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 On Hamilton-Connectivity and Detour Index of Certain Families of Convex Polytopes

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Sakander Hayat ◽  
Muhammad Yasir Hayat Malik ◽  
Ali Ahmad ◽  
Suliman Khan ◽  
Faisal Yousafzai ◽  
...  
Keyword(s):  
Convex Polytopes ◽  
Hamiltonian Path ◽  
Convex Polytope ◽  
Arbitrary Graph ◽  
Structural Graph ◽  
Complete Problems ◽  
Finite Set ◽  
Np Complete ◽  
Hamilton Connected ◽  
Set Of Points

A convex polytope is the convex hull of a finite set of points in the Euclidean space ℝ n . By preserving the adjacency-incidence relation between vertices of a polytope, its structural graph is constructed. A graph is called Hamilton-connected if there exists at least one Hamiltonian path between any of its two vertices. The detour index is defined to be the sum of the lengths of longest distances, i.e., detours between vertices in a graph. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering, whereas the detour index has important applications in chemistry. Checking whether a graph is Hamilton-connected and computing the detour index of an arbitrary graph are both NP-complete problems. In this paper, we study these problems simultaneously for certain families of convex polytopes. We construct two infinite families of Hamilton-connected convex polytopes. Hamilton-connectivity is shown by constructing Hamiltonian paths between any pair of vertices. We then use the Hamilton-connectivity to compute the detour index of these families. A family of non-Hamilton-connected convex polytopes has also been constructed to show that not all convex polytope families are Hamilton-connected.

scholarly journals The open-locating-dominating number of some convex polytopes

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 635-642
Author(s):  
Aleksandar Savic ◽  
Zoran Maksimovic ◽  
Milena Bogdanovic
Keyword(s):  
Planar Graphs ◽  
Convex Polytopes ◽  
Upper Bounds ◽  
Dominating Number

In this paper we will investigate the problem of finding the open-locating-dominating number for some classes of planar graphs - convex polytopes. We considered Dn, Tn, Bn, Cn, En and Rn classes of convex polytopes known from the literature. The exact values of open-locating-dominating number for Dn and Rn polytopes are presented, along with the upper bounds for Tn, Bn, Cn, and En polytopes.

scholarly journals ON THE BOUNDEDNESS OF AN ITERATION INVOLVING POINTS ON THE HYPERSPHERE

2012 ◽  
Vol 22 (06) ◽  
pp. 499-515
Author(s):  
THOMAS BINDER ◽  
THOMAS MARTINETZ
Keyword(s):  
Convex Hull ◽  
Upper Bound ◽  
Upper Bounds ◽  
Maximal Length ◽  
Unit Hypersphere ◽  
Finite Set ◽  
Set Of Points

For a finite set of points X on the unit hypersphere in ℝd we consider the iteration ui+1 = ui + χi, where χi is the point of X farthest from ui. Restricting to the case where the origin is contained in the convex hull of X we study the maximal length of ui. We give sharp upper bounds for the length of ui independently of X. Precisely, this upper bound is infinity for d ≥ 3 and [Formula: see text] for d = 2.

scholarly journals Computing Open Locating-Dominating Number of Some Rotationally-Symmetric Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1415
Author(s):  
Hassan Raza
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Upper Bound ◽  
Dominating Set ◽  
Convex Polytopes ◽  
Wireless Sensor ◽  
Environmental Processes ◽  
Dominating Number ◽  
Rotationally Symmetric ◽  
Location Detection

Location detection is studied for many scenarios, such as pointing out the flaws in multiprocessors, invaders in buildings and facilities, and utilizing wireless sensor networks for monitoring environmental processes. The system or structure can be illustrated as a graph in each of these applications. Sensors strategically placed at a subset of vertices can determine and identify irregularities within the network. The open locating-dominating set S of a graph G=(V,E) is the set of vertices that dominates G, and for any i,j∈ V(G) N(i)∩S≠N(j)∩S is satisfied. The set S is called the OLD-set of G. The cardinality of the set S is called open locating-dominating number and denoted by γold(G). In this paper, we computed exact values of the prism and prism-related graphs, and also the exact values of convex polytopes of Rn and Hn. The upper bound is determined for other classes of convex polytopes. The graphs considered here are well-known from the literature.

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

scholarly journals On extremums of sums of powered distances to a finite set of points

2012 ◽  
Vol 167 (1) ◽  
pp. 69-89 ◽  
Author(s):  
Nikolai Nikolov ◽  
Rafael Rafailov
Keyword(s):  
Finite Set ◽  
Set Of Points

scholarly journals Random Procedures for Dominating Sets in Graphs

10.37236/374 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Sarah Artmann ◽  
Frank Göring ◽  
Jochen Harant ◽  
Dieter Rautenbach ◽  
Ingo Schiermeyer
Keyword(s):  
Domination Number ◽  
Upper Bounds ◽  
Dominating Sets

We present and analyze some random procedures for the construction of small dominating sets in graphs. Several upper bounds for the domination number of a graph are derived from these procedures.

scholarly journals Analysis of Some Properties from Construction of Finite Projective Planes of Order 2 and 3

CAUCHY ◽  
2016 ◽  
Vol 4 (3) ◽  
pp. 131
Author(s):  
Vira Hari Krisnawati ◽  
Corina Karim
Keyword(s):  
Automorphism Group ◽  
Projective Plane ◽  
Symmetric Group ◽  
Block Design ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Steiner System ◽  
Incidence Relation ◽  
Finite Projective Planes ◽  
Set Of Points

<p class="abstract"><span lang="IN">In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system <em>S</em>(<em>t</em>, <em>k</em>, <em>v</em>) is a set of <em>v</em> points and <em>k</em> blocks which satisfy that every <em>t</em>-subset of <em>v</em>-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with <em>t</em> = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.</span></p><p class="abstract"><span lang="IN">In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.</span></p>

scholarly journals A hybrid VNS matheuristic for a bin packing problem with a color constraint

2021 ◽  
pp. 9-9
Author(s):  
Yury Kochetov ◽  
Arteam Kondakov
Keyword(s):  
Column Generation ◽  
Bin Packing ◽  
Packing Problem ◽  
Upper Bounds ◽  
Hard Problem ◽  
Generation Technique ◽  
Bin Packing Problem ◽  
New Variant ◽  
Finite Set ◽  
Np Hard Problem

We study a new variant of the bin packing problem with a color constraint. Given a finite set of items, each item has a set of colors. Each bin has a color capacity, the total number of colors for a bin is the unification of colors for its items and cannot exceed the bin capacity. We need to pack all items into the minimal number of bins. For this NP-hard problem we present approximability results and design a hybrid matheuristic based on the column generation technique. A hybrid VNS heuristic is applied to the pricing problem. The column generation method provides a lower bound and a core subset of the most promising bin patterns. Fast heuristics and exact solution for this core produce upper bounds. Computational experiments for test instances with number of items up to 500 illustrate the efficiency of the approach.

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