Hamiltonian Circuits and Paths Vocabulary

1. Hamiltonian Circuits and Paths Vocabulary AQR Unit VII Networks and Graphs

2. Objective: Use Hamiltonian graphs to solve problems such as scheduling or routing. A Hamiltonian path in a graph is a path that passes through everyvertex in the graph exactly once. A Hamiltonian path does not necessarily pass through all the edges of the graph, however. What is a Hamiltonian path? Trace a Hamiltonian path for the graph below.

3. A Hamiltonian path which ends in the same place in which it began is called a Hamiltonian circuit. What is a Hamiltonian circuit? Trace a Hamiltonian circuit for graph below. A weighted edge has a value assigned to the edge (i.e.- miles, time, gallons of gas, any unit of measure) What is a weighted edge?

4. Hamiltonian Circuits are often called the mail man circuit because the mailman goes to every mailbox but does not need to go over every street. What is often called “mail man circuit”? Remember: The path does not need to go over every edge but must pass through every vertex exactly once so no backtracking!

5. Summary and Review: Euler Paths • Passes over edge exactly once. • May pass through a vertex more than once. • Think “Snow Plow” problem • Think “E” for “Edges” and “Euler” • Two odd degree vertices Euler Circuits • Passes over edge exactly once & go back to starting vertex. • Zero odd degree vertices Eulerize • Find the odd degrees and the shortest path b/t them • Add a copy of those edges along that shortest path • Now you have an Euler Circuit!

6. Summary and Review Continued: • Passes through every vertex exactly once. • Not necessarily over every edge. • Have to guess and check to see if it’s Hamiltonian. Degrees don’t matter. Hamiltonian Paths • Passes through every vertex exactly once and ends at the same vertex where it started. • Have to guess and check to find it. Hamiltonian Circuits