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---
title: "Graphes - Généralités"
date: "2022-05-03"
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---
# Historique
**Un mini-peu d'histoire...**
## L. Euler, les 7 ponts de Koenigsberg:
* Existe-t-il une promenade sympa, passant **une seule fois** par les 7 ponts et revenant au point de départ?
![Les ponts c'est beau. Source: Wikipédia, <https://bit.ly/37h0yOG>](figs/Konigsberg_bridges.png){width=50%}
. . .
* Réponse: ben non!
# Utilisation quotidienne
## Réseau social
![Source, Wikipedia: <https://bit.ly/3kG6cgo>](figs/Social_Network.svg){width=40%}
* Chaque sommet est un individu.
* Chaque trait une relation d'amitié.
* Miam, Miam, Facebook.
# Utilisation quotidienne
## Moteurs de recherche
![Source, Wikipedia: <https://bit.ly/3kG6cgo>](figs/PageRanks-Example.svg){width=40%}
* Sommet est un site.
* Liens sortants;
* Liens entrants;
* Notion d'importance d'un site: combien de liens entrants, pondérés par l'importance du site.
* Miam, Miam, Google (PageRank).
# Introduction
## Définition, plus ou moins
* Un graphe est un ensemble de sommets, reliés par des lignes ou des flèches.
![Deux exemples de graphes.](figs/ex_graphes.png)
* Des sommets (numérotés 1 à 6);
* Connectés ou pas par des traits ou des flèches!
# Généralités
## Définitions
* Un **graphe** $G(V, E)$ est constitué
* $V$: un ensemble de sommets;
* $E$: un ensemble d'arêtes.
* Une **arête** relie une **paire** de sommets de $V$.
## Remarques
* Il y a **au plus** une arête $E$ par paire de sommets de $V$.
* La **complexité** d'un algorithme dans un graphe se mesure en terme de $|E|$ et $|V|$, le nombre d'éléments de $E$ et $V$ respectivement.
# Généralités
## Notations
* Une arête d'un graphe **non-orienté** est représentée par une paire **non-ordonnée** $(u,v)=(v,u)$, avec $u,v\in V$.
* Les arêtes ne sont pas orientées dans un graphe non-orienté (elles sont bi-directionnelles, peuvent être parcourues dans n'importe quel ordre).
## Exemple
::: columns
:::: column
![Un graphe non-orienté.](figs/ex_graphe_non_oriente.svg)
::::
:::: column
## Que valent $V$, $|V|$, $E$, et $|E|$?
. . .
\begin{align*}
V&=\{1, 2, 3, 4\},\\
|V|&=4,\\
E&=\{(1,2),(2,3),(2,4),(4,1)\},\\
|E|&=4.
\end{align*}
::::
:::
# Généralités
## Notations
* Une arête d'un graphe **orienté** est représentée par une paire **ordonnée** $(u,v)\neq(v,u)$, avec $u,v\in V$.
* Les arêtes sont orientées dans un graphe orienté (étonnant non?).
## Exemple
::: columns
:::: column
![Un graphe non-orienté.](figs/ex_graphe_oriente.svg)
::::
:::: column
## Que valent $V$, $|V|$, $E$, et $|E|$?
. . .
\begin{align*}
V&=\{1, 2, 3, 4\},\\
|V|&=4,\\
E&=\{(1,2),(2,3),(2,4),(4,1)\},\\
|E|&=4.
\end{align*}
::::
:::
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