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Module 5 – Networks and Decision Mathematics

Module 5 – Networks and Decision Mathematics. Chapter 23 – Undirected Graphs. 23.4 Euler and Hamilton Paths. A path can be considered as a sequence of edges of the form AB, BG, GE, EC, CD, DE, EG, GF for the graph below: Insert graph

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Module 5 – Networks and Decision Mathematics

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  1. Module 5 – Networks and Decision Mathematics Chapter 23 – Undirected Graphs

  2. 23.4 Euler and Hamilton Paths • A path can be considered as a sequence of edges of the form AB, BG, GE, EC, CD, DE, EG, GF for the graph below: • Insert graph • A circuit is a sequence of edges linking successive vertices that starts and ends at the same vertex. For the graph below: • Insert graph • A path that includes every edge just once is called an Euler Path. • An Euler Circuit is an Euler Path that starts and ends at the same vertex.

  3. Identifying Euler Paths & Circuits • To identify Euler Circuits, look for connected graphs where all vertices have an even degree. • An Euler Path exists if there are exactly 2 odd degree vertices. The path would start at one of these and end at the other. • Insert example • For an Euler path, you would start at B and end at C or vice versa. ie B-A-E-D-B-C-D-C • Note: With Euler paths/circuits, edges cannot be travelled over more than once but you can revisit a vertex.

  4. Identifying Hamilton Paths & Circuits • A Hamilton Path is a path through a graph that passes through each vertex exactly once. • Insert example • A Hamilton Path could be: A-F-G-E-B-C-D-H (there are many more) • A Hamilton Circuit is a Hamilton Path that starts and ends at the same vertex. For the previous graph, a Hamilton Circuit could be: C-D-H-G-F-A-E-B-C • Note: Not all edges must be covered but all vertices must be visited.

  5. Remember:Euler Paths/Circuits focus on EDGES and Hamilton Paths/Circuits focus on VERTICES. • Unlike Euler, there are no specific conditions (ie 2 odd vertices) that are used to identify Hamilton paths or circuits. It is simply a matter of trial and error.

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