topological sort bfs dfs biconnected components

1. Bhagwan Mahavir College Of Engineering &Technology

2.  BHAM RAHELA A (150063131003)
 AGRAVAT DIVYA H (150063131001)
 BODRA BHUMI A (150063131004)
 CHODVADIYA JYOTSNA B (150063131005)

3.  Traversing a Graph
 Breadth First search(BFS)
 Depth first search(DFS)
 Topological sort
 Connected component

4.  A graph search (or traversal) technique visits every
node exactly one in a systematic fashion.
 Two standard graph search techniques have been
widely used:
 DFS
 BFS
 In the case of rooted binary trees, three recursive
traversal techniques are widely used:
 InorderTraversal
 PreorderTraversal
 PostorderTraversal

5.  Can be used to attempt to visit all nodes of a
graph in a systematic manner
 The basic idea behind this algorithm is that it
traverses the graph using recursion
 In DFS, go as far as possible along a single
path until reach a dead end (a vertex with no
edge out or no neighbor unexplored) then
backtrack

6. DFS-iterative (G, s): //where G is graph and s is source vertex.
let S be stack
S.push( s ) // inserting s in stack
mark s as visited.
while ( S is not empty): // pop a vertex from stack to visit next
v = S.top( )
S.pop( )
//push all the neighbours of v in stack that are not visited
for all neighbours w of v in Graph G:
If w is not visited :
S.push( w )
mark w as visited DFS-recursive(G, s):
mark s as visited
for all neighbours w of s in Graph G:
if w is not visited:
DFS-recursive(G, w)

8.  In BFS, one explore a graph level by level away (explore all
neighbors first and then move on)
The breath first forest is a collection of a tree in which
the traversal starting vertex serve as a root of first tree.
 Rule 1 −Visit unvisited vertex. Mark it visited.
Display it. Insert it in a queue.
 Rule 2 − If no vertex found, remove the first
vertex from queue.
 Rule 3 − Repeat Rule 1 and Rule 2 until queue
is empty.

9. BFS (G, s)
//where G is graph and s is source node.
let Q be queue.
Q.enqueue( s )
//inserting s in queue until all its neighbor vertices are marked.
mark s as visited.
while ( Q is not empty)
// removing that vertex from queue, whose neighbor will be visited now.
v = Q.dequeue( )
//processing all the neighbors of v
for all neighbors w of v in Graph G
if w is not visited
Q.enqueue(w)
//stores w in Q to further visit its neighbor
mark w as visited.

11. node level[ node ] s(source node) 0 1 1 2 1 3 2 4 2 5 2 6 2 7 3node level[ node ] s(source node) 0 1 1 2 1 3 2 4 2 5 2 6 2 7 3node level[ node ] s(source node) 0 1 1 2 1 3 2 4 2 5 2 6 2 7 3node level[ node ] s(source node) 0 1 1 2 1 3 2 4 2 5 2 6 2 7 3
node Level
[ node ]
s(source node) 0
1 1
2 1
3 2
4 2
5 2
6 2
7 3

12.  Topological sorting of vertices of
a Directed Acyclic Graph is an
ordering of the vertices
v1,v2,...vn in such a way, that if
there is an edge directed
towards vertex vj from vertex vi,
then vi comes before vj

13.  To understand the sorting with the help of
two algorithm
 DFS based algorithm (using stack)
 Source removal algorithm (removing source)
 Use divide and conquer method

14.  Find node with no incoming vertex and
remove it with its out going edges(if more
then such vertex then select randomly)
 Note that vertex which is deleted
 All the recorded vertexes gives
topologically sorted list

16. A topological sorting
of this graph is:
0-1-2-3-4-5

17.  Articulation point: An Articulation point in a
connected graph is a vertex that, if delete,
would break the graph into two or more
pieces

18. 1
After removing articulation point 1
two disjoint graph

19.  Biconnected graph: A graph with no
articulation point called biconnected. In
other words, a graph is biconnected if and
only if any vertex is deleted, the graph
remains connected.

20.  Biconnected component:
A biconnected component of a graph is a
maximal biconnected subgraph- a
biconnected subgraph that is not properly
contained in a larger biconnected
subgraph.
A graph that is not biconnected can divide
into biconnected components, sets of
nodes mutually accessible via two distinct
paths.

21.  Steps to find articulation points
 The root of the DFS tree is an articulation
point.
 A leaf node of DFS tree is not an articulation
point
 If u is an internal node then it is not
articulation point if and only if from every
child w and u is possible to reach an ancestor
of u using only a pathmade up of descendant
of w and back edge

22.  www.hackerearth.com
 www.csie.ntu.edu.tw
 www.boost.org
 www.khanacademy.org