This document provides a summary of a university course on graph theory from the 2022-2023 academic year. It was created by 5 students and contains 4 sections: an introduction, definitions and terminology, algorithms for traversing graphs, and a conclusion. The document defines key graph theory concepts such as graphs, directed and undirected graphs, vertices, edges, trees, cycles, and matrices of adjacency and incidence. It also describes algorithms for breadth-first search and depth-first search traversal of graphs.

1. La théorie des graphes
Année universitaire : 2022-2023
Réalisé par:
 AIT HLIMA Khadija
 ELASRI Amina
 Fahmi Oussama
 ashraf
 mohamed

2. Introduction
01
02
03
conclusion
04
Définitions et terminologies
Algorithmes de parcours d’un graphe

4. On suppose qu'un distributeur des produits travailler dans une entreprise de
livraison au Maroc. Cet employeur aimé transmet des produit d'Agadir à Fès et
n'est pas beaucoup de temps. Il existe plusieurs routes entre les deux villes :
 Agadir , Marakech,Beni Mellal,Khenifra,Fes
 Agadir,Essaouiraa, Marakech,Beni Mellal,Khenifra,Fes
 Agadir,Essaouiraa, Marakech,safi, benguerir ,settat ,casablanca ,mouhamadia
Rabat ,Kenitra mekness ,Fes
 …

5. Les villes reliées entre elles

6. Un réseau d’ordinateurs

7. Un graphe
A
C
B
Une boucle
Les arêtes ou les arcs
Les sommets

8. Définition d’un graphe.
Un graphe est ensembles des points appelés
sommets reliés par des segments de droites
ou de courbes appelés arêtes ( arcs).

9. Définitions et concepts de base
• D1: Un graphe Orienté G=(S,A) est
défini par la donnée d'un ensemble S
(avec |S|> 1) dont les éléments sont
appelés des sommets ou des nœuds et
d'un ensemble U (avec|A|≥ 0) dont les
éléments a ∈ A sont des arcs et
correspondent à des couples ordonnés
de sommets.
C
A
B

10. La position n’a pas d’importance
A
C
B
A
C
B
A
C
B
= =

11. L’ordre et la taille
• R1: Dans toute la suite on ne considère que des graphes finis
pour lesquels S et A sont des ensembles finis.
Si |S|= n (resp. |A|= m), on dit que le graphe G est d'ordre n
(resp. de taille m).
C
A
B le graphe G est d'ordre
n=3 et de taille m=2

12. Graphe non oriente : on a chaque couple (Si,Sj) ∈A n’est pas oriente, c’est-à-dire que (Si,Sj) est
équivalent a (Si,Sj) . Le couple (Si,Sj) est appelé une arête, et il est représente graphiquement
par Si Sj
Le graphe non oriente G2 = (S,A) avec S={A,B,C,D,E,F}
Remarques :
Le graphe de la figure contient un nœud
isole ( sommet E )
Graphe non oriente
D
C
F
A E
B

13. Graphe oriente
Graphe oriente : les couples (Si , Sj ) ∈ A sont orientes, c’est-à-dire que (Si , Sj )est un couple
ordonne ou Si est le sommet initial (i est l'extrémité initiale ), et Sj le sommet terminal (j est
l'extrémité terminale ).
Un couple (Si , Sj ) est appelé un arc. Et il est représenté graphiquement par Si →Sj
i =I(u) et j =T(u). L'arc u est une boucle si i = j
4
1
2
5 6
3
Remarques :
Le graphe de la figure est compose de deux
composants L’arc (4,4) représente une boucle
Le graphe oriente G1= (S,A ) avec S={1,2,3,4,5,6} et A={(1,2), (2,4), ), (2,5), ), (4,1), ), (4,4), ), (4,5), ), (5,4), ), (6,3)}

14. L’ordre et la taille
•D4 : On dit que j est un successeur de i, s'il existe un arc ayant i comme extrémité initiale et j comme
extrémité terminale. L'ensemble des successeurs d'un sommet i ∈ X est noté : Γ+(i).
On dit que j est un prédécesseur de i, s'il existe un arc de la forme (j,i). L'ensemble des prédécesseurs de i ∈
X est noté : Γ-(i).
Γ+(i) ∪ Γ-(i) est appelé ensemble des voisins de i.
C
A
B
D4 : Etant donné un Graphe G = (S,A) et
un arc u = (i,j) on dit que :
i et j sont deux sommets adjacents.
i et j sont incidents à l'arc u = (i,j).
ex :
A et B sont deux sommets adjacents.
A et B sont incidents à l'arc u = (i,j).
u

15. Multiplicité
Entre deux sommets donnes, il peut y avoir plusieurs arêtes , ce que l’on appelle aussi une arête multiple
- Une boucle est un arc ou une arête reliant un sommet a lui-même
e
Un graphe non-orienté est dit simple s’il ne comporte pas de boucle , et s’il ne comporte jamais plus
d’une arête entre deux sommet . Un graphe non oriente qui n’est pas simple est un multigraphe.
D8 :Soient e et v deux arcs distincts et x,y deux sommets d'un
Graphe, alors si e = (x,y) = v on dit que e et v sont deux
arcs parallèles. Un graphe sera dit Simple s'il est sans
boucles et sans arcs parallèles.

16. Graphe pondéré
Graphe non orienté pondéré
• Un graphe pondéré est un graphe
étiqueté dont toutes les étiquettes sont
des nombres réels positifs ou nuls. Ces
nombres sont les poids des liaisons
(arêtes ou arcs) entre les sommets
Graphe orienté pondéré
• Un graphe pondéré est un graphe
étiqueté dont toutes les étiquettes sont
des nombres réels positifs ou nuls avec le
graphe est oriente. Ces nombres sont les
poids des liaisons (arêtes ou arcs) entre
les sommets

17. comparaison
non orienté orienté
Chaine Chemin
Cycle Circuit

18. Cycle
Cycle : dédié au
graphe non-orienté
(chaine fermée)
.
Circuit
Circuit : dédié au
graphe orienté
(chemin fermé)
Cycle et Circuit

19. Relations entre sommets
 Dans un graphe non oriente, nœud Si est dit adjacent a un autre nœud Sj s’il existe une arête
entre Si et Sj L’ensemble des nœud voisins a un nœud Si est défini par :
T(Si) =adj(Si) = {Sj / (Si,Sj) ∈A ou (Sj,Si) ∈A }
 Dans un graphe oriente, on distingue les sommets successeurs des sommets prédécesseurs :
succ(Si) = {Sj / (Si,Sj) ∈A }
pred(Sj) = {Si / (Si,Sj) ∈A }
Relation entre arcs et sommets
Dans le cas d’un graphe non oriente , le degré d’un sommet est le nombre d arêtes dont ce sommet
est une extrémité ( les boucles étant comptées deux fois). Ce degré vaut 0 si le sommet est isole

20. Matrices d’adjacence
On peut remarque que cette matrice est symétrique. C'est
le cas pour toutes les matrices d'adjacence d'un graphe
non-orienté
D 18 : La matrice d'adjacence X = [xij] d'un graphe G non
orienté et sans arêtes parallèles et comportant n sommets est
une matrice carrée de type n , n dont chaque rangée et chaque
colonne correspondent à un sommet et dont les éléments sont
définis comme suit :
xij =
• 1 s'il existe une arête entre i
et j
• 0 autrement

21. Matrices d’incidence
D 19 : La matrice d'incidence A =[aij] d'un graphe orienté G sans
boucles de n sommets et de m arcs est une matrice de type n ,
m dont chaque ligne correspond à un sommet et chaque colonne
à un arc et dont les éléments sont définis comme suit :
aij =
• 1 si le j ème arc sort du
sommet i
• -1 si le jème arc arrive au
sommet i
• 0 autrement
On peut aussi définir la matrice d'incidence d'un graphe
non-orienté. Dans un tel graphe, il n'y a plus de notion
d'origine et d'arrivée d'une arête. On met donc +1 là où
auparavant on mettait +1 ou -1, et on met 0 ailleurs.

22. Dans le cas un graphe oriente, on peut distinguer deux types de degré d’un sommet. On note d+(s) le
degré extérieur du sommet s , c’est-à-dire le nombre d’arcs ayant s comme extrémité initiale.
 On note d-(s) le degré intérieur du sommet s , c’est-à-dire le nombre d’arcs ayant s comme extrémité
final.
Le degré du somme S est : d(s) = d+(s) + d-(s)
 L’ordre d’un graphe est le nombre de ses sommets.

23. Chaîne : Une chaine de longueur n est une suite de n arêtes qui relient un sommet
i à un autre j ou à lui-même
Des chaînes de longueur 2 :
A D C
A B C
Des chaînes de longueur 3 :
A B C D
B A D C
Des chaînes de longueur 4 :
A B C D A
A B C A D

24. Cycle : Un cycle est une chaine qui permet de partir d’un sommet et revenir
a ce sommet en parcourant une et une seule fois les autres sommets.
Exemples :
A B C D A est un cycle
A B C A est un cycle
D C A D est un cycle
A B C A D C A n'est un cycle
Chemin : c’est une chaine bien orientée
Exemples :
A B C D est un chemin
A D C A B n'est pas un chemin
A B C A D n'est pas un chemin

25. Circuit : est un cycle "bien orienté", à la fois cycle et chemin.
Exemples :
A B C A est un circuit
A B C D A est un circuit
A D C A est un cycle mais pas un circuit

26. Graphe Connexe : Un graphe connexe est un graphe dont tout couple de
sommets peut être relie par une chaine de longueur n>=1
 Un graphe est connexe s’il possède un arbre couvrant.

27. Chaîne hamiltonienne : Chaîne passant une seule fois par tous les
sommets d’un graphe.
Exemples : ABCD, ABDC, ACBD,ACBD ACDB
Chaîne eulérienne : Chaîne passant une seule fois par toutes les arêtes
d’un graphe.
Exemples :
BACBDC est une chaîne eulérienne
ACDBACB n'est pas une chaîne eulérienne

28. Cycle hamiltonien : passant une seule fois par tous les sommets d’un
graphe et revenant au sommet de départ
Cycle eulérien : passant une seule fois par toutes les arêtes d’un
graphe et revenant au sommet de départ.

29. • Graphe hamiltonien : Graphe qui possède au moins un cycle hamiltonien
• Graphe eulérien : Graphe qui possède au moins un cycle
Eulérien
Un graphe est eulérien si tous les sommets du graphe ont un degré pair

31. Parcours en largeur
Le parcours en largeur consiste comme nous venons de le dire à parcourir le graphe en passant par
les sommets directement voisins puis ceux dont il faut passer par un sommet pour les atteindre, et
ainsi de suite.
Les informations à prendre en compte pour ce type de parcours sont :
• Un sommet déjà visité ne doit pas être revisité.
• On explore les sommets successeurs directs.
L'algorithme de base repose sur l'utilisation d'une pile qui va maintenir à jour la liste des sommets à
visiter.
• Prenons le graphe suivant :

32. Parcours en profondeur
Le parcours en profondeur est le deuxième type de parcours de graphes que nous verrons ici. Il s'agit
d'un parcours où l'on explore un chemin jusqu'a ce que l'on ne puisse plus avancer dans le graphe.
Prenons le graphe suivant :

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Text Here
Content Here
Text Here
Content Here
Text Here
Content Here
Text Here
Content Here
Text Here
Content Here
Easy to change
colors, photos
and Text.
Text Here
Easy to change
colors, photos
and Text.
Text Here
Easy to change
colors, photos
and Text.
Text Here
Easy to change
colors, photos
and Text.
Text Here

64. Cycle
Cycle : dédié au
graphe non-orienté
(chaine fermée)
.
Circuit
Circuit : dédié au
graphe orienté
(chemin fermé)
Cycle et Circuit

66. You can simply impress your audience
and add a unique zing and appeal to
your Presentations. Easy to change
colors, photos and Text. Get a modern
PowerPoint Presentation that is
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believe that this Template will your
Time, Money and Reputation.
P R E S E N TAT I O N
AWESOME

67. Infographic Style
40%
30%
20%
10%
40%
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appeal to your Presentations.
30%
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appeal to your Presentations.
20%
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appeal to your Presentations.
10%
You can simply impress your audience and add a unique zing and
appeal to your Presentations.

68. Infographic Style
1
2
4
6
3
5
Content Here
You can simply impress
your audience and add a
unique zing.
Content Here
You can simply impress
your audience and add a
unique zing.
Content Here
You can simply impress
your audience and add
a unique zing.
Content Here
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unique zing.
Content Here
You can simply impress
your audience and add a
unique zing.
Content Here
You can simply impress
your audience and add a
unique zing.

69. You can simply impress your
audience and add a unique zing and
appeal to your Presentations. Easy to
change colors, photos and Text. Get
a modern PowerPoint Presentation
that is beautifully designed.
You can simply impress your
audience and add a unique zing and
appeal to your Presentations. Easy to
change colors, photos and Text. Get
a modern PowerPoint Presentation
that is beautifully designed.
You can simply impress your
audience and add a unique zing and
appeal to your Presentations. Easy to
change colors, photos and Text. Get
a modern PowerPoint Presentation
that is beautifully designed.
LOREM IPSUM DOLOR SIT AMET,
CU USU AGAM INTEGRE IMPEDIT.
LOREM IPSUM
DOLOR SIT AMET,

70. Infographic Style
A
B
D
C
Content Here
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PowerPoint
Presentation that is
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Content Here
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Presentation that is
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Content Here
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Content Here
Get a modern
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beautifully designed.
Content Here
Get a modern PowerPoint
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Content Here
Get a modern PowerPoint
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Content Here
Get a modern PowerPoint
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71. Infographic Style
Content Here
You can simply impress your audience and add a unique zing and appeal to
your Presentations. Easy to change colors, photos and Text.
Content Here
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your Presentations. Easy to change colors, photos and Text.
Content Here
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your Presentations. Easy to change colors, photos and Text.
Content Here
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your Presentations. Easy to change colors, photos and Text.
Content Title
Here

72. You can simply impress your
audience and add a unique zing
and appeal to your
Presentations.
Your Text Here
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Your Text Here
Get a modern PowerPoint Presentation that is beautifully designed. I hope and I believe that this Template will your Time, Money and
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Infographic Style

73. Infographic Style
Text Text Text
Add Text Text Text Text
Add Text Text Text Text
Add Text Text Text Text
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74. Infographic Style
Simple
PowerPoint
Simple
PowerPoint
Simple
PowerPoint
Simple
PowerPoint
Simple
PowerPoint
Your Text Here
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Easy to change
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Text.
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Easy to change
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Text.
Your Text Here
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audience and add
a unique zing and
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Easy to change
colors, photos and
Text.

75. You can simply impress
your audience and add a
unique zing and appeal
to your Presentations. I
hope and I believe that
this Template will your
Time, Money and
Reputation. You can
simply impress your
audience and add a
unique zing and appeal
to your Presentations.
Easy to change colors,
photos and Text.
Get a modern PowerPoint Presentation that is beautifully designed. I
hope and I believe that this Template will your Time.
You can simply impress your audience and add a unique zing and appeal to your
Presentations. Easy to change colors, photos and Text. I hope and I believe that this
Template will your Time, Money and Reputation. You can simply impress your audience
and add a unique zing and appeal to your Presentations. Easy to change colors, photos
and Text. You can simply impress your audience and add a unique zing and appeal to
your Presentations. Easy to change colors, photos and Text. I hope and I believe that
this Template will your Time, Money and Reputation. You can simply impress your
audience and add a unique zing and appeal to your Presentations. Easy to change
colors, photos and Text. I hope and I believe that this Template will your Time, Money
and Reputation. You can simply impress your audience and add a unique zing and
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You can simply impress your audience and add a unique zing.
Column Infographic

76. Column
Infographic
LOREM IPSUM DOLOR SIT AMET, CU USU AGAM INTEGRE
IMPEDIT.
01
PowerPoint Presentation Templates
Get a modern PowerPoint Presentation that is beautifully designed. I hope
and I believe that this Template will your Time, Money and Reputation.
Easy to change colors, photos and Text. You can simply impress your
audience and add a unique zing and appeal to your Reports and
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Get a modern PowerPoint Presentation that is beautifully designed.
LOREM IPSUM DOLOR SIT AMET, CU USU AGAM INTEGRE
IMPEDIT.
02
PowerPoint Presentation Templates
Get a modern PowerPoint Presentation that is beautifully designed. I hope
and I believe that this Template will your Time, Money and Reputation.
Easy to change colors, photos and Text. You can simply impress your
audience and add a unique zing and appeal to your Reports and
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that is beautifully designed.
Get a modern PowerPoint Presentation that is beautifully designed.

77. Fully Editable Shapes

78. Fully Editable Shapes

79. You can Resize without
losing quality
You can Change Fill
Color &
Line Color
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Fully Editable Icon Sets: A

80. You can Resize without
losing quality
You can Change Fill
Color &
Line Color
www.allppt.com
FREE
PPT
TEMPLATES
Fully Editable Icon Sets: B

81. You can Resize without
losing quality
You can Change Fill
Color &
Line Color
www.allppt.com
FREE
PPT
TEMPLATES
Fully Editable Icon Sets: C