is iff is a complete graph
The remaining article is managed as follows. In Section 2, the partition dimension of kayak paddle graph is computed, which is constant for all three parameters given in the definition. In Section 3, the partition dimension of cycle graph with chord is discussed, and in Section 4, we study partition dimension of a family of graph generated by chain of cycles. In the end, conclusion and references have been given.
2. Kayak Paddle Graph
This section deals with the graph generated by cycle, known as kayak paddle graph, denoted by . This family of graphs can be constructed by considering two cycles of length and and a path of length with . The vertex set and edge set of kayak paddle graph are as follows, respectively:where and .
Figure 1 shows a kayak paddle graph with two cycles of length and and a path of length joining them.
2.1. Partition Dimension of Kayak Paddle Graph
In the following result, partition dimension of the kayak paddle graph generated by two cycles and a path graph has been discussed.
Theorem 3. Let be a kayak paddle graph with , , and . Then, .
Proof. To show that , we use double inequality. First of all, we prove that by constructing a resolving partition with three elements. For this, consider the following cases on and :(i)Case 1: when and and . We claim is a resolving set, where , , and , where is any of the vertex of the type , or other than the vertices of and . Then, Table 1 shows different representations of , and vertices with respect to resolving partition , where when , when , and , otherwise.(iii)Case 2: when , and . Then, we split into the following three subcases: Subcase 2.1: for , we claim is a resolving partition with , , and . Then, Table 2 shows different representations of , and with respect to partition , where when , when , and , otherwise. Subcase 2.2: for , we claim is a resolving partition with , , and . Then, Table 3 shows different representations of , and with respect to the partition , where when , when , and , otherwise. Subcase 2.3: for and , we claim is a resolving partition , , and . Then, Table 4 shows different representations of , and with respect to the partition , where when , when , and , otherwise.(iv)Case 3: when and , then we split and in the following 9 subcases: Subcase 3.1: for and , we claim is a resolving partition with , , and . Then, Table 5 shows different representations of , and with respect to the partition , where when and , otherwise. Subcase 3.2: for and , we claim is a resolving partition with , , . Then, Table 6 shows different representations of , and with respect to the partition , where when , when , and , otherwise. Subcase 3.3: for and and , we claim is a resolving partition, where , , and . Then, Table 7 shows different representations of , and with respect to the partition , where when , when , and , otherwise. Subcase 3.4: for and and , we claim is a resolving partition, where , , and . Then, Table 8 shows different representation of , and with respect to the partition , where when and otherwise are 0. Subcase 3.5: for and and , we claim is a resolving partition, where , , and . Then, Table 9 shows different representation of , and with respect to the partition , where when , when , and , otherwise. Subcase 3.6: for and and , we claim is a resolving partition, where , , and . Then, Table 10 shows different representation of , and with respect to the partition , where when , when , and , otherwise. Subcase 3.7: for and , we claim is a resolving partition, where , , and . Then, Table 11 shows different representations of , and with respect to the partition , where when and , otherwise. Subcase 3.8: for and and , we claim is a resolving with , , and . Then, Table 12 shows different representations of , and with respect to the partition , where when , when , and , otherwise. Subcase 3.9: for and and , we claim is a resolving 3-partition with , , and . Then, Table 13 shows different representation of , and with respect to the partition , where when , when , and , otherwise.Since all the vertices , , and have different representations with respect to the partition , we obtainConversely. To prove that , suppose, on contrary, . It is not possible because if and only is a path graph. Our assumption is wrong. Therefore,Hence, from inequalities (2) and (3), we conclude that
3. Partition Dimension of Cycle with Chord Graph
In this section, we consider a cycle with chord graph, denoted by . This graph is constructed by joining any two nonadjacent vertices of the cycle. A graph of cycle with chord with vertex set and is shown in Figure 2. To find the partition dimension of cycle with chord graph , it suffices to consider for given value of .
The partition dimension of the cycle with chord graph is given in the following result.
Theorem 4. Let be a cycle with chord graph, and . Then, .
Proof. To show that , first, we will prove by constructing a resolving partition having 3 elements. For this, we split and into the following two cases:(i)Case 1: for and , we claim is a resolving partition with , , and . Then, Table 14 shows different representations of with respect to the partition , where when and 0, otherwise.(iii)Case 2: when and , we claim that is a resolving partition with , , and . Then, Table 15 shows different representations of with respect to the partition , where when and 0, otherwise. Case 2.1: when and , where when and 0, otherwise. Table 16 shows different representations of with respect to the partition . Case 2.2: when and , where when and 0, otherwise. Table 17 shows different representations of with respect to the partition .Since all the vertices of have different representations with respect to the partition , therefore,Conversely. To prove that , suppose, on the contrary, . It is not possible because if and only is a path graph. Therefore,Hence, from inequalities (5) and (6), we can conclude that
4. Cycles’ Chain Graph
This section deals with the graph generated by taking and joining copies of cycles and copies of path of length 1 alternately. This graph is denoted by with vertex set and edge set , where represents the number of cycles and represents the number of vertices in a cycle, and for each cycle, , i.e., for and and for . The order and size of the chain graph is and , respectively. Figure 3 shows chain of cycles’ graph .
4.1. Partition Dimension of Chain of Cycles’ Graph
In this section, we study the partition dimension of graph generated by chain of cycles . The following theorem presents the partition dimension of the graph .
Theorem 5. If and , then .
Proof. First, we prove by constructing a resolving partition set , , and from the vertex set of . We assume the following cases on the vertex set of and on the copies of cycle graph, i.e., .
Table 18 shows different representations of with respect to the partition .
Tables 19–21 show different representations of with respect to the partition .
Third vector representations areHence, it follows from the above discussion that because all the vertices of have unique representations with respect to resolving partition set .
The reverse inequality can be easily followed from the fact that partition dimension of the graph is 2 if and only if is a path graph. Thus, we conclude that, for and ,
5. Conclusion and Discussion
In this paper, we computed partition dimension of cycle-related graph such as paddle graph and cycle graph with chord and chain of cycles’ graph. It has been shown that partition dimension of the aforementioned graph is 3. It was proved that the partition dimension of the cycle graph [1] is 3. We conclude that, by making small or significant changes in the cycle graph, do not affect its partition dimension.
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
All the authors contributed equally for the preparation of this article.
Acknowledgments
This paper was supported by Anhui Natural Science Research Project (2020) under Grant no. KJ2020A0696.