Maximum-weight independent set problem for a path graph

Question

While taking the Algorithms: Design and Analysis II class, one of the questions asks about the maximum-weight independent set problem for a path graph. shown below is a (blu

Answer 1

OP here: Here's a full answer for completeness sake, inspired by @robert-king's answer.

Consider the path 10-2-1-4. The vertices selected by the algorithm are 10, 1, where 1, the minimum, is selected. Thus, option 1 is incorrect.

Consider the path 1-3-10-9. The vertices selected by the algorithm are 3, 9, where the maximum 10 isn't selected. Thus, option 2 is incorrect.

Consider the path 1-9-7-1-5. The vertices selected by the algorithm are 1, 7, 5. However, 7 was not included in the optimal solution of the subproblem 1-9-7. Note that, 7 was not included in the optimal solution of the subproblem 1-9-7-1 either, because the its previous vertex was "heavier", and since all weights are positive, the sum of the next weight and the heavier vertex is certainly greater than 7. Thus, option 4 is incorrect.

Option 3 is correct. This follows from induction, since the optimal solution to a subproblem depends only on the solutions of the previous two subproblems.