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Vizing's Conjecture is a lower bound for the domination number of the Cartesian Product of two graphs in terms of the domination number of the separate graphs. This module addresses observations about certain graph properties that can be assumed for Vizing's Conjecture and a related conjecture on independent domination numbers.

Introduction

Preliminary definitions

Graph

A graph is a set G ( V , E ) with V a set of vertices and E a set of edges or vertex pairs. Two vertices v 1 , v 2 V are adjacent if the vertex pair ( v 1 , v 2 ) are in E . Graphs are a common model for networks.

A graph

Complete graph

A graph G on n vertices is a complete graph if for each pair v 1 , v 2 V ( v 1 , v 2 ) E . Call K n the complete graph on n vertices.

Complete graph on 4 vertices

Cartesian product graph

Given graphs G and H the Cartesian Product Graph is defined to be G H with

V ( G H ) = { ( v , w ) : v G , w H } E ( G H ) = { ( ( v 1 , w 1 ) , ( v 2 , w 2 ) ) : v 1 = v 2 and ( w 1 , w 2 ) E ( H ) or w 1 = w 2 and ( v 1 , v 2 ) E ( G ) }

The cartesian product of 2 complete graphs makes a "cheese block"

Neighbors

Given a graph G ( V , E ) and a set S V then we define the neighbors of S to be the set

N ( S ) = { v : v V and ( v , s ) E for some s S } S

and similarly the closed neighborhood is the set

N [ S ] = { v : v V and ( v , s ) E for some s S } S

Dominating set

Given a graph G ( V , E ) , a set D V is a dominating set if N [ D ] = V .

A star graph showing 2 dominating sets (red and cyan)

Domination number

Given a graph G ( V , E ) , the domination number of G is

γ ( G ) = min { | D | : N [ D ] = V }

K-critical graph

A graph G ( V , E ) , is called k-edge-critical (or k-critical , for short) if γ ( G ) = k , and, u , v V ( G ) such that u and v are not adjacent, γ ( G + u v ) < k .

Independent set

Given a graph G ( V , E ) , a set I V is independent if for all v , w I ( v , w ) E . An independent set is maximal if it is not a subset of any other independent set.

A maximal independent set (cyan)

Independence number

Given a graph G ( V , E ) , the independence number denoted i ( G ) is defined by

i ( G ) = m i n ( { | I | : I is a maximal independent set } )

Domination theory

Domination Theory is an emerging field in Graph Theory addressing how to find dominating sets for certain graphs and important models in the theory.

Applications

Domination Theory is a very interesting subfield of graph theory because it has many real-world applications. Finding a minimum set whose closed neighborhood encompasses a network has obvious implications for minimum-cost ways of altering a network, or cheaply distributing goods throughout a network. For example, dominating set theory can help cell phone companies place a minimum number of towers to insure coverage for all of its clients. Similarly, dominating set theory can be useful for social marketing, in order to succesfully spread news about a product by using a minimal number of advertisements. In a less business-minded view, domination theory can help modeling the squares which are connected by the moves of a chess piece (such as the queen). This can be useful for solving problems like the maximum-placement problem, for an arbitary chess board, or for pieces with different movements. Lastly, domination theory can also have applications in facility location problems, such as finding the minimum distance to travel to one out of a set of locations (such as a police station). [link]

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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