1. Graphs
Chapter 7
Visit for more Learning Resources

2. What is a graph?
A data structure that consists of a set of finite nodes (V) and a set of edges(E) that relate the nodes to each other
G=(V,E)
V(G)=Vertices of graph G ,e.g. {1,2,3,4}
E(G)=Edges of graph G ,e.g.{(1,2),(1,3),(3,4)} 1 4 2 3

3. Graphs u v Vertices are also called nodes and points.
E is the edge set.
Each edge connects two different vertices.
Edges are also called arcs and lines.
Directed edge has an orientation (u,v). u v

4. Graphs Undirected graph => no oriented edge.
Undirected edge has no orientation (u,v). u v
Undirected graph => no oriented edge.
Directed graph => every edge has an orientation.

5. Undirected Graph 2 3 8 10 1 4 5 9 11 6 7

6. Directed Graph (Digraph)2 3 8 10 1 4 5 9 11 6 7

7. Applications—Communication Network2 3 8 10 1 4 5 9 11 6 7
Internet connection. Vertices are computers. Send from 1 to 7.
Vertex = city, edge = communication link.

8. Driving Distance/Time Map2 3 8 10 1 4 5 9 11 6 7
Vertex = city, edge weight = driving distance/time.

9. Street Map 2 3 8 10 1 4 5 9 11 6 7
Some streets are one way.

10. Complete Undirected Graph
Has all possible edges.
n = 2
n = 4
n = 3
n = 1

11. Number Of Edges—Undirected Graph
Each edge is of the form (u,v)
Number of such pairs in an n vertex graph is n(n-1).
Since edge (u,v) is the same as edge (v,u), the number of edges in a complete undirected graph is n(n-1)/2.
Number of edges in an undirected graph is <= n(n-1)/2.

12. Number Of Edges—Directed Graph
Each edge is of the form (u,v), u != v.
Number of such pairs in an n vertex graph is n(n-1).
Since edge (u,v) is not the same as edge (v,u), the number of edges in a complete directed graph is n(n-1).
Number of edges in a directed graph is <= n(n-1).

13. Graph terminology
1.Adjacent nodes: two nodes are adjacent if they are connected by an edge 2.Path: A sequence of edges that connect two nodes in a graph 3.Path Length: Total no of edges included in path. 4.Complete graph: a graph in which every vertex is directly connected to every other vertex
5 is adjacent to 7
7 is adjacent from 5

14. Graph terminology (cont.)
What is the number of edges in a complete directed graph with N vertices? 
N * (N-1)

15. Graph terminology (cont.)
What is the number of edges in a complete undirected graph with N vertices? 
N * (N-1) / 2

16. 5. Vertex Degree OR Degree of Node2 3 8 10 1 4 5 9 11 6 7
Number of edges incident to vertex. degree(2) = 2, degree(5) = 3, degree(3) = 1

17. 6.In-Degree Of A Vertex 2 3 8 10 1 4 5 9 11 6 7
in-degree is number of incoming edges indegree(2) = 1, indegree(8) = 0

18. 7.Out-Degree Of A Vertex 2 3 8 10 1 4 5 9 11 6 7
out-degree is number of outbound edges outdegree(2) = 1, outdegree(8) = 2

19. Graph terminology (cont.)
8.Source:A Node which has only outgoing edges. 9.Sink: A Node which has only Incoming edges. 10. Pendant Node: if indegree of node is 1 and outdegree is 0 then it is pendant Node. 11. Reachable: If path exist between two nodes then they are reachable.

20. Graph terminology (cont.)
12. Sling or loop: An edge of graph which joins a node to itself is called sling or loop. 13.Parallel Edges: The two distinct edges between a pair to nodes which are opposite in a direction. 14.Weighted Edge: edge has Numerical Values. 15.Isolated Node: Not an Adjacent node of any other node

21. Graph terminology (cont.)
16. Weighted graph: a graph in which each edge carries a value 2 3 8 10 1 4 5 9 11 6 7

22. Graph terminology (cont.)
17. Isolated Graph: A Graph which contain only isolated node. 18. Directed Acyclic Graph: A diagraph with no cycles. 2 1 4 5

23. Graph implementation Array-based implementation
A 1D array is used to represent the vertices
A 2D array (adjacency matrix) is used to represent the edges

24. Graph Representation There are two ways to represent the Graph:
Sequential Representation
Adjacency Matrix
Path Matrix
Adjacency Lists/Linked List representation.

25. 1.Adjacency Matrix
0/1 n x n matrix, where n = # of vertices
A(i,j) = 1 iff (i,j) is an edge 2 3 1 4 5 1 2 3 4 5 1 1 1 1 1 1 1 1 1 1

26. Path Matrix

27. Path Matrix

29. Adjacency Lists aList[1] = (2,4) aList[2] = (1,5) aList[3] = (5)
Adjacency list for vertex i is a linear list of vertices adjacent from vertex i.
An array of n adjacency lists.
aList[1] = (2,4)
aList[2] = (1,5)
aList[3] = (5)
aList[4] = (5,1)
aList[5] = (2,4,3) 2 3 1 4 5

30. Adjacency Lists Representation
A graph of n nodes is represented by a one-dimensional array L of linked lists, where
L[i] is the linked list containing all the nodes adjacent from node i.
The nodes in the list L[i] are in no particular order

31. Example of List Helen 1 Mary 0 Joe 3 John 2 Tom 4 Paul 5
L[0]: empty L[1]: empty L[2]: 0, 1, 4, 5 L[3]: 0, 1, 4, 5 L[4]: 0, 1 L[5]: 0, 1
Helen 1
Mary 0
Joe 3
John 2
Tom 4
Paul 5

32. Linked Adjacency Lists
Each adjacency list is a chain. 2 3 1 4 5
aList[1]
aList[5]
[2]
[3]
[4] 2 4 1 5 5 5 1 2 4 3
Array length n simply means we need an array with n spots. A direct implementation using a Java array would need n+1 spots, because spot 0 would not be utilized. However, by using spot 0 for vertex 1, spot 1 for vertex 2, and so on, we could get by with a Java array whose length is actually n.
Array Length = n
# of chain nodes = 2e (undirected graph)
# of chain nodes = e (digraph)

33. Array Adjacency Lists aList[1] [2] [3] [4] aList[5] Array Length = n
Each adjacency list is an array list. 2 3 1 4 5
aList[1]
aList[5]
[2]
[3]
[4]
Array Length = n
# of list elements = 2e (undirected graph)
# of list elements = e (digraph)

34. Graph Search (traversal)
How do we search a graph?
At a particular vertices, where shall we go next?
Two common Methods:
Breadth-first search (BFS)
Depth-first search (DFS)
In BFS, one explore a graph level by level away (explore all neighbors first and then move on)
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
Data Structure and Algorithm

35. A vertex u is reachable from vertex v iff there is a path from v to u.
Graph Search Methods
A vertex u is reachable from vertex v iff there is a path from v to u. 2 3 8 10 1 4 5 9 11 6 7

36. Graph Search Methods
A search method starts at a given vertex v and visits/labels/marks every vertex that is reachable from v. 2 3 8 10 1 4 5 9 11 6 7
Visit, mark, and label used as synonyms.

37. Breadth-First Search Visit start vertex and put into a FIFO queue.
Repeatedly remove a vertex from the queue, visit its unvisited adjacent vertices, put newly visited vertices into the queue.

38. Breadth-First Search Example2 3 8 10 1 4 5 9 11 6 7
Start search at vertex 1.

39. Breadth-First Search Example
FIFO Queue 1 2 3 8 1 1 4 5 9 10 6 7 11
Visit/mark/label start vertex and put in a FIFO queue.

40. Breadth-First Search Example
FIFO Queue 1 2 3 8 1 1 4 5 9 10 6 7 11
Remove 1 from Q; visit adjacent unvisited vertices; put in Q.

41. Breadth-First Search Example2
FIFO Queue 2 3 2 4 8 1 1 4 4 5 9 10 6 7 11
Remove 1 from Q; visit adjacent unvisited vertices; put in Q.

42. Breadth-First Search Example2
FIFO Queue 2 3 2 4 8 1 1 4 4 5 9 10 6 7 11
Remove 2 from Q; visit adjacent unvisited vertices; put in Q.

43. Breadth-First Search Example2
FIFO Queue 2 3 3 4 5 3 6 8 1 1 4 4 5 5 9 10 6 6 7 11
Remove 2 from Q; visit adjacent unvisited vertices; put in Q.

44. Breadth-First Search Example2
FIFO Queue 2 3 3 4 5 3 6 8 1 1 4 4 5 5 9 10 6 6 7 11
Remove 4 from Q; visit adjacent unvisited vertices; put in Q.

45. Breadth-First Search Example2
FIFO Queue 2 3 3 5 3 6 8 1 1 4 4 5 5 9 10 6 6 7 11
Remove 4 from Q; visit adjacent unvisited vertices;
put in Q.

46. Breadth-First Search Example2
FIFO Queue 2 3 3 5 3 6 8 1 1 4 4 5 5 9 10 6 6 7 11
Remove 5 from Q; visit adjacent unvisited vertices;
put in Q.

47. Breadth-First Search Example2
FIFO Queue 2 3 3 3 6 9 7 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11
Remove 5 from Q; visit adjacent unvisited vertices;
put in Q.

48. Breadth-First Search Example2
FIFO Queue 2 3 3 3 6 9 7 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11
Remove 3 from Q; visit adjacent unvisited vertices;
put in Q.

49. Breadth-First Search Example2
FIFO Queue 2 3 3 6 9 7 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11
Remove 3 from Q; visit adjacent unvisited vertices;
put in Q.

50. Breadth-First Search Example2
FIFO Queue 2 3 3 6 9 7 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11
Remove 6 from Q; visit adjacent unvisited vertices;
put in Q.

51. Breadth-First Search Example2
FIFO Queue 2 3 3 9 7 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11
Remove 6 from Q; visit adjacent unvisited vertices;
put in Q.

52. Breadth-First Search Example2
FIFO Queue 2 3 3 9 7 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11
Remove 9 from Q; visit adjacent unvisited vertices;
put in Q.

53. Breadth-First Search Example2
FIFO Queue 2 3 3 7 8 8 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11
Remove 9 from Q; visit adjacent unvisited vertices;
put in Q.

54. Breadth-First Search Example2
FIFO Queue 2 3 3 7 8 8 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11
Remove 7 from Q; visit adjacent unvisited vertices;
put in Q.

55. Breadth-First Search Example2
FIFO Queue 2 3 3 8 8 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11
Remove 7 from Q; visit adjacent unvisited vertices;
put in Q.

56. Breadth-First Search Example2
FIFO Queue 2 3 3 8 8 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11
Remove 8 from Q; visit adjacent unvisited vertices;
put in Q.

57. Breadth-First Search Example2
FIFO Queue 2 3 3 8 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11
Queue is empty. Search terminates.

58. Breadth-First Search Property
All vertices reachable from the start vertex (including the start vertex) are visited.

59. Breadth-first Search (BFS)
Search for all vertices that are directly reachable from the root (called level 1 vertices)
After mark all these vertices, visit all vertices that are directly reachable from any level 1 vertices (called level 2 vertices), and so on.
Data Structure and Algorithm

60. Example r s t u         v w x y
Data Structure and Algorithm

61. Example        Q: s r s t u r s t u v w x y v w x y      v w x y v w x y Q: s
Data Structure and Algorithm

62. Example 1    1   Q: w r r r s s t t u u v v w w x x y y   1   v v w w x x y y Q: w r
Data Structure and Algorithm

63. Example 1 2   1 2  Q: r t x r s t u r s t u v w x y v w x y2   1 2  v w x y v w x y Q: r t x
Data Structure and Algorithm

64. Example 1 2  2 1 2  Q: t x v r s t u r s t u v w x y v w x y2  2 1 2  v w x y v w x y Q: t x v
Data Structure and Algorithm

65. Example 1 2 3 2 1 2  Q: x v u r s t u r s t u v w x y v w x y2 3 2 1 2  v w x y v w x y Q: x v u
Data Structure and Algorithm

66. Example 1 2 3 2 1 2 3 Q: v u y r s t u r s t u v w x y v w x y2 3 2 1 2 3 v w x y v w x y Q: v u y
Data Structure and Algorithm

67. Example 1 2 3 2 1 2 3 Q: u y r s t u r s t u v w x y v w x y2 3 2 1 2 3 v w x y v w x y Q: u y
Data Structure and Algorithm

68. Example 1 2 3 2 1 2 3 Q: y r s t u r s t u v w x y v w x y2 3 2 1 2 3 v w x y v w x y Q: y
Data Structure and Algorithm

69. All Vertices are Visited……..
Example r s t u r s t u 1 2 3 2 1 2 3 v w x y v w x y
All Vertices are Visited…….. Q: Ø
Data Structure and Algorithm

70. Depth-First Search (DFS)
The basic idea behind this algorithm is that it traverses the graph using recursion
Go as far as possible until you reach a dead end
Backtrack to the previous path and try the next branch
The graph below, started at node a, would be visited in the following order: a, b, c, g, h, i, e, d, f, j d a b h i c g e j f
Data Structure and Algorithm

71. Depth-First Search
depthFirstSearch(v) { Label vertex v as reached. for (each unreached vertex u adjacenct from v) depthFirstSearch(u); }
Note that vertices adjacent from v are examined one at a time. As soon as an unreached adjacent vertex u is found, a depthFirstSearch(u) is done. Remaining vertices adjacent from v are examined after depthFirstSearch(u) completes.

72. Depth-First Search Example2 3 8 10 1 4 5 9 11 6 7 2 1
Start search at vertex 1.
Label vertex 1 and do a depth first search from either 2 or 4.
Suppose that vertex 2 is selected.

73. Depth-First Search Example2 2 2 3 8 1 1 4 5 5 9 10 6 7 11
Label vertex 2 and do a depth first search from either 3, 5, or 6.
Suppose that vertex 5 is selected.

74. Depth-First Search Example2 2 2 3 8 1 1 4 5 5 5 9 9 10 6 7 11
Label vertex 5 and do a depth first search from either 3, 7, or 9.
Suppose that vertex 9 is selected.

75. Depth-First Search Example2 2 2 3 8 8 1 1 4 5 5 5 9 9 9 10 6 7 11
Label vertex 9 and do a depth first search from either 6 or 8.
Suppose that vertex 8 is selected.

76. Depth-First Search Example2 2 2 3 8 8 8 1 1 4 5 5 5 9 9 9 10 6 6 7 11
Label vertex 8 and return to vertex 9.
From vertex 9 do a dfs(6).

77. Depth-First Search Example2 2 2 3 8 8 8 1 1 4 4 5 5 5 9 9 9 10 6 6 6 7 11
Label vertex 6 and do a depth first search from either 4 or 7.
Suppose that vertex 4 is selected.

78. Depth-First Search Example2 2 2 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 11
Label vertex 4 and return to 6.
From vertex 6 do a dfs(7).

79. Depth-First Search Example2 2 2 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 7 11
Label vertex 7 and return to 6.
Return to 9.

80. Depth-First Search Example2 2 2 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 7 11
Return to 5.

81. Depth-First Search Example2 2 2 3 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 7 11
Do a dfs(3).

82. Depth-First Search Example2 2 2 3 3 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 7 11
Label 3 and return to 5.
Return to 2.

83. Depth-First Search Example2 2 2 3 3 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 7 11
Return to 1.

84. Depth-First Search Example2 2 2 3 3 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 7 11
Return to invoking method.
For more detail contact us