Consider using a dynamic graph algorithm that maintains reachability information efficiently, such as the "Fully Dynamic Transitive Closure" approach using matrix multiplication or BFS/DFS with optimizations.

Explore the use of "Incremental Transitive Closure" techniques, where edge insertions are processed in batches to update reachability without recomputing from scratch, possibly leveraging topological order or strongly connected components (SCCs).

Investigate advanced data structures like "Dynamic Trees" (e.g., Link-Cut Trees) or "Dynamic Connectivity" structures (e.g., Holm et al.'s algorithm) to maintain transitive closure under edge insertions with sublinear time complexity per operation.