Test On Euler And Hamilton Paths And Circuits

1)

List the vertices in the above graph:

{A,B,C,D,E,F,G,H}

{A,B,F,G,H}

{D,E}

{F,G,H}

The correct answer includes all the vertices in the given graph. The graph is not explicitly shown, but the answer {A,B,C,D,E,F,G,H} suggests that these are all the vertices present in the graph.

Explanation

The correct answer includes all the vertices in the given graph. The graph is not explicitly shown, but the answer {A,B,C,D,E,F,G,H} suggests that these are all the vertices present in the graph.

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3)

Use the above graph. The degree of Vertex B is:

The degree of a vertex in a graph is the number of edges that are connected to it. In the given graph, Vertex B is connected to three edges, namely BC, BD, and BF. Therefore, the degree of Vertex B is 3.

Explanation

The degree of a vertex in a graph is the number of edges that are connected to it. In the given graph, Vertex B is connected to three edges, namely BC, BD, and BF. Therefore, the degree of Vertex B is 3.

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4)

Refer to the above graph and choose the best answer:

Hamiltonian path and Hamiltonian circuit

Hamiltonian path

Hamiltonian circuit

Neither a Hamiltonian path nor Hamiltonian circuit

Hamiltonian path: A path that visits every vertex of a graph exactly once.

Hamiltonian circuit: A Hamiltonian path that starts and ends at the same vertex.

In the given graph, the following is a Hamiltonian path:

A -> B -> D -> C -> E -> A

This path visits every vertex exactly once. It is also a Hamiltonian circuit because it starts and ends at the same vertex (A). Therefore, the graph has both a Hamiltonian path and a Hamiltonian circuit.

Explanation

Hamiltonian path: A path that visits every vertex of a graph exactly once.

Hamiltonian circuit: A Hamiltonian path that starts and ends at the same vertex.

In the given graph, the following is a Hamiltonian path:

A -> B -> D -> C -> E -> A

This path visits every vertex exactly once. It is also a Hamiltonian circuit because it starts and ends at the same vertex (A). Therefore, the graph has both a Hamiltonian path and a Hamiltonian circuit.

5)

Refer to the above graph and choose the best answer:

Euler path and Euler circuit

Euler path only

Euler circuit only

Neither an Euler path nor Euler circuit

Based on the given graph, an Euler path is a path that visits every edge exactly once, while an Euler circuit is a path that visits every edge and returns to the starting point. The graph in the question has multiple paths that visit every edge exactly once, indicating the presence of an Euler path. Additionally, the graph also has a cycle that visits every edge and returns to the starting point, indicating the presence of an Euler circuit. Therefore, the correct answer is "Euler path and Euler circuit."

Explanation

Based on the given graph, an Euler path is a path that visits every edge exactly once, while an Euler circuit is a path that visits every edge and returns to the starting point. The graph in the question has multiple paths that visit every edge exactly once, indicating the presence of an Euler path. Additionally, the graph also has a cycle that visits every edge and returns to the starting point, indicating the presence of an Euler circuit. Therefore, the correct answer is "Euler path and Euler circuit."

6) Elizabeth is planning a trip. She wants to fly out of Charlotte, visit New York, visit Hollywood, and end up at her parent's home in Dallas. Which graph would be best for this scenario?

Hamiltonian path

Hamiltonian circuit

Euler path

Euler circuit

A Hamiltonian path is a path in a graph that visits every vertex exactly once. In this scenario, Elizabeth wants to visit each location (Charlotte, New York, Hollywood, and Dallas) exactly once, without revisiting any of them. Therefore, a Hamiltonian path would be the best graph for this scenario.

Explanation

A Hamiltonian path is a path in a graph that visits every vertex exactly once. In this scenario, Elizabeth wants to visit each location (Charlotte, New York, Hollywood, and Dallas) exactly once, without revisiting any of them. Therefore, a Hamiltonian path would be the best graph for this scenario.

7)

Refer to the above graph and choose the best answer:

Euler path and Euler circuit

Euler path only

Euler circuit only

Neither an Euler path nor Euler circuit

Based on the given graph, an Euler path is a path that visits every edge exactly once, while an Euler circuit is a path that starts and ends at the same vertex, visiting every edge exactly once. Since the graph in the question has a path that visits every edge exactly once and also starts and ends at the same vertex, it satisfies the conditions for both an Euler path and an Euler circuit. Therefore, the correct answer is "Euler path and Euler circuit".

Explanation

Based on the given graph, an Euler path is a path that visits every edge exactly once, while an Euler circuit is a path that starts and ends at the same vertex, visiting every edge exactly once. Since the graph in the question has a path that visits every edge exactly once and also starts and ends at the same vertex, it satisfies the conditions for both an Euler path and an Euler circuit. Therefore, the correct answer is "Euler path and Euler circuit".

8) A city is planning their snow plow route for next winter. They want to begin at the garage, go down each street only once, and end at the garage. Which graph would be best for this scenario?

Euler circuit

Euler graph

Hamiltonian circuit

Hamiltonian graph

An Euler circuit is the best graph for this scenario because it is a path that visits every edge exactly once and ends at the same vertex where it started. In this case, the city wants to start at the garage, go down each street only once, and end at the garage, which aligns with the properties of an Euler circuit.

Explanation

An Euler circuit is the best graph for this scenario because it is a path that visits every edge exactly once and ends at the same vertex where it started. In this case, the city wants to start at the garage, go down each street only once, and end at the garage, which aligns with the properties of an Euler circuit.

9)

Use the above graph. The degree of Vertex C is:

From the graph provided, we observe that Vertex C has the following edges connected to it:

Edge from C to B

Edge from C to F

Thus, Vertex C has two edges incident to it.

Therefore, the degree of Vertex C is 2.

Explanation

From the graph provided, we observe that Vertex C has the following edges connected to it:

Edge from C to B

Edge from C to F

Thus, Vertex C has two edges incident to it.

Therefore, the degree of Vertex C is 2.

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10) Roberta wants to deliver the mail. She begins at the Post Office where she picks up the mail each day, but wants to end close to her home. She only wants to go down each street once. Her home is 5 miles from the Post Office. Which graph would best meet this scenario?

Euler path

Euler circuit

Hamiltonian path

Hamiltonian circuit

An Euler path is a path in a graph that visits every edge exactly once. In this scenario, Roberta wants to go down each street once, which aligns with the concept of visiting each edge exactly once. Therefore, an Euler path would best meet this scenario.

Explanation

An Euler path is a path in a graph that visits every edge exactly once. In this scenario, Roberta wants to go down each street once, which aligns with the concept of visiting each edge exactly once. Therefore, an Euler path would best meet this scenario.

11) Bob has to check the stop signs at every intersection in his hometown. He parks his car in the parking lot of the transportation department and needs to end up back at his car. He only wants to check each intersection one time. Which graph would be best for this scenario?

Hamiltonian circuit

Hamiltonian path

Euler path

Euler circuit

A Hamiltonian circuit is the best graph for this scenario because it allows Bob to visit each intersection exactly once and end up back at his car. In a Hamiltonian circuit, every vertex is visited exactly once, and the last vertex is connected to the first vertex, forming a closed loop. This ensures that Bob can check all the stop signs at each intersection without missing any and without having to backtrack or visit any intersection multiple times.

Explanation

A Hamiltonian circuit is the best graph for this scenario because it allows Bob to visit each intersection exactly once and end up back at his car. In a Hamiltonian circuit, every vertex is visited exactly once, and the last vertex is connected to the first vertex, forming a closed loop. This ensures that Bob can check all the stop signs at each intersection without missing any and without having to backtrack or visit any intersection multiple times.

12)

Refer to the above graph and choose the best answer:

Euler path and Euler circuit

Euler path only

Euler circuit only

Neither an Euler path nor an Euler circuit

Based on the given graph, an Euler path is possible because it is a connected graph with exactly two vertices having an odd degree (A and C). An Euler path visits each edge exactly once and ends at a different vertex than where it started. However, an Euler circuit is not possible because there are more than two vertices with an odd degree. Therefore, the correct answer is "Euler path only".

Explanation

Based on the given graph, an Euler path is possible because it is a connected graph with exactly two vertices having an odd degree (A and C). An Euler path visits each edge exactly once and ends at a different vertex than where it started. However, an Euler circuit is not possible because there are more than two vertices with an odd degree. Therefore, the correct answer is "Euler path only".

13)

Refer to the above graph and choose the best answer:

Hamiltonian path only

Hamiltonian circuit only

Hamiltonian path and Hamiltonian circuit

Neither a Hamiltonian path nor a Hamiltonian circuit

Based on the given graph, a Hamiltonian path is possible as it is a path that visits each vertex exactly once. However, a Hamiltonian circuit is not possible as it is a closed path that visits each vertex exactly once and returns to the starting vertex. Therefore, the correct answer is the Hamiltonian path only.

Explanation

Based on the given graph, a Hamiltonian path is possible as it is a path that visits each vertex exactly once. However, a Hamiltonian circuit is not possible as it is a closed path that visits each vertex exactly once and returns to the starting vertex. Therefore, the correct answer is the Hamiltonian path only.