A basic tutorial on the elementary graph theory and its implementations using data structures. This tutorial deals with implementation of graphs in programs using data structures ,traversal algorithms: BFS, DFS, minimal spanning trees, Kruskal's algorithm, Prim's algorithm, Shortest path problem: Dijkastra's algorithm with graphical features.

1. GRAPHS
AUTHOR:
PRINCE KUMAR
MSC CS
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2. CONTENTS…
 INTRODUCTION
 START WITH GRAPHS
 BASIC TERMINOLOGIES OF GRAPHS
 VARIANCE OF GRAPHS
 MINIMAL SPANNING TREES
 IMPLEMENTATION OF GRAPHS
 TRAVERSAL IN GRAPHS: BFS, DFS
 KRUSKAL’s ALGORITHM
 PRIM’s ALGORITHM
 DIZKASTRA ALGORITHM OF SHORTEST PATH

3. INTRODUCTION
 This presentation is prepared as a tutorial package on basic graph
theory and algorithms and its implementation as a data structure.
 It also includes some assignments for practice on algorithms.
 Hope this presentation will help the students to understand the
concepts of elementary graph theory.
 PRINCE KUMAR

4. START WITH GRAPHS
 A Graph G(V,E) is defined as tuple where_
 V ≠ Φ is a finite set of vertices.
 E is set of edges between given vertices.
 Ex. G(V,E) is a graph as below_
V = {1,2,3,4,5}
E = {
{1,2},{2,3},{3,5},{5,4},{1,4},{2,5}
}
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5. BASIC TERMINOLOGIES OF GRAPHS
 Adjacent
 In a graph G(V,E) if u and v V such that { u , v } E then u and v are
called adjacent to each other.
 Example: in given figure:
2, 3, 4 are adjacent to 5.
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6. BASIC TERMINOLOGIES OF GRAPHS
 Degree of a vertex: deg(v)
 No. of adjacent to a given vertex is its degree.
 Example: in given figure:
vertices 2, 3 and 4 are adjacent to vertex 5
so it has degree 3 i.e. deg(5) = 3 .
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7. BASIC TERMINOLOGIES OF GRAPHS
 On the basis of degree, vertices are called_
 Isolated : 0 degree , e.g. Vertex 6,
 Pendent : 1 degree, e.g. Vertex 3,
 Odd : odd degree, e.g. Vertex 5,
 Even : even degree, e.g. Vertex 1.
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8. BASIC TERMINOLOGIES OF GRAPHS
 In a graph ,
The sum of degrees of all the vertices is twice of no. of
edges.
𝑖=1
𝑛
deg 𝑉𝑖 = 2|𝐸|

9. VARIANCE OF GRAPHS
 COMPLETE GRAPH
 A graph G(V,E) is called as complete graph and denoted as K|V| if
each vertices have degree |V|-1.
 Fig. K5
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10. VARIANCE OF GRAPHS
 N-REGULAR GRAPH
 A graph G(V,E) is called as N- regular graph if each vertices have
degree N.
 Fig: 4 Regular Graph
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11. VARIANCE OF GRAPHS
 CONNECTED GRAPH
 A graph G(V,E) is called connected if in between any two vertices there is a
path else it is disconnected.
Fig: (a) Connected (b) Disconnected
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12. VARIANCE OF GRAPHS
 WEIGHTED GRAPH
 A graph G(V,E) is called weighted if all the edges in the graph
having some cost.
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13. VARIANCE OF GRAPHS
 SUBGRAPH
 A graph S(V’,E’) is called subgraph of graph G(V,E) if V’ is subset of
V and E’ is subset of E.
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14. VARIANCE OF GRAPHS
 SPANNING SUBGRAPH
 A graph S(V,E’) is called spanning subgraph of graph G(V,E) if E’ is
subset of E.
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15. VARIANCE OF GRAPHS
 TREE
 A tree is a connected graph with no cycles.
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16. MINIMAL SPANNING TREES
 A spanning subgraph of a given graph is called its minimal
spanning tree if_
 It is a tree.
 The total weight on the all the edges in tree is minimal.
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17. IMPLEMENTATION OF GRAPHS
 A graph is represented in memory by_
 Adjacency Matrix : Through 2D arrays.
 The adjacency matrix of a graph G(V, E) is defined as _
A(G) = { a i,j }
where a i,j = { 1 or w if (vi, vj) E
0 else
}

18. IMPLEMENTATION OF GRAPHS
 Adjacency matrix for adjacent graph is as below_
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0 1 0 0 1
1 0 1 1 1
0 1 0 1 0
0 1 1 0 1
1 1 0 1 0

19. IMPLEMENTATION OF GRAPHS
 Adjacency List : Through Linked Lists.
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20. TRAVERSALS IN GRAPHS
 For searching an element in the graph generally
we use two types of traversals as below_
 Breadth First Search (BFS):
It searches a node in adjacent of a given node and so on.
It is used when it is known that the target is near to the
source.

21. TRAVERSALS IN GRAPHS
 Breadth First Search (BFS) Algorithm:
 Let the algorithm assumes A graph G(V,E) is connected is
given with source say root( r ).
 It maintains a queue to keep track of the nodes which are
traversed or which are not.
 Traversal status or say color of each node shows whether
 It is unseen : WHITE , source is yellow.
 It is seen only : YELLOW
 It is processed : BLACK

22. TRAVERSALS IN GRAPHS
 Put the root ‘r’ into the queue;
 While(queue ≠ Φ)
 u = dequeuer(queue)
 for all v in adj_list of u
 if color of v is white then
 Color v by Yelow.
 Enqueue v in queue.
 Color u by black

23. TRAVERSALS IN GRAPHS
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 Lets travers using BFS algorithm in following graph Let source is 1.
QUEUE12 5 3 4

24. TRAVERSALS IN GRAPHS
 Depth First Search (DFS):
It searches a node in the path starting of a given node
and until the end of the path, if the target is not in that
path keep backtrack on that path and select a new path
from predecessor nodes and so on .
It is used when it is known that the target is in the depth
of the path starting from the source.

25. TRAVERSALS IN GRAPHS
 Depth First Search (BFS) Algorithm:
 Let the algorithm assumes A graph G(V,E) is connected is
given with source say root( r ).
 It maintains a stack to keep track of the nodes which are
traversed or which are not .
 Traversal status or say color of each node shows whether
 It is unseen : WHITE , source is yellow.
 It is seen only : YELLOW
 It is processed : BLACK

26. TRAVERSALS IN GRAPHS
 DFS(G)
 for each u G.V
 Color u white
 Set u.Π as Null
 for each u G.V
 If color of u is white
 VISIT(u)
 VISIT(u)
 Color u yellow
 for each v adj_list of u
 If color of v is white
 Set v. Π as Null
 VISIT(v)
 Color u black

27. ASSIGNMENT
 Q. Traverse the following given graph using DFS algorithm assuming
‘F’ as a source vertex.
A
B
G
F D
E
C
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28. KRUSKAL’S ALGORITHM
 This algorithm gives MST for a given graph G(V,E).
 Steps
 Sort all the edges in ascending order by weight and keep then in a Queue.
 Make a spanning subgraph say S(V,Φ) of the given graph with no edges.
 Dequeue an edge from the queue and add in the S if it’s not making cycle
in the resultant graph.
 Follow above step until the queue will be empty.

29. KRUSKAL’S ALGORITHM
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30. PRIM’S ALGORITHM
 This algorithm also grows a MST of a given graph G(V,E) starting with
a source say root (r).
 It adds an edge of minimal weight and doesn’t making cycle in the
resultant graph to an isolated vertex.
 It uses a min heap(Q) structure to grow vertex with minimal distance
of connecting edge.
 A weight function(w) that gives the weight of the edge between two
vertices if exists.
 Initial configuration
 It assumes that the distance(d) of all other vertices are infinite from the
given source and their predecessor(Π) is NULL.

31. PRIM’S ALGORITHM
Prims_algo(G,w,r)
1. r.d <= 0
2. Q <= G.V
3. while Q is not empty
1. u <= Extract_min(Q)
2. for each v Adj_List[u]
1. If v Q and w(u,v) < v.d
1. v.Π <= u
2. v.d <= w(u,v)

32. ASSIGNMENT
 Q. Grow MST for the following graph using Prim’s algorithm assuming
the source is ‘C’.
A
B
G
F D
E
C
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33. DIZKASTRA’S ALGORITHM FOR
SHORTEST PATH
 It gives shortest path to all the
target vertices in a given
weighted, connected graph
G(V,E) with respect to a given
source vertex say ‘S’.
 It maintains a min heap let Q
to maintain the traversal in the
graph.
 DIJKASTRA_ALGO(G,S)
 Initialize the cost of all the target vertices ∞.
 Set the predecessor as Null.
 Set cost of S 0.
 Get a set R = Φ
 PUT(Q, G.V)
 WHILE Q is not Null
 u = Extract(Q)
 R = R U u
 For each vertex in adj_list of u
 Maintain the lesser cost of paths for the
adjacents of u.

34. ASSIGNMENT
 Q.Find out shortest distances for all the vertices in the following graph
assuming the source is ‘C’.
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F D
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35. THANKYOU
MAIL ME: KRPRINCE8888@GMAIL.COM