5.1 Vizing's conjecture and related graph properties

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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.

Graph

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

Complete graph

A graph G on $n$ vertices is a complete graph if for each pair ${v}_{1},{v}_{2}\in V⇒\left({v}_{1},{v}_{2}\right)\in E$ . Call ${K}_{n}$ the complete graph on $n$ vertices.

Cartesian product graph

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

$\begin{array}{ccc}\hfill V\left(G\square H\right)& =& \left\{\left(v,w\right):v\in G,w\in H\right\}\hfill \\ \hfill E\left(G\square H\right)& =& \left\{\left(\left({v}_{1},{w}_{1}\right),\left({v}_{2},{w}_{2}\right)\right):{v}_{1}={v}_{2}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\left({w}_{1},{w}_{2}\right)\in E\left(H\right)\hfill \\ & & \phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}{w}_{1}={w}_{2}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\left({v}_{1},{v}_{2}\right)\in E\left(G\right)\right\}\hfill \end{array}$

Neighbors

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

$N\left(S\right)=\left\{v:v\in V\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\left(v,s\right)\in E\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4.pt}{0ex}}s\in S\right\}\setminus S$

and similarly the closed neighborhood is the set

$N\left[S\right]=\left\{v:v\in V\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\left(v,s\right)\in E\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4.pt}{0ex}}s\in S\right\}\cup S$

Dominating set

Given a graph $G\left(V,E\right)$ , a set $D\subseteq V$ is a dominating set if $N\left[D\right]=V$ .

Domination number

Given a graph $G\left(V,E\right)$ , the domination number of $G$ is

$\gamma \left(G\right)=min\left\{|D|:N\left[D\right]=V\right\}$

K-critical graph

A graph $G\left(V,E\right)$ , is called k-edge-critical (or k-critical , for short) if $\gamma \left(G\right)=k$ , and, $\forall$ $u,v\in V\left(G\right)$ such that $u$ and $v$ are not adjacent, $\gamma \left(G+uv\right) .

Independent set

Given a graph $G\left(V,E\right)$ , a set $I\subseteq V$ is independent if for all $v,w\in I⇒\left(v,w\right)\notin E$ . An independent set is maximal if it is not a subset of any other independent set.

Independence number

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

$i\left(G\right)=min\left(\left\{|I|:I\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{a}\phantom{\rule{4.pt}{0ex}}\text{maximal}\phantom{\rule{4.pt}{0ex}}\text{independent}\phantom{\rule{4.pt}{0ex}}\text{set}\right\}\right)$

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]

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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Sherica
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Sherica
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Tamia
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Kim
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Al
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Asali
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China
Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Azam
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Prasenjit
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silver nanoparticles could handle the job?
Damian
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Azam
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