Leonhard Euler solved the Konigsberg Bridge Problem in 1736. The problem was to find a walk through the city of Konigsberg that would cross each of its bridges once and only once. Euler determined that such an Eulerian circuit is only possible if the graph representing the city is connected and has no nodes of odd degree. This established the foundations of graph theory and topology.

1. 1
GRAPH

2. INTRODUCTION
Introduced in 1736
Leonhard Euler
Solved “Konigsberg Bridge Problem”
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Eulerian Circuit or Euler Tour

3. KONIGSBERG BRIDGE
PROBLEM
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The problem was to find a walk through the city that would
cross each bridge once and only once. The islands could not
be reached by any route other than the bridges, and every
bridge must have been crossed completely every time; one
could not walk halfway onto the bridge and then turn around
and later cross the other half from the other side. The walk
need to start and end at the same spot.

4. 4
KONIGSBERG BRIDGE
PROBLEM

5. 5
DESIGN OF SOLUTION

6. 6
CLASS TASK

7. 7
SOLUTION

8. CONCLUSION
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Eularian circuit exists if,
and only if, the graph is
connected, and there
are no nodes of odd
degree at all.

9. BASIC TERMINOLOGY
Graph : Two sets V(G) & E(G)
Undirected Graph
(u,v)
Directed Graph
<u,v>
Self Edges
Multigraph
No. of edges = n(n-1)/2(undirected)
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No. of edges = n(n-1) (directed)

10. BASIC TERMINOLOGY
Adjacent Vertices
Subgraph
Path
Length of Path
Simple Path : all vertices are distinct except first & last
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Cycle

11. BASIC TERMINOLOGY
Connected Vertices
Connected graph
Connected Components : maximal connected subgraph
Acyclic Graph
Strongly Connected
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Strongly Connected Components

12. BASIC TERMINOLOGY
Degree
In-degree
Out-degree
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Weighted Edges

13. GRAPH
REPRESENTATION
Adjacency Matrix
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Adjacency List

14. OPERATIONS ON
GRAPH
Depth-First Search
Breadth-First Search
Connected Components
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Shortest Path Algorithm

15. DEPTH-FIRST SEARCH
Void Graph ::DFS( )
Visited = new boolean[n];
For(int I=0;I<n;I++) visited [I] = false;
DFS(0);
Delete [] visited;
Void Graph ::DFS(const int v)
visited [v] = true;
for( each vertex w adjacent to v)
if(! Visited [w]) DFS(w);
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•
•
•
•

16. BREADTH-FIRST SEARCH
Void Graph ::BFS(int v)
• Visited = new boolean[n];
• For(int I=0; I<n; I++) visited [I] = false;
• Visited [v] = true;
• Queue<int> q;
• Q.insert(v);
• While(!q.IsEmpty())
• V= *q.delete(v);
• For(all vertices w adjacent to v)
• If(!visited [w])
• q.Insert(w);
• Visited [w] = true;
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• Delete [] visited;

17. CONNECTED COMPONENT
Void Graph::Components( )
• Visited = new boolean [n];
• For(int I = 0;I < n; I++) visited [I] = false;
• For(I=0; I < n; I++)
• If (! Visited [I])
• DFS (I);
• OutputNewComponent();
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• Delete [ ] visited;

18. SHORTEST PATH
Void Graph::ShortestPath(int n, int v)
• For(int I=0; I<n; I++)
{s[I] =false;dist[I]=length[v][I];}
• S[v] = true;
• Dist[v] = 0;
• For(I=0; I<=n-2; I++)
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• Int u=choose(n);
• S[u] = true;
• For(int w=0; w<n; w++)
• If(!s[w])
• If(dist[u] + length[u][w] < dist[w])
• Dist[w] = dist[u] + length[u][w];

19. EXAMPLE
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