Abstract
Consider to be a connected graph with a vertex set that may be partitioned into any partition set If each vertex in has a separate representation with regard to and is an ordered partition, then the set with is a resolving partition of . A partition dimension of represented by is the minimal cardinality of resolving partitions of The partition dimension of various generalised families of graphs, such as the Harary graph, Cayley graph, and Pendent graph, is given as a sharp upper bound in this article.
1. Introduction and Preliminaries
Computation is the process of analysing anything using mathematical or logical principles. An algorithm’s implementation is a well-known example of computing. An issue is deemed inherently difficult if solving it requires significant resources regardless of the algorithm used. The problem discussed in this research work has computational cost nondeterministic polynomial time, and therefore, it is called as NP-hard problem in short. The partition dimension is introduced in [1] as a natural generalization of the metric dimension.
The concept of resolving set was introduced by the authors of [2] and later introduced by the authors of [3]. In [4], the authors worked on the star partition dimension of the cycle and complete graph by using a comb product. Authors worked on the partition dimension of different classes of circulant graph [5]. The authors worked on the cartesian product of different graphs and wrote a note on the partition dimension [6]. Recently [7, 8], the partition dimensions of the various families of convex polytopes and their types of graphs, such as with pendant edges, have been calculated.
The partition dimension is used in a variety of fields, including robot navigation, strategies for the mastermind game, network discovery and verification, pattern recognition, and image processing, the Djokovic-Winkler relation, chemistry for representing, chemical compounds, all of which involve the use of hierarchical data structures [9–17].
Literature on partition dimension and its bounds on the specific families of graphs are found in [18, 19]. The antivirus properties of COVID structures are discussed in terms of partition dimension and bounds in [20], novel chemical structures of diamonded structures are discovered in [21] in terms of partition dimension bounds, carbon nanotube, and its partition dimension bounds are described in [22], hexagonal Mobius ladder networks are discussed in terms of bounds on the partition dimension in [23], Harary graphs are found in terms of partition dimension bounds in [24], alpha-boron nanotubes are available in [25], a novel chemical structure is found in terms of metric-based parameters in [26], and different generalised families of graphs are discussed in terms of resolving set and partition dimension in [27–29].
Definition 1. “Let and be the vertex set and edge set (respectively) of a simple, connected graph The least count of the edges between two vertices is called the distance between and .”
Definition 2. “Let be an -ordered set of The distance between a vertex and a is defined as A planar graph is one that can be embedded in the plane; i.e., it can be drawn on the plane so that no two edges overlap. Let ordered set of vertices of . A vertex the representations denoted by , is the -tuple distances as If every vertices have the different representations with respect to then is called the resolving set of vertices .”
Definition 3. The least number of in the resolving set is known as the metric dimension of .
Definition 4. “Let be the -ordered partition set and be the -tuple distance representations of a vertex to If the representations of differ with respect to then is the resolving partition set of a graph’s vertex set.”
Definition 5. The partition dimension of is defined as the smallest count of sections in the resolving partition set of [1].
The and can be related for any nontrivial connected graph in [1]The theorems that follow are quite useful in determining the partition dimension of a graph.
Theorem 1. (see[1]). “Let be a resolving partition of and If for all vertices Then belong to different classes of ”
Theorem 2. (see[1]). “Let be a simple and connected graph. Then,(i) is iff is a path graph(ii) is iff is a complete graph.”
2. Harary Graph
Harary is an -regular graph with order the vertex set if is even, then for some integer For each we join to and to
The arrangement of the vertices is made like developed cycle, and then, its every vertex is attached to the -vertices that come right after and the vertices that move right away Here are some limits for the Harary graph’s partition dimension.
Theorem 3. With parametric values and Let is a generalised Harary graph. Then, the partition dimension of is
Proof. In order to prove we sorted the proof into the given below categories: Case 1. Suppose you have a resolving partition set where , the unique and distinct codes of the whole vertex set of with regard to are as follows: Case 2. Suppose the resolving partition set where , the unique and distinct codes of the whole vertex set of with regard to are as follows: Case 3. Suppose the resolving partition set where , the unique and distinct codes of the whole vertex set of with regard to are as follows:There are various representations for the complete vertex set of in relation to the resolving partition set . Hence,
3. The Partition Dimension of Barycentric Subdivision of Cayley Graphs
The Cayley graphs , is a cubic graph that can be obtained as the Cartesian product of a path on two vertices and cycle on vertices. The Cayley graphs consist of an outer -cycle and inner -cycle and set of -spokes with order, size, and faces , and , respectively.
The barycentric Cayley graph can be obtained by adding new vertices between and between and , and between and modulo which is shown in the Figure 1. Its order and size become and , respectively. The cycles’ arrangements in the graph like, called the inner cycle, named as outer cycle, set of interior vertices.
Theorem 4. Let denote the barycentric subdivisions of Cayley graphs. Then, for every
Proof. Case 1. When taken as an even, For this particular case, we assume Let be a resolving partition set where , and Representations of all vertices regarding the resolving partition set The vertices on the inner cycle are represented as follows: The internal cycle vertices are represented as follows: The following is vertices’ representations on the outer cycle: Case 2. When consider as an odd, For this particular case, we assume Let be a resolving set where , and Representations of all vertices regarding the resolving partition set The vertices on the inner cycle are represented as follows:The internal cycle vertices are represented as follows:The followings are vertices’ representations on the outer cycle:With respect to the resolving partition set , the complete vertex set of has unique representations.
4. The Partition Dimension of Pendent Graph
Theorem 5. Let denote the Pendent graph. Then, for every
Proof. Let be a resolving partition set of the graph shown in Figure 2, where , and Representations of all vertices regarding the resolving partition set Case 1. When even vertices Case 2. When odd verticesWith respect to the resolving partition set the complete vertex set of has unique representations. Hence,
Theorem 6. Let denote the Pendent graph. Then, for every
Proof. Let be a resolving partition set where , and Representations of all vertices regarding the resolving partition set The representations when are shown in Tables 1–3. Case 1. When even vertices Case 2. When odd verticesEvery vertex ofgains the unique representation with respect to the resolving set; hence,
Theorem 7. Let denote the Pendent graph. Then, for every
Proof. Let be a resolving partition set where , and Representations of all vertices regarding the resolving partition set Case 1. When even vertices Case 2. When odd verticesEvery vertex ofgains the unique representation with respect to the resolving set; hence,
Theorem 8. Let denote the Pendent graph. Then, for every
Proof. Let be a resolving partition set where , and Representations of all vertices regarding the resolving partition set Case 1. When even vertices, Case 2. When odd vertices,Every vertex ofgains the unique representation with respect to the resolving set; hence,
5. Conclusion and Discussion
This article investigates the generalised Harary graph’s partition dimension’s precise boundaries, and we concluded that
For the Barycentric subdivision of Cayley Graphs ,pt
Furthermore, we also discussed the graph named as pendent Graph and this also gave the same bounds of partition dimension which is
We proved all the chosen graphs have upper bounds of partition dimension 4.
Data Availability
There are no data associated with this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. G:476-611-1442. The authors, therefore, acknowledge with thanks to DSR for technical and financial support.