22-Euler, Hamilton, & Shortest Path Problems

22-Euler, Hamilton, & Shortest Path Problems - EECS 210...

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Euler, Hamilton, & Shortest Path Problems David O. Johnson EECS 210 (Fall 2016) 1 EECS 210 Discrete Structures David O. Johnson Fall 2016
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Reminders Connect 5 due: 11:59 PM, Thursday, November 17 (today) Thanksgiving Holiday, Thursday, November 24 – No Assignment! Homework 6 due: Thursday, December 1 at the beginning of your lecture Euler, Hamilton, & Shortest Path Problems David O. Johnson EECS 210 (Fall 2016) 2
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Any Questions? Euler, Hamilton, & Shortest Path Problems 3 David O. Johnson EECS 210 (Fall 2016)
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Euler & Hamiltonian Graphs (Section 10.5) Euler Paths and Circuits Applications of Euler Paths and Circuits Hamilton Paths and Circuits Applications of Hamilton Paths and Circuits Euler, Hamilton, & Shortest Path Problems 4 David O. Johnson EECS 210 (Fall 2016)
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Euler Paths and Circuits The town of Kӧnigsberg, Prussia (now Kalingrad, Russia) was divided into four sections by the branches of the Pregel river. In the 18th century seven bridges connected these regions. People wondered whether it was possible to follow a path that crosses each bridge exactly once and returns to the starting point. The Swiss mathematician Leonard Euler proved that no such path exists. This result is often considered to be the first theorem ever proved in graph theory. The 7 Bridges of Kӧnigsberg Multigraph Model of the Bridges of K ӧ nigsberg Euler, Hamilton, & Shortest Path Problems 5 David O. Johnson EECS 210 (Fall 2016)
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Euler Paths and Circuits Definition : An Euler circuit in a graph G is a simple (i.e., it does not contain the same edge more than once) circuit containing every edge of G . An Euler path in G is a simple path containing every edge of G . An Euler circuit is also an Euler path. An Euler path may or may not be an Euler circuit. Example : Which of the undirected graphs G 1 , G 2 , and G 3 has a Euler circuit? Of those that do not, which has an Euler path? Solution : The graph G 1 has an Euler circuit (e.g., a , e , c , d , e , b , a ), and therefore an Euler path. But, as can easily be verified by inspection, neither G 2 nor G 3 has an Euler circuit. Note that G 3 has an Euler path (e.g., a , c , d , e , b , d, a , b ), but there is no Euler path in G 2 , which can be verified by inspection. Euler, Hamilton, & Shortest Path Problems 6 David O. Johnson EECS 210 (Fall 2016)
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Necessary and Sufficient Conditions for Euler Circuits and Paths Theorem : A connected multigraph with at least two vertices has an Euler circuit if and only if each of its vertices has an even degree and it has an Euler path if and only if it has exactly two vertices of odd degree. Example : 4 of the vertices in the multigraph model of the Kӧnigsberg bridge problem have odd degree and 0 have an even degree. Hence, there is no Euler circuit in this multigraph and it is impossible to start at a given point, cross each bridge exactly once, and return to the starting point.
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