Download - Bridge problem : Discrete Structure
BRIDGE PROBLEM
By:-
Mitul K. Desai - 23
WHAT IS IT ?• In the early eighteenth century, the mediaeval town of
Königsberg in Prussia had a central island (the Kneiphof) around which the Pregel river flowed before dividing in two.
• The four parts of the town were linked by seven bridges as shown on the diagram.
7 bridges of Koenigsberg (1736)
HISTORY• The official birth of graph theory goes
back precisely to the year 1736, when mathematician Leonard Euler (1707-1783) was asked the following question at the court of the king of Prussia:
QUESTION ?• Is it possible for a person to take a walk around town,
starting and ending at the same location, and crossing each of the seven bridges exactly once ?
SOLUTION• Euler studied this problem using the multigraph obtained
where:
• the four regions are represented by vertices.
• the bridges by edges.
DEFINATION• An Euler circuit in a graph G is a simple circuit containing
every edge of G. An Euler path in G is a simple path containing every edge of G.
• Path: a path from v to w is a walk from v to w that does not have any repeated edges.
• Circuit: a circuit is a closed walk, with no repeated edges.
Euler’s Theorem 1
• If a graph has any odd vertices, then it cannot have an Euler circuit.
• If a graph is connected and every vertex is even, then it has at least one Euler circuit.
CONCLUSION
• What we can show is that every vertex must have even degree.
• First note that an Euler circuit begins with a vertex a and continues with an edge incident with a, say {a, b}.
• The edge {a, b} contributes one to degree(a).
CONCLUSION• Each time the circuit passes through a vertex it contributes two
to the vertex’s degree, because the circuit enters via an edge incident with this vertex and leaves via another such edge.
• Finally, the circuit terminates where it started, contributing one to degree(a).
• Therefore, degree(a) must be even, because the circuit contributes one when it begins, one when it ends, and two every time it passes through a
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