Constraints will alter critical path by adding slack. Try an experiment. Add three tasks linked F to S. On the last task add a Must Finish On constraint with a date 1 day after the current scheduled date.
You'll see only the task with the constraint is critical. Remove the constraint.
Critical path appears for all tasks.
Add a Finish no Earlier Than constraint on the second task with a date 3 days later than its schedule. The first task drops off critical path.