scholarly journals On Edge Covering Chain of a Graph

2021 ◽  
Vol 23 (10) ◽  
pp. 157-160
Author(s):  
L. Sathikala ◽  
Keyword(s):  
Independent Sets ◽  
Dominating Sets ◽  
Edge Cover ◽  
Graph Theoretic ◽  
Edge Covering ◽  
A Chain ◽  
Maximal Independent Sets ◽  
Domination In Graphs

Abstract- In the study of domination in graphs, relationships between the concepts of maximal independent sets, minimal dominating sets and maximal irredundant sets are used to establish what is known as domination chain of parameters. 0 ir(G)  (G)  i(G)   (G)  (G)  IR(G) In this paper, starting from the concept of edge cover, six graph theoretic parameters are introduced which obey a chain of inequalities, called as the edge covering chain of the graph G

scholarly journals Minimum connected dominating sets and maximal independent sets in unit disk graphs

2006 ◽  
Vol 352 (1-3) ◽  
pp. 1-7 ◽  
Author(s):  
Weili Wu ◽  
Hongwei Du ◽  
Xiaohua Jia ◽  
Yingshu Li ◽  
Scott C.-H. Huang
Keyword(s):  
Unit Disk ◽  
Independent Sets ◽  
Dominating Sets ◽  
Connected Dominating Sets ◽  
Unit Disk Graphs ◽  
Maximal Independent Sets

scholarly journals Self-stabilizing algorithms for minimal dominating sets and maximal independent sets

2003 ◽  
Vol 46 (5-6) ◽  
pp. 805-811 ◽  
Author(s):  
S.M. Hedetniemi ◽  
S.T. Hedetniemi ◽  
D.P. Jacobs ◽  
P.K. Srimani
Keyword(s):  
Independent Sets ◽  
Dominating Sets ◽  
Maximal Independent Sets

Trees without twin-leaves with smallest number of maximal independent sets

2020 ◽  
Vol 30 (1) ◽  
pp. 53-67 ◽  
Author(s):  
Dmitriy S. Taletskii ◽  
Dmitriy S. Malyshev
Keyword(s):  
Independent Sets ◽  
Adjacent Vertex ◽  
Maximal Independent Sets

AbstractFor any n, in the set of n-vertex trees such that any two leaves have no common adjacent vertex, we describe the trees with the smallest number of maximal independent sets.

On graphs with the third largest number of maximal independent sets

2009 ◽  
Vol 109 (4) ◽  
pp. 248-253 ◽  
Author(s):  
Hongbo Hua ◽  
Yaoping Hou
Keyword(s):  
Independent Sets ◽  
The Third ◽  
Maximal Independent Sets

Modern Applications of Graph Theory

2021 ◽  
Author(s):  
Vadim Zverovich
Keyword(s):  
Graph Theory ◽  
Molecular Epidemiology ◽  
Emergency Response ◽  
Real Life ◽  
Dominating Sets ◽  
Postgraduate Students ◽  
Graph Theoretic ◽  
Electric Vehicle Charging ◽  
Advanced Students ◽  
Doctoral Dissertations

This book discusses many modern, cutting-edge applications of graph theory, such as traffic networks and Braess’ paradox, navigable networks and optimal routing for emergency response, backbone/dominating sets in wireless sensor networks, placement of electric vehicle charging stations, pedestrian safety and graph-theoretic methods in molecular epidemiology. Because of the rapid growth of research in this field, the focus of the book is on the up-to-date development of the aforementioned applications. The book will be ideal for researchers, engineers, transport planners and emergency response specialists who are interested in the recent development of graph theory applications. Moreover, this book can be used as teaching material for postgraduate students because, in addition to up-to-date descriptions of the applications, it includes exercises and their solutions. Some of the exercises mimic practical, real-life situations. Advanced students in graph theory, computer science or molecular epidemiology may use the problems and research methods presented in this book to develop their final-year projects, master’s theses or doctoral dissertations; however, to use the information effectively, special knowledge of graph theory would be required.

scholarly journals Maximal independent sets on a grid graph

2017 ◽  
Vol 340 (12) ◽  
pp. 2762-2768 ◽  
Author(s):  
Seungsang Oh
Keyword(s):  
Independent Sets ◽  
Grid Graph ◽  
Maximal Independent Sets

Combinatorial aspects of the Löwenstein avoidance rule. Part I: the independence polynomial

2021 ◽  
Vol 77 (6) ◽  
Author(s):  
Montauban Moreira de Oliveira Jr ◽  
Jean-Guillaume Eon
Keyword(s):  
Propositional Calculus ◽  
Independent Sets ◽  
Companion Paper ◽  
Theoretical Interpretation ◽  
Vector Method ◽  
Independence Polynomial ◽  
Calculation Technique ◽  
Independence Polynomials ◽  
Maximal Independent Sets

According to Löwenstein's rule, Al–O–Al bridges are forbidden in the aluminosilicate framework of zeolites. A graph-theoretical interpretation of the rule, based on the concept of independent sets, was proposed earlier. It was shown that one can apply the vector method to the associated periodic net and define a maximal Al/(Al+Si) ratio for any aluminosilicate framework following the rule; this ratio was called the independence ratio of the net. According to this method, the determination of the independence ratio of a periodic net requires finding a subgroup of the translation group of the net for which the quotient graph and a fundamental transversal have the same independence ratio. This article and a companion paper deal with practical issues regarding the calculation of the independence ratio of mainly 2-periodic nets and the determination of site distributions realizing this ratio. The first paper describes a calculation technique based on propositional calculus and introduces a multivariate polynomial, called the independence polynomial. This polynomial can be calculated in an automatic way and provides the list of all maximal independent sets of the graph, hence also the value of its independence ratio. Some properties of this polynomial are discussed; the independence polynomials of some simple graphs, such as short paths or cycles, are determined as examples of calculation techniques. The method is also applied to the determination of the independence ratio of the 2-periodic net dhc.

Graph Models for Backbone Sets and Limited Packings in Networks

2021 ◽  
pp. 213-274
Author(s):  
Vadim Zverovich
Keyword(s):  
Fault Tolerant ◽  
Dominating Set ◽  
Facility Location Problem ◽  
Upper Bounds ◽  
Sensor Nodes ◽  
Theoretic Approach ◽  
Dominating Sets ◽  
Packing Number ◽  
Graph Theoretic ◽  
Threshold Functions

Here, a graph-theoretic approach is applied to some problems in networks, for example in wireless sensor networks (WSNs) where some sensor nodes should be selected to behave as a backbone/dominating set to support routing communications in an efficient and fault-tolerant way. Four different types of multiple domination (k-, k-tuple, α‎- and α‎-rate domination) are considered and recent upper bounds for cardinality of these types of dominating sets are discussed. Randomized algorithms are presented for finding multiple dominating sets whose expected size satisfies the upper bounds. Limited packings in networks are studied, in particular the k-limited packing number. One possible application of limited packings is a secure facility location problem when there is a need to place as many resources as possible in a given network subject to some security constraints. The last section is devoted to two general frameworks for multiple domination: <r,s>-domination and parametric domination. Finally, different threshold functions for multiple domination are considered.

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