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Routes and Networks
A traveler wishes to go from City A to City F in a network of six cities: A, B, C, D, E, and F. The roads connecting these cities and the time (in hours) required to travel along each road segment are provided below. Roads are directional as specified; if a road exists from X to Y, it does not necessarily imply a road from Y to X. The traveler cannot traverse any road segment more than once. $$ \begin{array}{|c|c|c|} \hline \text{From City} & \text{To City} & \text{Time (hours)} \\ \hline \text{A} & \text{B} & 2 \\ \text{A} & \text{C} & 3 \\ \text{B} & \text{D} & 4 \\ \text{B} & \text{E} & 5 \\ \text{C} & \text{D} & 1 \\ \text{C} & \text{E} & 2 \\ \text{D} & \text{E} & 1 \\ \text{E} & \text{D} & 1 \\ \text{D} & \text{F} & 3 \\ \text{E} & \text{F} & 2 \\ \hline \end{array} $$ What is the minimum total travel time (in hours) from City A to City F, if the traveler *must* visit both City D and City E at least once? A) 8 hours B) 9 hours C) 7 hours D) 6 hours
The correct answer is C) 7 hours. To find the minimum total travel time from City A to City F, while visiting both City D and City E, and without traversing any road segment more than once, we need to enumerate all valid paths that satisfy these conditions and calculate their total times. There are two primary sequences in which City D and City E can be visited sequentially as part of the overall journey from A to F: 1. City D is visited before City E (A $\rightarrow$ ... $\rightarrow$ D $\rightarrow$ E $\rightarrow$ ... $\rightarrow$ F) 2. City E is visited before City D (A $\rightarrow$ ... $\rightarrow$ E $\rightarrow$ D $\rightarrow$ ... $\rightarrow$ F) Let's analyze each case: **Case 1: Paths where City D is visited before City E (A $\rightarrow$ ... $\rightarrow$ D $\rightarrow$ E $\rightarrow$ ... $\rightarrow$ F)** First, consider the possible sub-paths from A to D: * Path 1a: A $\rightarrow$ B $\rightarrow$ D. Total time = $2 + 4 = 6$ hours. * Path 1b: A $\rightarrow$ C $\rightarrow$ D. Total time = $3 + 1 = 4$ hours. Next, the path from D to E (this segment is mandatory for this case): * Path 1c: D $\rightarrow$ E. Total time = $1$ hour. Finally, the path from E to F: * Path 1d: E $\rightarrow$ F. Total time = $2$ hours. Combining these sub-paths to form complete paths from A to F under Case 1: * **Path 1.1**: (A $\rightarrow$ B $\rightarrow$ D) + (D $\rightarrow$ E) + (E $\rightarrow$ F) This complete path is A $\rightarrow$ B $\rightarrow$ D $\rightarrow$ E $\rightarrow$ F. Total time = $6 + 1 + 2 = 9$ hours. * **Path 1.2**: (A $\rightarrow$ C $\rightarrow$ D) + (D $\rightarrow$ E) + (E $\rightarrow$ F) This complete path is A $\rightarrow$ C $\rightarrow$ D $\rightarrow$ E $\rightarrow$ F. Total time = $4 + 1 + 2 = 7$ hours. **Case 2: Paths where City E is visited before City D (A $\rightarrow$ ... $\rightarrow$ E $\rightarrow$ D $\rightarrow$ ... $\rightarrow$ F)** First, consider the possible sub-paths from A to E: * Path 2a: A $\rightarrow$ B $\rightarrow$ E. Total time = $2 + 5 = 7$ hours. * Path 2b: A $\rightarrow$ C $\rightarrow$ E. Total time = $3 + 2 = 5$ hours. Next, the path from E to D (this segment is mandatory for this case): * Path 2c: E $\rightarrow$ D. Total time = $1$ hour. Finally, the path from D to F: * Path 2d: D $\rightarrow$ F. Total time = $3$ hours. Combining these sub-paths to form complete paths from A to F under Case 2: * **Path 2.1**: (A $\rightarrow$ B $\rightarrow$ E) + (E $\rightarrow$ D) + (D $\rightarrow$ F) This complete path is A $\rightarrow$ B $\rightarrow$ E $\rightarrow$ D $\rightarrow$ F. Total time = $7 + 1 + 3 = 11$ hours. * **Path 2.2**: (A $\rightarrow$ C $\rightarrow$ E) + (E $\rightarrow$ D) + (D $\rightarrow$ F) This complete path is A $\rightarrow$ C $\rightarrow$ E $\rightarrow$ D $\rightarrow$ F. Total time = $5 + 1 + 3 = 9$ hours. All road segments used in these paths are distinct, satisfying the condition of not traversing any segment more than once. Comparing the total times of all four valid paths: * Path 1.1 (A $\rightarrow$ B $\rightarrow$ D $\rightarrow$ E $\rightarrow$ F): 9 hours * Path 1.2 (A $\rightarrow$ C $\rightarrow$ D $\rightarrow$ E $\rightarrow$ F): 7 hours * Path 2.1 (A $\rightarrow$ B $\rightarrow$ E $\rightarrow$ D $\rightarrow$ F): 11 hours * Path 2.2 (A $\rightarrow$ C $\rightarrow$ E $\rightarrow$ D $\rightarrow$ F): 9 hours The minimum among these calculated total times is 7 hours. Thus, the minimum total travel time from City A to City F, visiting both D and E, is 7 hours.
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