1. Graphs

2. Graph The Graph Abstract Data Type Graph Application Graph Types
Path and Subgraph in Graph
Graph Connection types
Graph Operations
Graph Method
Graph Storage Structures
Graph implementations

3. The Graph Abstract Data Type
A graph is a way of representing relationships that exist between pairs of objects.
a graph is a set of objects, called vertices .
together with a collection of pairwise connections between them .
a graph G is simply a set V of vertices
and a collection E of pairs of vertices from V, called edges

4. The Graph Abstract Data Type
Graphs is a data structure that differ from all the others in one major concept , each node may have multiple predecessors as well as multiple successors .
Graphs are very useful structure . They can be used to solve complex routing problems ,such as :
designing and routing airlines among the air ports they serve .
route messages over a computer network from one node to another .

5. Graph Applications Graphs have applications :
a host of different domains
Mapping
Transportation
electrical engineering
computer networks.

6. Graph Types Directed Graph Undirected Graph mixed graph.
An edge (u, v) is said to be directed from u to v if the pair (u, v) is ordered, with u preceding v.
It contains ordered pair where (u,v) is not equal (v,u).
Undirected Graph
An edge (u, v) is said to be undirected if the pair (u, v) is not ordered.
It contain unordered pair as {u,v} and {u,v} is equal {v,u}.
mixed graph.
A graph that has both directed and undirected edges is often called a mixed graph.

7. Graph Types
Example:
World wide web
Example:
Social Network “Facebook”

8. Graph
The two vertices joined by an edge are called the end vertices (or endpoints) of the edge.
If an edge is directed, its first endpoint is its origin and the other is the destination of the edge.
Two vertices u and v are said to be adjacent if there is an edge whose end vertices are u and v
An edge is said to be incident on a vertex if the vertex is one of the edge's endpoints.

9. Graph cont.
The outgoing edges of a vertex are the directed edges whose origin is that vertex.
The incoming edges of a vertex are the directed edges whose destination is that vertex.
The degree of a vertex v, denoted deg(v), is the number of incident edges of v.
The in-degree and out-degree of a vertex v are the number of the incoming and outgoing edges of v, and are denoted indeg(v) and outdeg(v), respectively.

10. Graph cont.
Neighbors
Two vertices are said to be neighbors if an edge directly connects them
Path
A sequence of vertices in which each vertex is adjacent to the other
Cycle
A path that start and ends with the same vertex
Loop
A single Arc that begins and ends with the same vertex
subgraph of a graph G is a graph H whose vertices and edges are subsets of the vertices and edges of G

11. Path and Subgraph (BOS, NW 35, JFK, AA 1387, DFW)
Directed Simple Path:
(BOS, NW 35, JFK, AA 1387, DFW)
Directed Simple Cycle:
(LAX, UA 120, ORD, UA 877, DFW, AA 49, LAX)
Sub Graph
vertices ( BOS, JFK, and MIA) , and edges ( AA 903 and DL 247 )

12. Connectivity
Definition: An directed graph is strongly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph.
Definition: An directed graph is weakly connected if there is a path between any two vertices in the underlying undirected graph.

13. Connectivity
Example: Are the following directed graphs strongly or weakly connected? a b c d
Weakly connected, because, for example, there is no path from b to d. a b c d
Strongly connected, because there are paths between all possible pairs of vertices.

14. Graph connection types cont.
Disjoint
A graph is disjoint if it not connected d a b c e

15. Graph operations Add Vertex Insert a new vertex in the graph
When a vertex is added it is disjoint , it is not connected to any other vertices in the list . B A B A
AddVertix C C E E

16. Graph operations cont. Delete Vertex
Removes a vertex from the graph .When a vertex is deleted , all connecting edges are removed . B A B A C E C E

17. Graph operations cont. Add edge
Connect a vertex to a destination vertex .
If the graph is a digraph , then one of the edges must be the source and the other is the destination . B A B A C E C E

18. Graph operations cont. Find Vertex
Traverse the graph looking for a specified vertex.
If the vertex is found , its data are returned . If it is not founded , an error is indicated .

19. Graph Methods vertices(): edges(): areAdjacent(v,w):
Return an iterable collection of all the vertices of the graph.
edges():
Return an iterable collection of all the edges of the graph.
areAdjacent(v,w):
Test whether vertices v and w are adjacent.
replace(v,x):
Replace the element stored at vertex v with x.

20. Graph Methods replace(e,x): insertVertex(x): insertEdge(v, w,x):
Replace the element stored at edge e with x.
insertVertex(x):
Insert and return a new vertex storing element x.
insertEdge(v, w,x):
Insert and return a new undirected edge with end vertices v and w and storing element x.
removeVertex(v):
Remove vertex v and all its incident edges and return the element stored at v.
removeEdge(e):
Remove edge e and return the element stored at e.

21. Graph Storage Structures
To represent a graph , we need to store two sets :
The vertices of the graph
The edges or arcs .
The two common structures used to store these sets are array and linked list .

22. Graph implementations
Uses a vector ( one –dimensional array ) for the vertices and a matrix (two - dimensional array) to store the edges .

23. Graph implementations
Uses a two- dimensional ragged array to store the edges .
The vertex list is a singly linked list of the vertices in the list .

24. Adjacency Matrix

25. Adjacency List