The document provides an introduction to graphs and graph algorithms. It defines graphs as consisting of vertices/nodes and edges connecting the nodes. It discusses different graph types including undirected graphs, directed graphs, and weighted graphs. It then describes various graph traversal algorithms like depth-first search, breadth-first search, and their applications to problems like finding paths, cycles, and connected components. Finally, it discusses algorithms for shortest paths in weighted graphs.

1. INTRODUCTION TO GRAPHS
Partha P Chakrabarti
Indian Institute of Technology Kharagpur

2. Graphs
A Graph G = (V, E) consists of the
following:
• A set of Vertices or Nodes V
– Nodes may have one or more labels
• A set of Edges E where each edge
connects vertices of V
– An edge usually defines a connection or
relationship between vertices or nodes
– The edges can be undirected or directed
– Each edge can have one or more labels
– Usually there is at most one edge between
vertices, there could be multiple edges
between the same nodes.
– Normally an edge connects two vertices, but
in general we could have hyper-edges

3. Graphs
A Graph G = (V, E) consists of the
following:
• A set of Vertices or Nodes V
– Nodes may have one or more labels
• A set of Edges E where each edge
connects vertices of V
– An edge usually defines a connection or
relationship between vertices or nodes
– The edges can be undirected or directed
– Each edge can have one or more labels
– Usually there is at most one edge between
vertices, there could be multiple edges
between the same nodes.
– Normally an edge connects two vertices, but
in general we could have hyper-edges

4. Some Applications of Graphs
• Maps, Routes
• Layouts
• Circuits and Networks
• Relationships
• Constraints
• Dependencies
• Flow Charts
• State Machines

5. Some Applications of Graphs
• Maps, Routes
• Layouts
• Circuits and Networks
• Relationships
• Constraints
• Dependencies
• Flow Charts
• State Machines

6. Some Applications of Graphs
• Maps, Routes
• Layouts
• Circuits and Networks
• Relationships
• Constraints
• Dependencies
• Flow Charts
• State Machines

7. Some Applications of Graphs
• Maps, Routes
• Layouts
• Circuits and Networks
• Relationships
• Constraints
• Dependencies
• Flow Charts
• State Machines

8. Some Applications of Graphs
• Maps, Routes
• Layouts
• Circuits and Networks
• Relationships
• Constraints
• Dependencies
• Flow Charts
• State Machines

9. Some Applications of Graphs
• Maps, Routes
• Layouts
• Circuits and Networks
• Relationships
• Constraints
• Dependencies
• Flow Charts
• State Machines

10. Some Applications of Graphs
• Maps, Routes
• Layouts
• Circuits and Networks
• Relationships
• Constraints
• Dependencies
• Flow Charts
• State Machines

11. Some Applications of Graphs
• Maps, Routes
• Layouts
• Circuits and Networks
• Relationships
• Constraints
• Dependencies
• Flow Charts
• State Machines

12. Some Applications of Graphs
• Maps, Routes
• Layouts
• Circuits and Networks
• Relationships
• Constraints
• Dependencies
• Flow Charts
• State Machines

13. Graph Representation
Adjacency Matrix
Adjacency List

14. Graph Representation
Adjacency Matrix
Adjacency List

15. Some Algorithms on Graphs
• Paths
• Reachability
• Connected
Components
• Trees, Cycles,
ordering
• Costs & Distances
• Spanning Trees
• Shortest Paths
• Flows

16. Some Algorithms on Graphs
• Paths
• Reachability
• Trees, Cycles,
ordering
• Connected
Components
• Costs & Distances
• Spanning Trees
• Shortest Paths
• Flows

17. Some Algorithms on Graphs
• Paths
• Reachability
• Cycles, ordering
• Connected
Components
• Costs & Distances
• Spanning Trees
• Shortest Paths
• Flows

18. Some Algorithms on Graphs
• Paths
• Reachability
• Cycles, ordering
• Connected
Components
• Costs & Distances
• Spanning Trees
• Shortest Paths
• Flows

19. Some Algorithms on Graphs
• Paths
• Reachability
• Cycles, ordering
• Connected
Components
• Costs & Distances
• Spanning Trees
• Shortest Paths
• Flows

20. Thank you

21. TRAVERSAL OF UNDIRECTED GRAPHS
Partha P Chakrabarti
Indian Institute of Technology Kharagpur

22. Undirected Graph
An Undirected Graph G = (V, E) consists of
the following:
• A set of Vertices or Nodes V
• A set of Edges E where each edge
connects two vertices of V
Example: Figure 1
V = {0,1,2,3,4,5,6,7,8}
E = {(0,1), (0,8), (0,3),(1,7), (2,3), (2,5), (2,7),
(3,4), (4,8), (5,6)}
Successor Function: succ(i) = {set of nodes
to which node i is connected}
Example: Succ(2) = {3,5,7}
Weighted Undirected Graphs: Such Graphs
may have weights on edges (Figure 2)
Figure 1
Figure 2

23. Problems on Undirected Graphs
Reachability, Path, Cycle / Tree Detection,
Connected Components, Bi-Connected
Components, Spanning Tree, Shortest
Path, Maximum Flow, Vertex Cover, Edge
Cover, Travelling Salesperson,
Figure 1
Figure 2

24. Basic Traversal Algorithm (Depth First Search) I
Global Data: G = (V,E)
visited [i] indicates if node i is visited. For
all nodes j visited [j] is initialized to 0
succ(i) = {set of nodes to which node i is
connected}
Dfs(node) {
visited[node] = 1;
for each j in succ(node) do {
if (visited [j] ==0) Dfs(j)
}
}

25. Basic Traversal Algorithm (Depth First Search) II
Global Data: G = (V,E)
visited [i] indicates if node i is visited. For
all nodes j visited [j] is initialized to 0
succ(i) = {set of nodes to which node i is
connected}
Dfs(node) {
visited[node] = 1;
for each j in succ(node) do {
if (visited [j] ==0) Dfs(j)
}
}

26. Basic Traversal Algorithm (Depth First Search) III
Global Data: G = (V,E)
visited [i] indicates if node i is visited. For
all nodes j visited [j] is initialized to 0
succ(i) = {set of nodes to which node i is
connected}
Dfs(node) {
visited[node] = 1;
for each j in succ(node) do {
if (visited [j] ==0) Dfs(j)
}
}

27. Cycle Detection
Global Data: G = (V,E)
visited [i] indicates if node i is visited. For all
nodes j visited [j] is initialized to 0
succ(i) = {set of nodes to which node i is
connected}
Dfs(node) {
visited[node] = 1;
for each j in succ(node) do {
if (visited [j] ==0) Dfs(j)
}
}
// Cycle Detection //

28. Path Finding
Global Data: G = (V,E)
visited [i] indicates if node i is visited. For all nodes
j visited [j] is initialized to 0
succ(i) = {set of nodes to which node i is
connected}
Dfs(node) {
visited[node] = 1;
for each j in succ(node) do {
if (visited [j] ==0) {
Dfs(j) }
}
}
// Tree Edge, Back Edge, Parent Links, Tracing
Paths //

29. Connected Components
Global Data: G = (V,E)
Visited[i], comp[i] all initialized to 0
count = 0;
Algorithm components() {
for each node k do {
if visited [k] == 0 { count = count + 1;
DfComp_S(k) }
DfComp(node) {
visited[node] = 1; comp[node] = count;
for each j in succ(node) do {
if (visited [j] ==0) DfComp(j)
}
}

30. Depth-First Numbering & Time Stamping
Global Data: G = (V,E)
Visited[i], comp[i] all initialized to 0
count = 0;
Algorithm components() {
for each node k do {
if visited [k] == 0 { count = count + 1;
DfComp_S(k) }
DfComp(node) {
visited[node] = 1; comp[node] = count;
for each j in succ(node) do {
if (visited [j] ==0) {
DfComp(j)
}
}
}

31. Breadth-First Search
Global Data: G = (V,E)
Visited[i] all initialized to 0
Queue Q initially {}
BFS(k) {
visited [k] = 1; Q = {k};
While Q != {} {
j = DeQueue (Q);
if visited[j] == 0 {
visited [j] = 1;
For each k in succ (j) EnQueue(Q,k);
}
}
/Parent links, Shortest Length Path Finding in
unweighted graphs/

32. Pathfinding in Weighted Undirected Graphs I
Global Data: G = (V,E)
Visited[i] all initialized to 0,
Cost[j] all initialized to INFINITY
Ordered Queue Q initially {}
BFSW(k) {
visited [k] = 1; cost [k] = 0; Q = {k};
While Q != {} {
j = DeQueue (Q);
if visited[j] == 0 {
visited [j] = 1;
For each k in succ (j) {
if cost[k] > cost[j] + c[j,k]
cost[k] = cost[j] + c[j,k];
EnQueue(Q,k);}
}
}

33. Pathfinding in Weighted Undirected Graphs II
Global Data: G = (V,E)
Visited[i] all initialized to 0,
Cost[j] all initialized to INFINITY
Ordered Queue Q initially {}
BFSW(k) {
visited [k] = 1; cost [k] = 0; Q = {k};
While Q != {} {
j = DeQueue (Q);
if visited[j] == 0 {
visited [j] = 1;
For each k in succ (j) {
if cost[k] > cost[j] + c[j,k]
cost[k] = cost[j] + c[j,k];
EnQueue(Q,k);}
}
}

34. Thank you

35. TRAVERSAL OF DIRECTED GRAPHS
Partha P Chakrabarti
Indian Institute of Technology Kharagpur

36. Directed Graphs
An Undirected Graph G = (V, E) consists of
the following:
• A set of Vertices or Nodes V
• A set of DIRECTED Edges E where each
edge connects two vertices of V. The
edge is an ORDERED pair of vertices
Successor Function: succ(i) = {set of nodes
to which node i is connected}
Directed Acyclic Graphs (DAGs): Such
Graphs have no cycles (Figure 2)
Weighted Undirected Graphs: Such Graphs
may have weights on edges (Figure 3). We
can also have Weighted DAGs
Figure 1
Figure 3
Figure 2

37. Basic Traversal Algorithm (Depth First Search)
Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially
NULL /
succ(i) = {set of nodes to which node i is
connected}
Dfs(node) {
visited[node] = 1;
for each j in succ(node) do {
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
}
}

38. Traversing the Complete Graph by DFS
Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially
NULL /
succ(i) = {set of nodes to which node i is
connected}
Dfs(node) {
visited[node] = 1;
for each j in succ(node) do {
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
}
}

39. Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially
NULL /
Entry[i] = node entry sequence / initially 0 /
Exit[i] = node exit sequence / initially 0 /
succ(i) = {set of nodes to which node i is connected}
numb = 0;
Dfs(node) {
visited[node] = 1; numb = numb+1;
Entry[node] = numb;
for each j in succ(node) do
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
numb = numb + 1;
Exit[node] = numb;
}
Entry-Exit Numbering

40. Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially NULL
/
Entry[i] = node entry sequence / initially 0 /
Exit[i] = node exit sequence / initially 0 /
succ(i) = {set of nodes to which node i is connected}
numb = 0;
Dfs(node) {
visited[node] = 1; numb = numb+1;
Entry[node] = numb;
for each j in succ(node) do
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
numb = numb + 1;
Exit[node] = numb;
}
Tree Edge, Back Edge, Forward Edge, Cross Edge
Edge (u,v) is
Tree Edge or Forward Edge: if & only if
Entry[u] < Entry[v] < Exit[v] < Exit[u]
Back Edge: if & only if
Entry[v] < Entry [u] < Exit [u] < Exit [v]
Cross Edge: if & only if
Entry [v] < Exit [v] < Entry [u] < Exit [u]

41. Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially
NULL /
Entry[i] = node entry sequence / initially 0 /
Exit[i] = node exit sequence / initially 0 /
succ(i) = {set of nodes to which node i is connected}
numb = 0;
Dfs(node) {
visited[node] = 1; numb = numb+1;
Entry[node] = numb;
for each j in succ(node) do
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
numb = numb + 1;
Exit[node] = numb;
}
Reachability, Paths, Cycles, Components
Edge (u,v) is
Tree Edge or Forward Edge: if & only if
Entry[u] < Entry[v] < Exit[v] < Exit[u]
Back Edge: if & only if
Entry[v] < Entry [u] < Exit [u] < Exit [v]
Cross Edge: if & only if
Entry [v] < Exit [v] < Entry [u] < Exit [u]

42. Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially
NULL /
Entry[i] = node entry sequence / initially 0 /
Exit[i] = node exit sequence / initially 0 /
succ(i) = {set of nodes to which node i is connected}
numb = 0;
Dfs(node) {
visited[node] = 1; numb = numb+1;
Entry[node] = numb;
for each j in succ(node) do
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
numb = numb + 1;
Exit[node] = numb;
}
Directed Acyclic Graphs

43. Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially
NULL /
Entry[i] = node entry sequence / initially 0 /
Exit[i] = node exit sequence / initially 0 /
succ(i) = {set of nodes to which node i is connected}
numb = 0; numb1 = 0;
Dfs(node) {
visited[node] = 1; numb = numb+1;
Entry[node] = numb;
for each j in succ(node) do
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
numb1 = numb1 + 1;
Exit[node] = numb1;
}
Topological Ordering, Level Values

44. Shortest Cost Path in Weighted DAGs

45. Breadth-First Search
Global Data: G = (V,E)
Visited[i] all initialized to 0
Queue Q initially {}
BFS(k) {
visited [k] = 0; Q = {k};
While Q != {} {
j = DeQueue (Q);
if visited[j] == 0 {
visited [j] = 1;
For each k in succ (j) {
if (visited[k]==0) EnQueue(Q,k); }
}
}
/Parent links, Shortest Length Path Finding in
unweighted directed graphs/

46. Pathfinding in Weighted Directed Graphs
Global Data: G = (V,E)
Visited[i] all initialized to 0,
Cost[j] all initialized to INFINITY
Ordered Queue Q initially {}
BFSW(k) {
visited [k] = 0; cost [k] = 0; Q = {k};
While Q != {} {
j = DeQueue (Q);
if visited[j] == 0 {
visited [j] = 1;
For each k in succ (j) {
if cost[k] > cost[j] + c[j,k]
cost[k] = cost[j] + c[j,k];
EnQueue(Q,k);}
}
}

47. Thank you