Graph theory

1. SUBITTED BY:-AKHILESH SINGH
ROLL NUMBER:-51910079

2. GRAPH
 A data structure that consists of a set of
nodes(vertices) a set of edges that
relate the nodes to each other.
 The of edged describes relationships
among the vertices.
 A graph G is defined as follows.
G=(V,E)
V(G): a finite ,nonempty set of vertices
E(G): a set of edges (pairs of vertices)

3. Examples of Graphs
 V={0,1,2,3,4}
 E={(0,1),(1,2),(0,3),(3,0),(2,2),(4,3)}
0
4
1
3
2

4. Directed graph and undirected
graph
 Directed graph
 directed edge
 Undirected graph
 undirected edge
1 2
1
2

5. Weighted graph
1 2
3
4
5
6
8
3
4
7
53
7
9
2
2
4

6. Graph terminology
 Adjacent nodes: two nodes are said to adjacent if they are
connected by an edge
 Path: a sequence of vertices that connect two
 Complete graph : a graph in which every vertex is directly
connected to every other vertex.
4 5 is adjacent to 45

7.  Cycle : a cycle is a simple path with the same
start and end vertex.
 Degree : the degree of vertex is the no. of
edges incident on vertex.
e.g ., d(a)=3, d(d)=2.
a
s
r
d
v t h

8.  loops: edges connect a vertex to itself
 Simple graph a simple graph has no loop ,no
multiedge.
 Complete graph : a graph in which every vertex is
directly connected to every other vertex.
5

9. Connections
 An undirected graph are connected if there is a
path between any two vertices
 Directed graph are strongly connected if there is a
path from any one vertex to any other
 Directed graph are weakly connected if there is a
path between any two vertices
ignoring direction .
A complete graph has an edge between every pair of
vertices.

10. Graph searching
 Problem: find a path between two nodes
of the graph .
 Method:
1: depth-first-search (DFS)
2: breadth-first-search (BFS)

11. Depth-first-search (DFS)
 What is the idea behind DFS?
travel as far as you can down a path.
back up as little as possible when you reach a “dead
end”(i.e., next vertex has been “marked” or there is no next vertex)
 DFS can be implemented efficiently using a
stack

12. DFS…..
set found to false
stack.Push(stackVertex)
DO
stack.Pop(vertex)
IF vertex == endVertex
set found to true
ELSE
push all adjacent vertices onto stack
WHILE !stack.IsEmpty()AND !found
IF(!found)
Write “path does not exist”.

13. Breadth-first-searching
(BFS)
 what is idea behind BFS ?
 Look at all possible paths at the same the same depth
You go at a level
 Back up as far as possible when you reach a “dead end”
(i.e., next vertex has been “marked” or there is no next vertex)

14. Implementation of graph

37. Thank you
mam