# 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]

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