The document provides information on graph traversals including depth-first search (DFS) and breadth-first search (BFS). It explains that DFS uses a stack and visits nodes in a recursive, preorder manner. BFS uses a queue and visits nodes level-by-level starting from the root. Examples of DFS and BFS on sample graphs are given to illustrate the traversal order.

1. Graph Traversals
Depth-First Search
• Think Stack
Breadth-First Search
• Think Queue

2. Overview
 Goal
 To systematically visit the nodes of a graph
 A tree is a directed, acyclic, graph (DAG)
 If the graph is a tree,
 DFS is exhibited by preorder, postorder, and (for binary trees)
inorder traversals
 BFS is exhibited by level-order traversal

3. Depth-First Search
// recursive, preorder, depth-first search
void dfs (Node v) {
if (v == null)
return;
if (v not yet visited)
visit&mark(v); // visit node before adjacent nodes
for (each w adjacent to v)
if (w has not yet been visited)
dfs(w);
} // dfs

4. Depth-First Search
// recursive, postorder, depth-first search
void dfs (Node v) {
if (v == null)
return;
tag(v); // mark v as having been considered
for (each w adjacent to v)
if (w has not yet been tagged)
dfs(w);
visit(v); // postorder traversal: visit node after
// adjacent nodes
} // dfs

5. Depth-First Search
// non-recursive, preorder, depth-first search
void dfs (Node v) {
if (v == null)
return;
push(v);
while (stack is not empty) {
pop(v);
if (v has not yet been visited)
mark&visit(v);
for (each w adjacent to v)
if (w has not yet been visited)
push(w);
} // while
} // dfs

6. Depth-First Search
// non-recursive, postorder, depth-first search
void dfs (Node v) {
// Lex 20 (Do not need to submit)
} // dfs

7. Example
0
7
1
5
4
3
2
6
Policy: Visit adjacent nodes in increasing index order

8. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0 3 2 7 6 4

9. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5
Push 5

10. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5
Pop/Visit/Mark 5

11. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5
1
2
Push 2, Push 1

12. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1
2
Pop/Visit/Mark 1

13. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1
0
2
4
2
Push 4, Push 2,
Push 0

14. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0
2
4
2
Pop/Visit/Mark 0

15. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0
3
7
2
4
2
Push 7, Push 3

16. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0 3
7
2
4
2
Pop/Visit/Mark 3

17. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0 3
2
7
2
4
2
Push 2

18. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0 3 2
7
2
4
2
Pop/Mark/Visit 2

19. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0 3 2 7
2
4
2
Pop/Mark/Visit 7

20. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0 3 2 7
6
2
4
2
Push 6

21. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0 3 2 7 6
2
4
2
Pop/Mark/Visit 6

22. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0 3 2 7 6
4
2
Pop (don’t visit) 2

23. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0 3 2 7 6 4
2
Pop/Mark/Visit 4

24. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0 3 2 7 6 4
Pop (don’t visit) 2

25. Preorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
5 1 0 3 2 7 6 4
Done

26. Preorder DFS: Start with Node 5
Note: edge (0,3) removed
0
7
1
5
4
3
2
6
5 1 0 7 6 2 4 3

27. Depth-First Search
Policy: Don’t push nodes twice// non-recursive, preorder, depth-first search
void dfs (Node v) {
if (v == null)
return;
push(v);
while (stack is not empty) {
pop(v);
if (v has not yet been visited)
mark&visit(v);
for (each w adjacent to v)
if (w has not yet been visited && not yet stacked)
push(w);
} // while
} // dfs

28. Preorder DFS (Don’t push nodes
twice). Start with Node 5
0
7
1
5
4
3
2
6
5 1 0 3 7 6 4 2

29. Postorder DFS: Start with Node 5
0
7
1
5
4
3
2
6
2 3 6 7 0 4 1 5

30. Breadth-first Search
 Ripples in a pond
 Visit designated node
 Then visited unvisited nodes a distance i away, where i = 1,
2, 3, etc.
 For nodes the same distance away, visit nodes in systematic
manner (eg. increasing index order)

31. Breadth-First Search
// non-recursive, preorder, breadth-first search
void bfs (Node v) {
if (v == null)
return;
enqueue(v);
while (queue is not empty) {
dequeue(v);
if (v has not yet been visited)
mark&visit(v);
for (each w adjacent to v)
if (w has not yet been visited && has not been queued)
enqueue(w);
} // while
} // bfs

32. BFS: Start with Node 5
7
1
5
4
3
2
6
5 1 2 0 4 3 7 6
0

33. BFS: Start with Node 5
7
1
5
4
3
2
6
5
0

34. BFS: Node one-away
7
1
5
4
3
2
6
5
0

35. BFS: Visit 1 and 2
7
1
5
4
3
2
6
5 1 2
0

36. BFS: Nodes two-away
7
1
5
4
3
2
6
5 1 2
0

37. BFS: Visit 0 and 4
7
1
5
4
3
2
6
5 1 2 0 4
0

38. BFS: Nodes three-away
7
1
5
4
3
2
6
5 1 2 0 4
0

39. BFS: Visit nodes 3 and 7
7
1
5
4
3
2
6
5 1 2 0 4 3 7
0

40. BFS: Node four-away
7
1
5
4
3
2
6
5 1 2 0 4 3 7
0

41. BFS: Visit 6
7
1
5
4
3
2
6
5 1 2 0 4 3 7 6
0

42. Biconnectivity
A connected undirected graph is biconnected if
there are no vertices whose removal disconnects
the rest of the graph.
Example :
If the nodes are computers and the edges are
links, then if any computer goes down, network
mail is unaffected, except, of course, at the down
computer.

43. Biconnectivity
 If a graph is not biconnected, the vertices whose removal
would disconnect the graph are known as articulation
points. These nodes are critical in many applications.

44. Biconnectivity
The graph below is not biconnected: C and D
are articulation points. The removal of C
would disconnect G, and the removal of D
would disconnect E and F, from the rest of the
graph

45. Biconnectivity
• Depth-first search provides a linear-time algorithm to find all
articulation points in a connected graph.
1.Starting at any vertex, perform a depth-first
search and number the nodes as they are visited:
num(v).
2.For every vertex v in the depth-first spanning
tree, compute the lowest-numbered vertex called
low(v), that is reachable from v by taking zero or
more tree edges and then possibly one back
edge (in that order).

46. Biconnectivity
 The depth-first spanning tree shows the preorder number
first, and then the lowest-numbered vertex reachable
(For each node num/low is given in the tree)

47. Biconnectivity
• The lowest-numbered vertex , low(v), can be efficiently
computed by performing a postorder traversal of the
depth-first spanning tree, and it is the minimum of
1)num(v)
2)the lowest num(w) among all back edges (v, w)
3)the lowest low(w) among all tree edges (v, w)

48. Biconnectivity: How to decide
Articulation Points
The root is an articulation point if and only if it has
more than one child (because if it has two children,
removing the root disconnects nodes in different
subtrees, and if it has only one child, removing the
root merely disconnects the root)
Any other vertex v is an articulation point if and only if
v has some child w such that low(w) ≥ num(v).
Example:
Vertex D is an articulation point
 low(E)= 4 ≥ num(D)=4
Vertex C is an articulation point
 low(G)= 7 ≥ num(C)=3

49. How many squares can you create in this figure by connecting any 4
dots (the corners of a square must lie upon a grid dot?
TRIANGLES:
How many triangles are located in the image below?

50. There are 11 squares total; 5 small, 4 medium, and 2 large.
27 triangles. There are 16 one-cell triangles, 7 four-cell triangles, 3 nine-cell
triangles, and 1 sixteen-cell triangle.

51. GUIDED READING

53. ASSESSMENT

54. Do the DFS and BFS for this graph
12
43
57
6

55. DIFFERENT TRAVERSALS
STARTING FROM VERTEX 1
PRE ORDER TRAVERSAL (ALSO CALLED
DEPTH FIRST SEARCH)
1 2 3 4 5 6 7
POST ORDER TRAVERSAL
6 5 4 7 3 2 1
BREADTH FIRST SEARCH
1 2 4 3 5 6 7

56. DEPTH FIRST SPANNING TREE
6•1
•4
•2
•5
•3
•6
•7

57. DEPTH FIRST SPANNING TREE
6•1
•4
•2
•5
•3
•6
•7
1
2
3
6
5
4
7

58. DEPTH FIRST SPANNING TREE
6•1
•4
•2
•5
•3
•6
•7
1
2
3
6 4
5
4
7

59. DEPTH FIRST SPANNING TREE
6•1
•4
•2
•5
•3
•6
•7
1
2
3
6 4
5 4
4
7

60. DEPTH FIRST SPANNING TREE
6•1
•4
•2
•5
•3
•6
•7
1
2
3
6 4
5 4
4 1
7

61. DEPTH FIRST SPANNING TREE
6•1
•4
•2
•5
•3
•6
•7
1
2
3
6 4
5 4
4 1
7 7

62. DEPTH FIRST SPANNING TREE
6•1
•4
•2
•5
•3
•6
•7
1
2
3 1
6 4
5 4
4 1
7 7

63. DEPTH FIRST SPANNING TREE
6•1
•4
•2
•5
•3
•6
•7
1
2 1
3 1
6 4
5 4
4 1
7 7

64. DEPTH FIRST SPANNING TREE
6•1
•4
•2
•5
•3
•6
•7
1 1
2 1
3 1
6 4
5 4
4 1
7 7

65. DEPTH FIRST SPANNING TREE
6•1
•4
•2
•5
•3
•6
•7
1 1
2 1
3 1
6 4
5 4
4 1
7 7