Eulerian Trails and Circuits 2
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Question 1 of 4
1. Question
Jacob has a network of streets for his pizza delivery. Which sequence can he use so that he will past all the streets only once?
Hint
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Excellent!
Incorrect
Check each given sequence to see which one will cover all the streets, represented by the edges on the network.`A-E-F-B-D-E-A`
This sequence will not pass through all edges. Therefore, it is not the correct sequence.`E-F-B-C-D-E-A-F-B-A`
This sequence will not pass through all edges. Therefore, it is not the correct sequence.`E-F-B-C-D-E-F-B-A`
This sequence will not pass through all edges. Therefore, it is not the correct sequence.`A-F-D-C-B-A-E-F-B-D-E`
This sequence will pass through all edges once. Therefore, this is the correct sequence.`A-F-D-C-B-A-E-F-B-D-E` -
Question 2 of 4
2. Question
Jacob will be delivering pizza to each of the places marked by a vertex on the network below. Answer the following questions:
Answer `Y` for yes or `N` for no-
`(i)` Can Jacob deliver the pizzas while passing all the streets once?`=` (Y, y)`(ii)` Can Jacob start and finish on the same place?`=` (N, n)
Hint
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An Eulerian circuit is a circuit with all of its vertices having even degrees, starts and ends on the same vertex, and passes on all the edges only once.`(i)` Can Jacob deliver the pizzas while passing all the streets once?The previous question confirmed that we can use the sequence `A-F-D-C-B-A-E-F-B-D-E` to pass through all the streets once.
Therefore, Jacob can deliver the pizza while passing all the streets only once.`(ii)` Can Jacob start and finish on the same place?Check if the sequence is an Eulerian trail or circuit by counting the edges of each vertex.
This network has exactly 2 odd edges and start at one of the odd edges and ends on the other. Therefore, it is an Eulerian trail.Since an Eulerian trail starts and finishes on two different edges, this means that Jacob cannot start and finish at the same place.`(i)` Can Jacob deliver the pizzas while passing all the streets once? `\text(Yes)``(ii)` Can Jacob start and finish on the same place? `\text(No)` -
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Question 3 of 4
3. Question
Moe delivers leaflets to each location marked on the network below. Can he deliver leaflets to all the locations while passing on each street once?
Hint
Help VideoCorrect
Excellent!
Incorrect
An Eulerian circuit is a circuit with all of its vertices having even degrees, starts and ends on the same vertex, and passes on all the edges only once.For Moe to deliver the leaflets to all locations by passing all the streets only once, the network should have either an Eulerian trail or Eulerian circuit.Check if the sequence is an Eulerian trail or circuit by counting the edges of each vertex.
This network has exactly 2 odd edges and start at one of the odd edges and ends on the other. Therefore, it is an Eulerian trail.Next, further check if a Eulerian trail can be used to in this network.You can mark the starting vertex with `S` and the finishing vertex with `F`.
The diagram illustrates that you can start on vertex `T` and end on vertex `S`. This means that the network has an Eulerian trail.Therefore, Moe can deliver leaflets in all locations while passing all the streets once. -
Question 4 of 4
4. Question
Moe delivers leaflets to each location marked on the network below. He discovered a new path from `T` to `S`. Which sequence can he take so he can start and finish from the same place?
Hint
Help VideoCorrect
Keep Going!
Incorrect
For Moe to pass through all the streets while starting and finishing from the same place, the sequence needs to be an Eulerian circuit.First, check if it’s possible to have an Eulerian circuit in the network by counting the degrees of each edges
The edges all have even degrees. Therefore, this network can have an Eulerian circuit.Next, check each given sequence to see which one will cover all the streets, represented by the edges on the network.`T-S-R-Q-P-T-Q-S-T`
This sequence will pass through all edges once. Therefore, this is the correct sequence.`T-S-R-Q-P-T-Q-S-T`
Quizzes
- Vertices and Edges
- Degrees 1
- Degrees 2
- Degrees 3
- Drawing A Network 1
- Drawing A Network 2
- Completing a Table from a Network Diagram
- Network from Maps and Plans
- Identifying Paths and Cycles
- Eulerian Trails and Circuits 1
- Eulerian Trails and Circuits 2
- Identifying Spanning Trees
- Minimum Spanning Trees 1
- Minimum Spanning Trees 2
- Shortest Path 1
- Shortest Path 2