Depth first traversal and connected components(data structure algorithms)

1. By:
A.Bhuvaneshwari,M.SC(cs)

2. • A Graph is a non-linear data structure consisting of nodes
and edges. The nodes are sometimes also referred to as
vertices and the edges are lines or arcs that connect any
two nodes in the graph. More formally a Graph can be
defined as, In the above graph,
• V = {a, b, c, d, e}
• E = {ab, ac, bd, cd, de}

3. • Depth First Traversal (or Search) for a graph is similar
to Depth First Traversal of a tree. The only catch here is,
unlike trees, graphs may contain cycles, a node may be
visited twice. To avoid processing a node more than
once, use a boolean visited array.
• Depth First Search (DFS) algorithm traverses a graph in
a depthward motion and uses a stack to remember to get
the next vertex to start a search, when a dead end occurs
in any iteration. (It will pop up all the vertices from the
stack, which do not have adjacent vertices.

4. • Undirected graph
• Biconnectivity
• Euler circuits
• Directed graph
• Findind strong components

5. • Depth First Search (DFS) algorithm traverses a graph in
a depthward motion and uses a stack to remember to get
the next vertex to start a search, when a dead end occurs
in any iteration.

6. • As in the example given above, DFS algorithm traverses
from S to A to D to G to E to B first, then to F and lastly to
C. It employs the following rules.
• Rule 1 − Visit the adjacent unvisited vertex. Mark it as
visited. Display it. Push it in a stack.
• Rule 2 − If no adjacent vertex is found, pop up a vertex
from the stack. (It will pop up all the vertices from the
stack, which do not have adjacent vertices.)
• Rule 3 − Repeat Rule 1 and Rule 2 until the stack is
empty.

7. • Initialize the stack.

8. • Mark S as visited and put it onto the stack.Explore any
unvisited adjacent node from S. We have three nodes
and we can pick any of them. For this example, we shall
take the node in an alphabetical order.

9. • Mark A as visited and put it onto the stack. Explore any
unvisited adjacent node from A. Both S and D are
adjacent to A but we are concerned for unvisited nodes
only.

10. • Visit D and mark it as visited and put onto the stack.
Here, we have B and C nodes, which are adjacent
to D and both are unvisited. However, we shall again
choose in an alphabetical order.

11. • We choose B, mark it as visited and put onto the stack.
Here B does not have any unvisited adjacent node. So,
we pop B from the stack.

12. • We check the stack top for return to the previous node
and check if it has any unvisited nodes. Here, we
find D to be on the top of the stack.

13. • Only unvisited adjacent node is from D is C now. So we
visit C, mark it as visited and put it onto the stack.
As C does not have any unvisited adjacent node so we
keep popping the stack until we find a node that has an
unvisited adjacent node. In this case, there's none and
we keep popping until the stack is empty.

14. • Depth-first search, or DFS, is a way to traverse the
graph. Initially it allows visiting vertices of the graph only,
but there are hundreds of algorithms for graphs, which
are based on DFS. Therefore, understanding the
principles of depth-first search is quite important to move
ahead into the graph theory. The principle of the
algorithm is quite simple: to go forward (in depth) while
there is such possibility, otherwise to backtrack.
• For most algorithms boolean classification unvisited /
visited is quite enough, but we show general case here.

15. • Initially all vertices are white (unvisited). DFS starts in
arbitrary vertex and runs as follows:
• Mark vertex u as gray (visited).
• For each edge (u, v), where u is white, run depth-first
search for u recursively.
• Mark vertex u as black and backtrack to the parent.
• Example. Traverse a graph shown below, using DFS.
Start from a vertex with number 1.

16. • Source graph.
• Mark a vertex 1 as gray.
• There is an edge (1, 4) and a vertex 4 is unvisited. Go
there.
• Mark the vertex 4 as gray.
• There is an edge (4, 2) and vertex a 2 is unvisited. Go
there.
• Mark the vertex 2 as gray
• There is an edge (2, 5) and a vertex 5 is unvisited. Go
there.
• Mark the vertex 5 as gray.
• There is an edge (5, 3) and a vertex 3 is unvisited. Go
there.
• Mark the vertex 3 as gray.
• There are no ways to go from the vertex 3. Mark it as

17. • There are no ways to go from the vertex 5. Mark it as black
and backtrack to the vertex 2.
• There are no more edges, adjacent to vertex 2. Mark it as
black and backtrack to the vertex 4.
• There is an edge (4, 5), but the vertex 5 is black.
• There are no more edges, adjacent to the vertex 4. Mark it as
black and backtrack to the vertex 1.
• There are no more edges, adjacent to the vertex 1. Mark it as
black. DFS is over.
• As you can see from the example, DFS doesn't go through all
edges. The vertices and edges, which depth-first search has
visited is a tree. This tree contains all vertices of the graph (if it
is connected) and is called graph spanning tree. This tree
exactly corresponds to the recursive calls of DFS.
• If a graph is disconnected, DFS won't visit all of its vertices.
For details, see finding connected components algorithm.

18. • An undirected graph is said to be a biconnected graph, if
there are two vertex-disjoint paths between any two
vertices are present. In other words, we can say that
there is a cycle between any two vertices.
• We can say that a graph G is a bi-connected graph if it is
connected, and there are no articulation points or cut
vertex are present in the graph.

19. • To solve this problem, we will use the DFS traversal.
Using DFS, we will try to find if there is any articulation
point is present or not. We also check whether all
vertices are visited by the DFS or not, if not we can say
that the graph is not connected.
• Input and Output
• Input: The adjacency matrix of the graph.
• 0 1 1 1 0
• 1 0 1 0 0
• 1 1 0 0 1
• 1 0 0 0 1
• 0 0 1 1 0
• Output: The Graph is a biconnected graph.

20. • Euler Graph - A connected graph G is called an Euler
graph, if there is a closed trail which includes every edge
of the graph G. Euler Path - An Euler path is a path that
uses every edge of a graph exactly once. An Euler
path starts and ends at different vertices.
• Euler Graph - A connected graph G is called an Euler
graph, if there is a closed trail which includes every edge
of the graph G.
• Euler Path - An Euler path is a path that uses every edge
of a graph exactly once. An Euler path starts and ends at
different vertices.

21. • Euler Circuit - An Euler circuit is a circuit that uses every
edge of a graph exactly once. An Euler circuit always
starts and ends at the same vertex. A connected graph G
is an Euler graph if and only if all vertices of G are of
even degree, and a connected graph G is Eulerian if and
only if its edge set can be decomposed into cycles.

22. • The above graph is an Euler graph as a 1 b 2 c 3 d 4 e 5
c 6 f 7 g covers all the edges of the graph.
• Non-Euler Graph
• Here degree of vertex b and d is 3, an odd degree and
violating the euler graph condition.

23. • A directed graph is graph, i.e., a set of objects (called
vertices or nodes) that are connected together, where all
the edges are directed from one vertex to another.
A directed graph is sometimes called a digraph or
a directed network.

24. • Connectivity in an undirected graph means that every
vertex can reach every other vertex via any path. If the
graph is not connected the graph can be broken down
into Connected Components.
• Strong Connectivity applies only to directed graphs. A
directed graph is strongly connected if there is a directed
path from any vertex to every other vertex. This is same
as connectivity in an undirected graph, the only
difference being strong connectivity applies to directed
graphs and there should be directed paths instead of just
paths. Similar to connected components, a directed
graph can be broken down into Strongly Connected
Components.

25. • 1) Create an empty stack 'S' and do DFS traversal of a
graph. In DFS traversal, after calling recursive DFS for
adjacent vertices of a vertex, push the vertex to stack. ...
• 2) Reverse directions of all arcs to obtain the transpose
graph.
• 3) One by one pop a vertex from S while S is not empty.
Let the popped vertex be 'v'.

26. • We can find all strongly connected components in
O(V+E) time using Kosaraju's algorithm. Following is
detailed Kosaraju's algorithm. 1) Create an empty stack
'S' and do DFS traversal of a graph. In DFS traversal,
after calling recursive DFS for adjacent vertices of a
vertex, push the vertex to stack.