A cautionary note. Minimal respiration in meditation can be attained by sheer effort of will, leaving the meditator red faced and gasping, the blood roaring in the ears and the heart racing. Not a good idea, your body will be running short of oxygen. So this kind of practice should not be aimed for, attempted or tried at. It should not be a goal. With steady practice, something like it might just occur... one is breathing all one wishes to, but it just happens that relaxation is so complete that the breath only arises a couple of times a minute or even becomes so fine it can no longer be perceived.
In my experience, the quiet gentle way is the right way. Willfully applying the brakes to the breathing doesn't work.
Normal resting respiration rate for an adult is 12-18 breaths per minute. By being very still and calm, one reduces the energy requirement of the body substantially, and by being still, one reduces cooling by air currents, which reduces the need for metabolic heating. Then by calming the mind we switch off some 10-20 watts of power required by the brain, or part of it at least. As the respiration rate falls, heat lost by breathing falls too. All these add up to a massive drop in energy requirements and oxygen requirements.
Ironically, the mindful meditator cannot help but be aware that his respiration rate is dropping. If he finds this either exciting or alarming, his heart rate will spike and everything else with it. So mindfulness in this case can destroy the balance! With much practice, one becomes accustomed to low respiration rates,(and high rates, and variable rates) and literally no longer gives them a second thought. At that point mindfulness stops tripping over its own feet, so to speak.
Sergeant Schultz knew everything there was to know.