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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

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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