For the hybrid flow shop scheduling problem (HFSP), this study addresses its high computational burden associated with insertion operations due to over-much objective function calculations as the number of jobs increases. Unfortunately, the search solution space and existing accelerated local search methods exhibit limitations. We find that there are many operations on the critical path that do not affect the completion time, especially when there are multiple critical paths. However, deleting mandatory operations must make the original makespan value smaller. Moreover, only specific insertion positions have the potential to decrease the makespan of the changed sequence. In view of this, we emphasize the need for improved accelerated local search techniques tailored to overcome these limitations, and propose the theoretical analysis and methods of an accelerated local search based on effective insertion of mandatory operations in the graph for the HFSP with makespan criterion. More specifically, first, we represent a solution using a solution graph, and introduce four definitions related to mandatory operations in the graph space. Second, we propose five theorems and their corresponding proofs. Third, we present a mandatory operations-based accelerated iterated greedy algorithm (MOAIG) in the graph space according to the above theorems. Finally, we conduct many experiments on two well-known benchmarks (a total of 576 instances). Through statistical analysis, it becomes evident that the computational time required for calculating the makespan associated with operations-based insertion has decreased significantly. The development of the mandatory operations-based effective insertion accelerated local search is a promising approach to significantly mitigate the time complexity by reducing the numbers of operations perturbed and insertion positions involved in the calculation, and makes up for gaps in existing accelerated insertion methods for HFSP.