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Travelling Networks

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Travelling Networks

A transport planner is checking routes between five service hubs. Use the directed, weighted network below. An arrow shows the permitted direction of travel; its number is the travel time in minutes. A walk may repeat vertices or edges. A circuit is a walk that starts and ends at the same vertex. For Questions 1–8, use this network: P→Q (4), P→R (7), Q→R (2), Q→S (6), R→Q (3), R→S (5), S→P (8), and S→T (4), T→P (3), T→R (6).

Read and reason about the network

Show clear working for route totals. When a question asks for a route, write the vertices in order.

1.Which route is a valid directed walk from P to S?

P→Q→R→S
P→S→Q
P→R→Q→P
P→Q→S→R

2.Write a directed circuit that starts at P, uses the edge P→Q, and returns to P without repeating an edge. Include its total travel time.

3.Decide whether P→Q→R→Q→S is a walk, a trail (no repeated edges), both, or neither. Explain your choice.

4.True or false: R→Q→S→P is a valid directed route from R to P with total travel time 17 minutes. Check both the directions and the sum.

TrueFalse

5.A technician must travel from P to T. Find the least-time route and its total travel time. Show enough comparison to justify that it is least.

6.Because it has no repeated edges, the route T→R→S→P is a directed (the more specific term than walk) from T to P.

7.The travel time for the route T→R→S→P is minutes.

8.A courier claims that Q→R→Q→S→P is a circuit. Is the claim correct? Explain using the definition of a circuit.

9.Match each network term to its description.

Directed edge
Weight
Circuit

A number attached to an edge, such as travel time
A connection that can be travelled only in its arrow direction
A walk that begins and ends at the same vertex

Plan a route and test the network

Use the same directed network. A route can be valid even when it repeats a vertex; however, any repeated edge must still be followed in its permitted direction.

10.Can a courier start at S, visit T, and return to S using only directed edges? Give one route and its total time.

11.The edge Q→S is temporarily closed. Is there still a directed route from P to T? If so, give one and calculate its time.

12.The planner wants a circuit starting at P that visits every other hub at least once. Give a valid circuit with no repeated edge and calculate its travel time.