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Graph Theory Chapter 4 sec. 1. Koenigsberg bridge problem. It is the Pregel River divided Koenigsberg into four distinct sections. Seven bridges connected the four portions of Koenigsberg. .
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Koenigsberg bridge problem • It is the Pregel River divided Koenigsberg into four distinct sections. Seven bridges connected the four portions of Koenigsberg.
It was a popular pastime for the citizens of Koenigsberg to start in one section of the city and take a walk visiting all sections of the city, trying to cross each bridge exactly once and to return to the original starting point.
How did it start? • In 1735, a Swiss Mathematician Leonhard Euler became the first person to work in graph theory by solving the Koenigsburg bridge problem. • Discovered a simple way to determine when a graph can be traced.
Definition • Trace-to begin at some vertex and draw the entire graph without lifting your pencil and without going over any edge more than once.
Exercise 1 • Place your pencil on any dot and trace the figure completely without lifting your pencil and without tracing any part of any line twice. • Which of the two can be done?
Solution • Fig. A can be traced. • Fig. B cannot be traced.
Definitions • Graph- consists of a finite set of points • Vertices – are points on the graph • Edges- are lines that join pairs of vertices • Connected- if it is possible to travel from any vertex to any other vertex of the graph by moving along successive edges. • Bridge- in a connected graph is an edge such that if it were removed the graph is no longer connected.
Odd and Even Vertex • Odd – The graph is odd if it is an endpoint of an odd number of edges of the graph. • Even- The graph is even if it is an endpoint of an even number of edges of the graph.
Solution • Vertex A is odd • Vertex B is odd • Vertex C is odd • Vertex D is odd
Solution • Vertex A is odd • Vertex B is odd • Vertex C is even • Vertex D is even
Euler’s Theorem • A graph can be traced if it is connected and has zero or two odd vertices.
Solution • Fig. 1 Cannot be traced. (all odd) • Fig. 2 Can be traced by Euler’s theorem.
Note • If a graph has 2 odd vertices, the tracing must begin at one of these and end at the other. • If all vertices are even, then the graph tracing must begin and end at the same vertex. It does not matter at which vertex this occurs.
Definitions • Path- in a graph is a series of consecutive edges in which no edge is repeated. • Euler path- A path containing all the edges of a graph.
Euler circuit- An Euler path that begins and ends at the same vertex. • Eulerian graph-A graph with all even vertices contains an Euler circuit
Find Euler’s path and Euler’s circuit for the two fig. below.
Solution • Fig. 1 (star) • Euler’s path - ADBECA • Euler ‘s circuit - ADBECA • Fig. 2 • Euler’s path – CABCDEHIDFG • Euler’s circuit – There is none, because G and C are both odd vertices, we must begin at one and end at the other.
What is Euler’s circuit used for? • How many of you ride the pubic transportation? • Efficient routes. • Map Coloring
Eulerizing a Graph • 1. The graph must have all even vertices. • 2. If a graph has an odd vertex, then we will add some edges to make that vertex an even vertex. • 3. We want to begin and end at the same vertex. • 4. We do not want to travel on the same edge twice.
