Bounds on the partition dimension of convex polytopes

2020 ◽  
Vol 23 ◽  
Author(s):  
Jia-Bao Liu ◽  
Muhammad Faisal Nadeem ◽  
Mohammad Azeem
Keyword(s):  
Combinatorial Optimization ◽  
Computer Science ◽  
Facility Location ◽  
Robot Navigation ◽  
Convex Polytopes ◽  
Pharmaceutical Chemistry ◽  
Resolving Set ◽  
Minimum Number ◽  
Partition Dimension ◽  
Np Problem

Aims and Objective: The idea of partition and resolving sets plays an important role in various areas of engineering, chemistry and computer science such as robot navigation, facility location, pharmaceutical chemistry, combinatorial optimization, networking, and mastermind game. Method: In a graph to obtain the exact location of a required vertex which is unique from all the vertices, several vertices are selected this is called resolving set and its generalization is called resolving partition, where selected vertices are in the form of subsets. Minimum number of partitions of the vertices into sets is called partition dimension. Results: It was proved that determining the partition dimension a graph is nondeterministic polynomial time (NP) problem. In this article, we find the partition dimension of convex polytopes and provide their bounds. Conclusion: The major contribution of this article is that, due to the complexity of computing the exact partition dimension we provides the bounds and show that all the graphs discussed in results have partition dimension either less or equals to 4, but it cannot been be greater than 4.

On fault-tolerant partition dimension of graphs

2021 ◽  
Vol 40 (1) ◽  
pp. 1129-1135
Author(s):  
Kamran Azhar ◽  
Sohail Zafar ◽  
Agha Kashif ◽  
Zohaib Zahid
Keyword(s):  
Connected Graph ◽  
Intelligent Systems ◽  
Fault Tolerant ◽  
Robot Navigation ◽  
Natural Extension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Sensor Networking ◽  
Partition Dimension

Fault-tolerant resolving partition is natural extension of resolving partitions which have many applications in different areas of computer sciences for example sensor networking, intelligent systems, optimization and robot navigation. For a nontrivial connected graph G (V (G) , E (G)), the partition representation of vertex v with respect to an ordered partition Π = {Si : 1 ≤ i ≤ k} of V (G) is the k-vector r ( v | Π ) = ( d ( v , S i ) ) i = 1 k , where, d (v, Si) = min {d (v, x) |x ∈ Si}, for i ∈ {1, 2, …, k}. A partition Π is said to be fault-tolerant partition resolving set of G if r (u|Π) and r (v|Π) differ by at least two places for all u ≠ v ∈ V (G). A fault-tolerant partition resolving set of minimum cardinality is called the fault-tolerant partition basis of G and its cardinality the fault-tolerant partition dimension of G denoted by P ( G ) . In this article, we will compute fault-tolerant partition dimension of families of tadpole and necklace graphs.

scholarly journals Sharp bounds for partition dimension of generalized Möbius ladders

2018 ◽  
Vol 16 (1) ◽  
pp. 1283-1290 ◽  
Author(s):  
Zafar Hussain ◽  
Junaid Alam Khan ◽  
Mobeen Munir ◽  
Muhammad Shoaib Saleem ◽  
Zaffar Iqbal
Keyword(s):  
Lower Bounds ◽  
Metric Space ◽  
Robot Navigation ◽  
Upper Bounds ◽  
Pivotal Role ◽  
Resolving Set ◽  
Sharp Bounds ◽  
Partition Dimension

AbstractThe concept of minimal resolving partition and resolving set plays a pivotal role in diverse areas such as robot navigation, networking, optimization, mastermind games and coin weighing. It is hard to compute exact values of partition dimension for a graphic metric space, (G, dG) and networks. In this article, we give the sharp upper bounds and lower bounds for the partition dimension of generalized Möbius ladders, Mm, n, for all n≥3 and m≥2.

scholarly journals Dimensi Matriks Dan Dimensi Partisi Pada Graf Hasil Operasi Korona

2017 ◽  
Vol 4 (5) ◽  
Author(s):  
Yuni Listiana
Keyword(s):  
Ordered Set ◽  
Metric Dimension ◽  
Resolving Set ◽  
Product Graphs ◽  
Minimum Number ◽  
Partition Dimension ◽  
Corona Product

Let𝐺(𝑉,𝐸)is a connected graph.For an ordered set 𝑊={𝑤1,𝑤2,…,𝑤𝑘} of vertices, 𝑊⊆𝑉(𝐺), and a vertex 𝑣∈𝑉(𝐺), the representation of 𝑣 with respect to 𝑊 is the ordered k-tuple 𝑟(𝑣|𝑊)={𝑑(𝑣,𝑤1),𝑑(𝑣,𝑤2),…,𝑑(𝑣,𝑤𝑘)|∀𝑣∈𝑉(𝐺)}. The set W is called a resolving set of G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for 𝐺. The metric dimension of 𝐺, denoted by 𝑑𝑖𝑚(𝐺), is the number of vertices in a basis of 𝐺. Then, for a subset S of V(G), the distance between u and S is 𝑑(𝑣,𝑆)=𝑚𝑖𝑛{𝑑(𝑣,𝑥)|∀𝑥∈𝑆,∀𝑣∈𝑉(𝐺)}. Let Π=(𝑆1,𝑆2,…,𝑆𝑙)be an ordered l-partition of V(G), for∀𝑆𝑙⊂𝑉(𝐺) dan𝑣∈𝑉(𝐺), the representation of v with respect to Π is the l-vector 𝑟(𝑣|Π)=(𝑑(𝑣,𝑆1),𝑑(𝑣,𝑆2),…,𝑑(𝑣,𝑆𝑙)). The set Π is called a resolving partition for G if the 𝑙−vector 𝑟(𝑣|Π),∀𝑣∈𝑉(𝐺)are distinct. The minimum l for which there is a resolving l-partition of V(G) is the partition dimension of G, denoted by 𝑝𝑑(𝐺). In this paper, we determine the metric dimension and the partition dimension of corona product graphs 𝐾𝑛⨀𝐾𝑛−1, and we get some result that the metric dimension and partition dimension of 𝐾𝑛⨀𝐾𝑛−1respectively is𝑛(𝑛−2) and 2𝑛−1, for𝑛≥3.Keyword: Metric dimention, partition dimenstion,corona product graphs

scholarly journals Power graphs and exchange property for resolving sets

2019 ◽  
Vol 17 (1) ◽  
pp. 1303-1309 ◽  
Author(s):  
Ghulam Abbas ◽  
Usman Ali ◽  
Mobeen Munir ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Shin Min Kang
Keyword(s):  
Present Article ◽  
Linear Algebra ◽  
Finite Groups ◽  
Robot Navigation ◽  
Simple Graph ◽  
Exchange Property ◽  
Metric Dimension ◽  
Sufficient Condition ◽  
Resolving Set ◽  
Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

scholarly journals On Sharp Bounds on Partition Dimension of Convex Polytopes

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 224781-224790
Author(s):  
Yu-Ming Chu ◽  
Muhammad Faisal Nadeem ◽  
Muhammad Azeem ◽  
Muhammad Kamran Siddiqui
Keyword(s):  
Convex Polytopes ◽  
Sharp Bounds ◽  
Partition Dimension

scholarly journals Computation of the Double Metric Dimension in Convex Polytopes

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Liying Pan ◽  
Muhammad Ahmad ◽  
Zohaib Zahid ◽  
Sohail Zafar
Keyword(s):  
Source Localization ◽  
Computer Network ◽  
Convex Polytopes ◽  
The Internet ◽  
Metric Dimension ◽  
Human Beings ◽  
Source Detection ◽  
Resolving Set ◽  
Detection Problem ◽  
Rumor Spreading

A source detection problem in complex networks has been studied widely. Source localization has much importance in order to model many real-world phenomena, for instance, spreading of a virus in a computer network, epidemics in human beings, and rumor spreading on the internet. A source localization problem is to identify a node in the network that gives the best description of the observed diffusion. For this purpose, we select a subset of nodes with least size such that the source can be uniquely located. This is equivalent to find the minimal doubly resolving set of a network. In this article, we have computed the double metric dimension of convex polytopes R n and Q n by describing their minimal doubly resolving sets.

scholarly journals Computational Complexity Theory and the Philosophy of Mathematics†

2019 ◽  
Vol 27 (3) ◽  
pp. 381-439
Author(s):  
Walter Dean
Keyword(s):  
Computational Complexity ◽  
Computer Science ◽  
Complexity Theory ◽  
Philosophy Of Mathematics ◽  
Computability Theory ◽  
Computational Complexity Theory ◽  
Mathematical Problems ◽  
Np Problem

Abstract Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof.

On the metric dimension and diameter of circulant graphs with three jumps

2018 ◽  
Vol 10 (01) ◽  
pp. 1850008
Author(s):  
Muhammad Imran ◽  
A. Q. Baig ◽  
Saima Rashid ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Metric Spaces ◽  
Metric Dimension ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Infinite Class ◽  
Metric Properties ◽  
Minimum Number ◽  
Metric Basis

Let [Formula: see text] be a connected graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The diameter of [Formula: see text] is defined as [Formula: see text] and is denoted by [Formula: see text]. A subset of vertices [Formula: see text] is called a resolving set for [Formula: see text] if for every two distinct vertices [Formula: see text], there is a vertex [Formula: see text], [Formula: see text], such that [Formula: see text]. A resolving set containing the minimum number of vertices is called a metric basis for [Formula: see text] and the number of vertices in a metric basis is its metric dimension, denoted by [Formula: see text]. Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let [Formula: see text] be a family of connected graphs [Formula: see text] depending on [Formula: see text] as follows: the order [Formula: see text] and [Formula: see text]. If there exists a constant [Formula: see text] such that [Formula: see text] for every [Formula: see text] then we shall say that [Formula: see text] has bounded metric dimension, otherwise [Formula: see text] has unbounded metric dimension. If all graphs in [Formula: see text] have the same metric dimension, then [Formula: see text] is called a family of graphs with constant metric dimension. In this paper, we study the metric properties of an infinite class of circulant graphs with three generators denoted by [Formula: see text] for any positive integer [Formula: see text] and when [Formula: see text]. We compute the diameter and determine the exact value of the metric dimension of these circulant graphs.

scholarly journals P vs NP: P is Equal to NP: Desired Proof

2021 ◽  
pp. 1-11
Author(s):  
Zulfia A. Chotchaeva
Keyword(s):  
Artificial Intelligence ◽  
Computer Science ◽  
Structure Prediction ◽  
Complexity Classes ◽  
Open Problems ◽  
Exponential Time ◽  
Running Time ◽  
Algorithmic Problems ◽  
Np Problems ◽  
Np Problem

Computations and computational complexity are fundamental for mathematics and all computer science, including web load time, cryptography (cryptocurrency mining), cybersecurity, artificial intelligence, game theory, multimedia processing, computational physics, biology (for instance, in protein structure prediction), chemistry, and the P vs. NP problem that has been singled out as one of the most challenging open problems in computer science and has great importance as this would essentially solve all the algorithmic problems that we have today if the problem is solved, but the existing complexity is deprecated and does not solve complex computations of tasks that appear in the new digital age as efficiently as it needs. Therefore, we need to realize a new complexity to solve these tasks more rapidly and easily. This paper presents proof of the equality of P and NP complexity classes when the NP problem is not harder to compute than to verify in polynomial time if we forget recursion that takes exponential running time and goes to regress only (every problem in NP can be solved in exponential time, and so it is recursive, this is a key concept that exists, but recursion does not solve the NP problems efficiently). The paper’s goal is to prove the existence of an algorithm solving the NP task in polynomial running time. We get the desired reduction of the exponential problem to the polynomial problem that takes O(log n) complexity.

scholarly journals Sparse complete sets for coNP: Solution of the P versus NP problem

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Computer Science ◽  
Fundamental Question ◽  
Complexity Class ◽  
Open Problems ◽  
Precise Statement ◽  
Complete Sets ◽  
Np Problem

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is coNP. Whether NP = coNP is another fundamental question that it is as important as it is unresolved. In 1979, Fortune showed that if any sparse language is coNP-complete, then P = NP. We prove there is a possible sparse language in coNP-complete. In this way, we demonstrate the complexity class P is equal to NP.

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