On The Partition Dimension of Cm + Pn Graph

Vertana, Hidra; Kusmayadi, Tri Atmojo

doi:10.1088/1742-6596/855/1/012058

Cited by 4 publications

(4 citation statements)

References 2 publications

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“…For example, in [4] graphs with partition dimension n − 3 are discussed. The graphs obtained by sum operation of cycle and path graphs are studied in [41]. [18] provided bounds of partition dimension of wheel related graphs.…”

Section: Introduction and Some Basic Definitionsmentioning

confidence: 99%

On the Partition Dimension of Tri-Hexagonal α-Boron Nanotube

Shabbir

Azeem

2021

IEEE Access

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The production of low-cost, small in size, and high in efficiency objects is the topic of research in almost all scientific fields, especially of engineering. In this scenario, nanotechnology becomes of great importance. To achieve these tasks, one needs to study the different physical and chemical aspects of a chemical compound. In mathematical graph theory, chemical compounds are transformed in a unique mathematical representation and then examined under various parameters for these purposes. The Partition Dimension is also one of these parametric tools, which are used to identify each atom or vertex of a chemical structure under some chosen conditions. In the present paper, we use this tool to show the unique identification of each atom of a tri-hexagonal lattice of the α-boron nanotube.INDEX TERMS α-boron nanotube, tri-hexagonal boron nanotube, molecular graph, resolving partition set, partition dimension.

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Section: Introduction and Some Basic Definitionsmentioning

confidence: 99%

On the Partition Dimension of Tri-Hexagonal α-Boron Nanotube

Shabbir

Azeem

2021

IEEE Access

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“…e concept of resolving partition sets and partition dimensions extensively appeared in the literature. For example, the graph with partition dimension |V| − 3 is discussed [6] and the graph obtained by graph operations and its corresponding partition dimension is studied in [7]. e bounds on the partition dimension for convex polytopes are studied in [8][9][10] and bounds of partition on the circulant and multipartite discussed in [11,12].…”

Section: Introductionmentioning

confidence: 99%

“…For more recent literature and results, we refer to see [18][19][20][21][22][23][24]. e graph obtained by graph operations and its corresponding partition dimension is considered in [7]. e relation between the metric dimension and partition dimension in a connected graph G is represented as follows.…”

Section: Introductionmentioning

confidence: 99%

Partition Dimension of Generalized Petersen Graph

et al. 2021

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Let G = V G , E G be the connected graph. For any vertex i ∈ V G and a subset B ⊆ V G , the distance between i and B is d i ; B = min d i , j | j ∈ B . The ordered k -partition of V G is Π = B 1 , B 2 , … , B k . The representation of vertex i with respect to Π is the k -vector, that is, r i | Π = d i , B 1 , d i , B 2 , … , d i , B k . The partition Π is called the resolving (distinguishing) partition if r i | Π ≠ r j | Π , for all distinct i , j ∈ V G . The minimum cardinality of the resolving partition is called the partition dimension, denoted as pd G . 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.

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“…As, the problem of finding the partition dimension is a generalize variant of metric dimension, therefore partition dimension is also a NPhard problem. For the discussion of graphs with partition dimension n − 3 we refer [5], graphs obtained by sum operation of cycle and path graph and its partition dimension studied in [17], [39] provide bounds of partition dimension, [14], [28] discussed circulant graph's partitioning and for complete multipartite graph [36], strong partition dimension discussed in [25], [34], whereas its local version dealt in [1], brief and detailed review of resolving partition and partition dimension, we referred the literature [2], [13], [16], [21], [31]- [33], [35], [40]. The applications of resolving partition can be found in different fields such as robot navigation [24], Djokovic-Winkler relation [8], network discovery and verification [6], in chemistry for representing chemical compounds [22], [23], strategies for the mastermind game [12] and in problems of pattern recognition and image processing, some of which involve the use of hierarchical data structures [30] for more applications see [9], [15].…”

Section: Introductionmentioning

confidence: 99%