I had these problems as homework questions, and I\'m trying to determine whether
ID: 3027244 • Letter: I
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I had these problems as homework questions, and I'm trying to determine whether my answers were correct. Please let me know if they are, and if not, provide a clear explanation/ work steps of why not. Thank you!
My answers were:
a) The graph should have a Euler circuit, because all of the vertices have an even degree. THe path produced by Fleury's algorithm is always a Euler circuit. Fleury’s algorithm has 3 basic rules to follow. First, you must make sure the graph has either 0 or 2 odd vertices. This graph has no odd vertices, so it meets this rule, and because there are zero, you can start from anywhere in the graph. Second, you have to follow the edges one at a time. When given the choice between a bridge and a non-bridge, choose the non-bridge, so you can come back a vertex and traverse remaining edges. Finally, stop when you run out of edges.
Following this algorithm to determine the Euler circuit, we can find the Euler Circuit:
FD->DB->BA->AE->EC->CA->AD->DC->CF
b) A Hamiltonian cycle is a graph cycle through a closed loop that visits each vertex exactly once, and begins and ends on the same vertex. In this problem, a Hamiltonian cycle is D-> B-> A-> E-> C-> F-> D (begins and ends on the same vertex(D))
4. a) The graph should have a Euler circuit, because all of the vertices have an even degree. THe path produced by Fleury's algorithm is always a Euler circuit. Fleury’s algorithm has 3 basic rules to follow. First, you must make sure the graph has either 0 or 2 odd vertices. This graph has no odd vertices, so it meets this rule, and because there are zero, you can start from anywhere in the graph. Second, you have to follow the edges one at a time. When given the choice between a bridge and a non-bridge, choose the non-bridge, so you can come back a vertex and traverse remaining edges. Finally, stop when you run out of edges. Following this algorithm to determine the Euler circuit, we can find the Euler Circuit: FD->DB->BA->AE->EC->CA->AD->DC->CF b) A Hamiltonian cycle is a graph cycle through a closed loop that visits each vertex exactly once, and begins and ends on the same vertex. In this problem, a Hamiltonian cycle is D-> B-> A-> E-> C-> F-> D (begins and ends on the same vertex(D))
Problem 4. (20 points In lecture, we saw the following graph: (a) Explain why this graph should have an Euler circuit, and find it using Fleury's algorithm. (Write out the Euler circuit by referring to edges through their end vertices, as in "AB BD etc.) (b) There is a Hamiltonian cycle in this graph, beginning and ending at the same vertex Find it, and list the vertices in the path in order.Explanation / Answer
Both of your answers(Eulerian circuit and hamiltonian cycles) are correct. There is absolutely no problem in your solution.
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