1. Discrete Mathematics and
Graph Traversal
Discrete Mathematics and
Its Applications
Baojian Hua

2. BFS and DFS BFS: breath first searching DFS: depth first searching
start from one vertex, near to far
generates BFS forest
flat
DFS: depth first searching
recursion and back-tracking
generates DFS forest
narrow

3. “graph” ADT in C: Interface
// in file “graph.h”
#ifndef GRAPH_H
#define GRAPH_H
typedef struct graph *graph;
typedef void (*tyVisit)(poly);
graph newGraph ();
void insertVertex (graph g, poly data);
void insertEdge (graph g, poly from, poly to);
void dfs (graph g, poly start, tyVisit visit);
void bfs (graph g, poly start, tyVisit visit);
// we’d see more later …
#endif

4. Sample Graph
For BFS, associate each vertex with a “distance” property.
distance(v): the number of edges from the vertex “start” to vertex “v”,
with
distance(start)=0 a b c d e f

5. Sample Graph BFS
bfs (g, “a”, strOutput); a b c d e f

6. Sample Graph BFS
bfs (g, “a”, strOutput);
print a; a b c d e f

7. Sample Graph BFS bfs (g, “a”, strOutput); print a; // a choice
print b; a b 1 c d e f

8. Sample Graph BFS bfs (g, “a”, strOutput); print a; // a choice
print b;
print d; a b 1 c d 1 e f

9. Sample Graph BFS bfs (g, “a”, strOutput); print a; // a choice
print b;
print d;
print e; a b 1 c d 1 e 2 f

10. Sample Graph BFS bfs (g, “a”, strOutput); print a; // a choice
print b;
print d;
print e;
print c; a b 1 c d 1 e 2 f

11. Sample Graph BFS bfs (g, “a”, strOutput); print a; // a choice
print b;
print d;
print e;
print c;
print f; a b 1 c d 1 e 2 f 1

12. BFS Algorithm bfs (vertex start, tyVisit visit){
queue q = newQueue ();
setDistance (start, 0);
//Invariant: all vertices in q have distance property
enQueue (q, start);
while (q not empty) {
vertex current = deQueue (q);
int dist = getDistance (current);
visit (current);
for (each adjacent vertex u of “current”){
if (not visited u){
setDistance (u, dist+1);
enQueue (q, u); }
}}}

13. BFS Algorithm void bfsMain (graph g, poly start, tyVisit visit) {
vertex startV = searchVertex (g, start);
bfs (startV, visit);
for (each vertex u in graph g)
if (not visited u)
bfs (q, u); }

14. Sample Graph BFS // color convention: not visited, inQueue, deQueued
bfs (g, “a”, strOutput); a b c d e f
Queue: a

15. Sample Graph BFS // color convention: not visited, inQueue, deQueued
bfs (g, “a”, strOutput);
print a; a b 1 c d 1 e f
Queue: a
Queue: b, d

16. Sample Graph BFS // color convention: not visited, inQueue, deQueued
bfs (g, “a”, strOutput);
print a;
// a choice
print b; a b 1 c d 1 e 2 f
Queue: a
Queue: b, d
Queue: d, e

17. Sample Graph BFS // color convention: not visited, inQueue, deQueued
bfs (g, “a”, strOutput);
print a;
// a choice
print b;
print d; a b 1 c d 1 e 2 f
Queue: a
Queue: b, d
Queue: d, e
Queue: e

18. Sample Graph BFS // color convention: not visited, inQueue, deQueued
bfs (g, “a”, strOutput);
print a;
// a choice
print b;
print d;
print e; a b 1 c d 1 e 2 f
Queue: a
Queue: c
Queue: b, d
Queue: d, e
Queue: e
Queue:

19. Sample Graph BFS // color convention: not visited, inQueue, deQueued
bfs (g, “a”, strOutput);
print a;
// a choice
print b;
print d;
print e;
print c; a b 1 c d 1 e 2 f 1
Queue: a
Queue: c
Queue: b, d
Queue: d, e
Queue: f
Queue: e
Queue:

20. Sample Graph BFS // color convention: not visited, inQueue, deQueued
bfs (g, “a”, strOutput);
print a;
// a choice
print b;
print d;
print e;
print c;
print f; a b 1 c d 1 e 2 f 1
Queue: a
Queue: c
Queue: b, d
Queue: d, e
Queue: f
Queue: e
Queue:
Queue:

21. Sample Graph DFS
Associate a “discover time” and a “finish time” with each vertex v
with:
discover (start) = 0 a b c d e f

22. Sample Graph DFS
dfs (g, “a”, strOutput); a b c d e f

23. Sample Graph DFS
dfs (g, “a”, strOutput);
print a; a
(0, ) b c d e f

24. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice
print b; a
(0, ) b
(1, ) c d e f

25. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice
print b;
print e; a
(0, ) b
(1, ) c d e
(2, ) f

26. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice
print b;
print e;
print d; a
(0, ) b
(1, ) c d
(3, ) e
(2, ) f

27. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice
print b;
print e;
print d; a
(0, ) b
(1, ) c d
(3, 4) e
(2, ) f

28. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice
print b;
print e;
print d; a
(0, ) b
(1, ) c d
(3, 4) e
(2, 5) f

29. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice
print b;
print e;
print d; a
(0, ) b
(1, 6) c d
(3, 4) e
(2, 5) f

30. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice
print b;
print e;
print d; a
(0, 7) b
(1, 6) c d
(3, 4) e
(2, 5) f

31. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice
print b;
print e;
print d;
print c a
(0, 7) b
(1, 6) c
(8, ) d
(3, 4) e
(2, 5) f

32. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice
print b;
print e;
print d;
print c
print f a
(0, 7) b
(1, 6) c
(8, ) d
(3, 4) e
(2, 5) f
(9, )

33. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice
print b;
print e;
print d;
print c a
(0, 7) b
(1, 6) c
(8, ) d
(3, 4) e
(2, 5) f
(9, 10)

34. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice
print b;
print e;
print d;
print c a
(0, 7) b
(1, 6) c
(8, 11) d
(3, 4) e
(2, 5) f
(9, 10)

35. DFS Algorithm dfs (vertex start, tyVisit visit, time time) {
visit (start);
setDiscover (start, time++);
for (each adjacent vertex u of “start”)
if (not visited u)
dfs (u, visit, time);
setFinish (start, time++); }

36. DFS Algorithm void dfsMain (graph g, poly start, tyVisit visit) {
vertex startV = searchVertex (g, start);
time time = newTime ();
dfs (startV, visit, time);
for (each vertex u in graph g)
if (not visited u)
dfs (u, visit, time); }

37. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a; a
(0, ) b c d e f
dfs(a)

38. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b; a
(0, ) b
(1, ) c d e f
dfs(a) => dfs(b)

39. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e; a
(0, ) b
(1, ) c d e
(2, ) f
dfs(a) => dfs(b) => dfs(e)

40. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d; a
(0, ) b
(1, ) c d
(3, ) e
(2, ) f
dfs(a) => dfs(b) => dfs(e) => dfs(d)

41. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, natOutput);
print a;
// a choice
print b;
print e;
print d; a
(0, ) b
(1, ) c d
(3, ) e
(2, ) f
dfs(a) => dfs(b) => dfs(e) => dfs(d) => dfs(b)???

42. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d; a
(0, ) b
(1, ) c d
(3, 4) e
(2, ) f
dfs(a) => dfs(b) => dfs(e)

43. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d; a
(0, ) b
(1, ) c d
(3, 4) e
(2, 5) f
dfs(a) => dfs(b)

44. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d; a
(0, ) b
(1, 6) c d
(3, 4) e
(2, 5) f
dfs(a)

45. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d; a
(0, ) b
(1, 6) c d
(3, 4) e
(2, 5) f
dfs(a) =>dfs(d)???

46. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d; a
(0, ) b
(1, 6) c d
(3, 4) e
(2, 5) f
dfs(a)

47. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d; a
(0, 7) b
(1, 6) c d
(3, 4) e
(2, 5) f
empty!

48. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d;
print c; a
(0, 7) b
(1, 6) c
(8, ) d
(3, 4) e
(2, 5) f
dfs(c)

49. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d;
print c;
print f; a
(0, 7) b
(1, 6) c
(8, ) d
(3, 4) e
(2, 5) f
(9, )
dfs(c)=>dfs(f)

50. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d;
print c;
print f; a
(0, 7) b
(1, 6) c
(8, ) d
(3, 4) e
(2, 5) f
(9, )
dfs(c)=>dfs(f)=>dfs(f)???

51. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d;
print c;
print f; a
(0, 7) b
(1, 6) c
(8, ) d
(3, 4) e
(2, 5) f
(9, )
dfs(c)=>dfs(f)=>dfs(f)???

52. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d;
print c;
print f; a
(0, 7) b
(1, 6) c
(8, ) d
(3, 4) e
(2, 5) f
(9, 10)
dfs(c)

53. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d;
print c;
print f; a
(0, 7) b
(1, 6) c
(8, ) d
(3, 4) e
(2, 5) f
(9, 10)
dfs(c)=>dfs(e)???

54. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d;
print c;
print f; a
(0, 7) b
(1, 6) c
(8, 11) d
(3, 4) e
(2, 5) f
(9, 10)
dfs(c)

55. Sample Graph DFS // color convention: not visited, discover, finish
dfs (g, “a”, strOutput);
print a;
// a choice
print b;
print e;
print d;
print c;
print f; a
(0, 7) b
(1, 6) c
(8, 11) d
(3, 4) e
(2, 5) f
(9, 10)
empty!

56. Edge Classification
Once we obtain the DFS (or BFS) spanning trees (forests), the graph edges could be classified according to the trees:
tree edges: edges in the trees
forward edges: ancestors to descants
back edges: descants to ancestors
cross edges: others

57. Edge Classification Example
tree edges:
a->b, b->e, e->d, c->f
forward edges:
a->d
back edges:
d->b, f->f
cross edges:
c->e a b c d e f

58. Edge Classification Algorithm
Based on discover and finish time, for each edge e=(u, v):
if v not visited, e is tree edge
if v not finished, e is back edge
if v finished
if discover(u)<discover(v), e is forward edge
if discover(u)>discover(v), e is cross edge

59. Edge Classification Algorithm
dfs (vertex start, tyVisit visit, time time) {
visit (start);
setDiscover (start, time++);
for (each edge e=(start, v)) {
if (not visited v) {
dfs (v, visit, time);
classifyEdge (e, “TreeEdge”);
} else {
if (not setFinish v)
classifyEdge (e, “BackEdge”);
else {
// leave to you }
setFinish (start, time++);

60. DFS Algorithm void dfsMain (graph g, poly start, tyVisit visit) {
vertex startV = searchVertex (g, start);
time time = newTime ();
dfs (startV, visit, time);
for (each vertex u in graph g)
if (not visited u)
dfs (u, visit, time); }

61. BFS and DFS Application #1:
Topological Sorting

62. An Example: Hasse Diagram12 20 4 2 5 1
From Rosen’s book

63. An Example: Hasse Diagram12 20 4 2 5 1
Sorted Sequence: 1

64. An Example: Hasse Diagram12 20 4 2 5 1
Sorted Sequence: 1, 2

65. An Example: Hasse Diagram12 20 4 2 5 1
Sorted Sequence: 1, 2, 4

66. An Example: Hasse Diagram12 20 4 2 5 1
Sorted Sequence: 1, 2, 4, 12

67. An Example: Hasse Diagram12 20 4 2 5 1
Sorted Sequence: 1, 2, 4, 12, 5

68. An Example: Hasse Diagram12 20 4 2 5 1
Sorted Sequence: 1, 2, 4, 12, 5, 20

69. Source-queue Topological Sorting Algorithm
topoSortBfs (graph g) {
for (each vertex v)
calculate in-degree for v;
// vertices in queue q are candidates for
// deletion
queue q = newQueue ();
if (in-degree of v ==0)
enQueue (q, v);

70. Source-queue Topological Sorting Algorithm (cont’)
while (q not empty) {
vertex current = deQueue (q);
for (each edge e=(current, v))
if (not visited v) {
in-degree of v --;
if (in-degree of v ==0)
enQueue (q, v); }
// BFS-based algorithm

71. An Example: Hasse Diagram12 20 4 2 5 1
Source queue: 1
// color convention: not visited, enQueue, deQueue

72. An Example: Hasse Diagram12 20 4 2 5 1
Source queue: 2, 5
Sorted Sequence: 1

73. An Example: Hasse Diagram12 20 4 2 5 1
Source queue: 5, 4
Sorted Sequence: 1, 2

74. An Example: Hasse Diagram12 20 4
// Note that we don’t
// enQueue 20!! 2 5 1
Source queue: 4
Sorted Sequence: 1, 2, 5

75. An Example: Hasse Diagram12 20 4 2 5 1
Source queue: 12, 20 // a chance
Sorted Sequence: 1, 2, 5, 4

76. An Example: Hasse Diagram12 20 4 2 5 1
Source queue: 20
Sorted Sequence: 1, 2, 5, 4, 12

77. An Example: Hasse Diagram12 20 4 2 5 1
Source queue: empty!
Sorted Sequence: 1, 2, 5, 4, 12, 20

78. DFS-based Algorithm 12 20 4 2 5 1
DFS from 4:

79. DFS-based Algorithm 12 (1, 2) 20 (3, 4) 4 (0, 5) 2 (7, 8) 5 (9, 10) 1
(6, 11)
DFS from 4:
When each vertex v finishes, insert v onto head of a linked list

80. Topological Sorting Algorithm
dfs (vertex start, tyVisit visit, linkedList list) {
visit (start);
for (each edge e=(start, v))
// as before …
linkedListInsertHead (list, start) }

81. DFS Algorithm linkedList topoSortDfs (graph g, poly start,
tyVisit visit) {
vertex startV = searchVertex (g, start);
linkedList list = newLinkedList ();
dfs (startV, visit, list);
for (each vertex u in graph g)
if (not visited u)
dfs (u, visit, list);
return list; }

82. Some Extra Programming Assignments
Simple path: is there a simple path from vertex u to v?
Or, are vertices u and v connected?
Is an undirected g connected?
Or, how many connected components are there in g?
Cycle detection: is there a cycle in a digraph g?
Two colorability: is it possible to color all the vertices in a digraph g using two colors, such that no adjacent vertices are of the same color?