To handle dynamic updates to the set of pre-existing comparisons in the topological heapsort algorithm, we can implement a mechanism that allows for efficient modifications to the comparison outcomes without significantly affecting the overall sorting process. One approach could involve maintaining a data structure that tracks the changes in comparisons and updates the heap structure accordingly. When a new comparison is added or an existing one is modified, the algorithm can adjust the heap to reflect these changes while minimizing the impact on the sorting process. By efficiently updating the heap structure based on dynamic comparison updates, the algorithm can adapt to changing input conditions without compromising its overall efficiency.