Resolvability and fault-tolerant resolvability structures of convex polytopes

Siddiqui, Hafız Muhammad Afzal; Hayat, Sakander; Khan, Asad; Imran, Muhammad; Razzaq, Abdul; Liu, Jia‐Bao

doi:10.1016/j.tcs.2019.08.032

Cited by 41 publications

(22 citation statements)

References 18 publications

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“…As is well known, discrete mathematics is a field of mathematics that deals with countable processes and components. One of the most significant and intriguing disciplines in discrete mathematics is graph theory [1][2][3]. Graph theory is the study of structural models called graphs, which are made up of a collection of vertices and edges.…”

Section: Introductionmentioning

confidence: 99%

Decompositions of Circulant-Balanced Complete Multipartite Graphs Based on a Novel Labelling Approach

El-Mesady

Bazighifan

2022

Journal of Function Spaces

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For applied scientists and engineers, graph theory is a strong and vital tool for evaluating and inventing solutions for a variety of issues. Graph theory is extremely important in complex systems, particularly in computer science. Many scientific areas use graph theory, including biological sciences, engineering, coding, and operational research. A strategy for the orthogonal labelling of a bipartite graph G with n edges has been proposed in the literature, yielding cyclic decompositions of balanced complete bipartite graphs K n , n by the graph G . A generalization to circulant-balanced complete multipartite graphs K n , n , ⋯ , n ⏟ m ; m , n ≥ 2 , is our objective here. In this paper, we expand the orthogonal labelling approach used to generate cyclic decompositions for K n , n to a generalized orthogonal labelling approach that may be used for decomposing K n , n , ⋯ , n ⏟ m . We can decompose K n , n , ⋯ , n ⏟ m into distinct graph classes based on the proposed generalized orthogonal labelling approach.

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Section: Introductionmentioning

confidence: 99%

Decompositions of Circulant-Balanced Complete Multipartite Graphs Based on a Novel Labelling Approach

El-Mesady

Bazighifan

2022

Journal of Function Spaces

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“…Later, this idea of metric dimension attracted much intention from the researchers, and several articles related to this concept were published; some of them are, for instance, in robot navigation [14], and applications in chemistry [5], [6]. Moreover, some of the recent articles for the reader's convenience are [11], [17], [18], [21], [22].…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On Mixed Metric Dimension of Rotationally Symmetric Graphs

Raza

Liu

2020

IEEE Access

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A vertex u ∈ V (G) resolves (distinguish or recognize) two elements (vertices or edges) v, w ∈ E(G) ∪ V (G) if d G (u, v) = d G (u, w). A subset L m of vertices in a connected graph G is called a mixed metric generator for G if every two distinct elements (vertices and edges) of G are resolved by some vertex set of L m. The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dim m (G). In this paper, we studied the mixed metric dimension for three families of graphs D n , A n , and R n , known from the literature. We proved that, for D n the dim m (D n) = dim e (D n) = dim(D n), when n is even, and for A n the dim m (A n) = dim e (A n), when n is even and odd. The graph R n has mixed metric dimension 5.

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“…Zheng et al in [15] calculated the precise values for the FTMD of the generalized wheels and some families of convex polytopes. Afzal and others in [16] gave some important results and proved that there exist some families of convex polytopes, which have unbounded MD and FTMD. Basak et al in [17] calculated the FTMD for the circulant graph.…”

Section: Introductionmentioning

confidence: 99%

Fault‐Tolerant Resolvability in Some Classes of Subdivision Graphs

Faheem

Zahid

Alrowaili

et al. 2022

Journal of Mathematics

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The concept of resolving sets (RSs) and metric dimension (MD) invariants have a wide range of applications in robot navigation, computer networks, and chemical structure. RS has been used as a sensor in an indoor positioning system to find an interrupter. Many terminologies in machine learning have also been used to diagnose the interrupter in the systems of marine and gas turbines using sensory data. We proposed a fault-tolerant self-stable system that allows for the detection of an interrupter even if one of the sensors in the chain fails. If the elimination of any element from a RS is still a RS, then the RS is considered as a fault-tolerant resolving set (FTRS), and the fault-tolerant metric dimension (FTMD) is its minimum cardinality. In this paper, we calculated the FTMD of the subdivision graphs of the necklace and prism graphs. We also found that this invariant has constant values for both graphs.

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Resolvability and fault-tolerant resolvability structures of convex polytopes

Cited by 41 publications

References 18 publications

Decompositions of Circulant-Balanced Complete Multipartite Graphs Based on a Novel Labelling Approach

Decompositions of Circulant-Balanced Complete Multipartite Graphs Based on a Novel Labelling Approach

On Mixed Metric Dimension of Rotationally Symmetric Graphs

Fault‐Tolerant Resolvability in Some Classes of Subdivision Graphs

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