scholarly journals On the metric dimension of rotationally-symmetric convex polytopes

2016 ◽  
Vol 3 (2) ◽  
Author(s):  
Muhammad Imran ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
A. Q. Baig
Keyword(s):  
Convex Polytopes ◽  
Metric Dimension ◽  
Symmetric Convex ◽  
Rotationally Symmetric

scholarly journals Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 727 ◽  
Author(s):  
Hassan Raza ◽  
Sakander Hayat ◽  
Xiang-Feng Pan
Keyword(s):  
Convex Polytopes ◽  
Convex Polytope ◽  
Upper Bounds ◽  
Symmetric Convex ◽  
Dominating Sets ◽  
Incidence Relation ◽  
Dominating Number ◽  
Rotationally Symmetric ◽  
Finite Set ◽  
Set Of Points

A convex polytope or simply polytope is the convex hull of a finite set of points in Euclidean space R d . Graphs of convex polytopes emerge from geometric structures of convex polytopes by preserving the adjacency-incidence relation between vertices. In this paper, we study the problem of binary locating-dominating number for the graphs of convex polytopes which are symmetric rotationally. We provide an integer linear programming (ILP) formulation for the binary locating-dominating problem of graphs. We have determined the exact values of the binary locating-dominating number for two infinite families of convex polytopes. The exact values of the binary locating-dominating number are obtained for two rotationally-symmetric convex polytopes families. Moreover, certain upper bounds are determined for other three infinite families of convex polytopes. By using the ILP formulation, we show tightness in the obtained upper bounds.

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 Non-symmetric convex polytopes and Gabor orthonormal bases

2018 ◽  
Vol 146 (12) ◽  
pp. 5147-5155
Author(s):  
Randolf Chung ◽  
Chun-kit Lai
Keyword(s):  
Convex Polytopes ◽  
Symmetric Convex ◽  
Orthonormal Bases

Metric dimension of heptagonal circular ladder

2020 ◽  
pp. 2050095
Author(s):  
Sunny Kumar Sharma ◽  
Vijay Kumar Bhat
Keyword(s):  
Open Problem ◽  
Geodesic Distance ◽  
Radial Symmetry ◽  
Metric Dimension ◽  
Plane Graphs ◽  
Path Graph ◽  
Symmetric Plane ◽  
Rotationally Symmetric ◽  
Circular Ladder ◽  
Multiple Edges

Let [Formula: see text] be an undirected (i.e., all the edges are bidirectional), simple (i.e., no loops and multiple edges are allowed), and connected (i.e., between every pair of nodes, there exists a path) graph. Let [Formula: see text] denotes the number of edges in the shortest path or geodesic distance between two vertices [Formula: see text]. The metric dimension (or the location number) of some families of plane graphs have been obtained in [M. Imran, S. A. Bokhary and A. Q. Baig, Families of rotationally-symmetric plane graphs with constant metric dimension, Southeast Asian Bull. Math. 36 (2012) 663–675] and an open problem regarding these graphs was raised that: Characterize those families of plane graphs [Formula: see text] which are obtained from the graph [Formula: see text] by adding new edges in [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, by answering this problem, we characterize some families of plane graphs [Formula: see text], which possesses the radial symmetry and has a constant metric dimension. We also prove that some families of plane graphs which are obtained from the plane graphs, [Formula: see text] by the addition of new edges in [Formula: see text] have the same metric dimension and vertices set as [Formula: see text], and only 3 nodes appropriately selected are sufficient to resolve all the nodes of these families of plane graphs.

scholarly journals Computing the edge metric dimension of convex polytopes related graphs

2020 ◽  
Vol 22 (02) ◽  
pp. 174-188
Author(s):  
Muhammad Ahsan ◽  
Zohaib Zahid ◽  
Sohail Zafar ◽  
Arif Rafiq ◽  
Muhammad Sarwar Sindhu ◽  
...  
Keyword(s):  
Convex Polytopes ◽  
Metric Dimension

scholarly journals On Constant Metric Dimension of Some Generalized Convex Polytopes

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Xuewu Zuo ◽  
Abid Ali ◽  
Gohar Ali ◽  
Muhammad Kamran Siddiqui ◽  
Muhammad Tariq Rahim ◽  
...  
Keyword(s):  
Image Processing ◽  
Euclidean Space ◽  
Global Positioning System ◽  
Metric Space ◽  
Network Theory ◽  
Convex Polytopes ◽  
Metric Dimension ◽  
Positioning System ◽  
The Family ◽  
Global Positioning

Metric dimension is the extraction of the affine dimension (obtained from Euclidean space E d ) to the arbitrary metric space. A family ℱ = G n of connected graphs with n ≥ 3 is a family of constant metric dimension if dim G = k (some constant) for all graphs in the family. Family ℱ has bounded metric dimension if dim G n ≤ M , for all graphs in ℱ . Metric dimension is used to locate the position in the Global Positioning System (GPS), optimization, network theory, and image processing. It is also used for the location of hospitals and other places in big cities to trace these places. In this paper, we analyzed the features and metric dimension of generalized convex polytopes and showed that this family belongs to the family of bounded metric dimension.

scholarly journals On Mixed Metric Dimension of Rotationally Symmetric Graphs

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 11560-11569 ◽  
Author(s):  
Hassan Raza ◽  
Jia-Bao Liu ◽  
Shaojian Qu
Keyword(s):  
Metric Dimension ◽  
Symmetric Graphs ◽  
Rotationally Symmetric ◽  
Mixed Metric

On the edge metric dimension of convex polytopes and its related graphs

2019 ◽  
Vol 39 (2) ◽  
pp. 334-350 ◽  
Author(s):  
Yuezhong Zhang ◽  
Suogang Gao
Keyword(s):  
Convex Polytopes ◽  
Metric Dimension
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