As a medical professional, I can explain that it is possible to have a pulse without breathing, although it is an unusual and critical situation that requires immediate attention. The pulse is a measure of the heart's activity, while breathing is a function of the respiratory system. In some cases, such as during certain medical emergencies or under the influence of specific drugs, a person's heart may continue to beat with a pulse, but they may not be breathing or may be breathing inadequately. This condition is known as agonal respiration or gasping and is often a sign of severe distress. Immediate medical intervention is necessary in such cases. read more >>

If there is no sign of breathing or pulse, begin CPR starting with compressions. ... If the patient definitely has a pulse but is not breathing adequately, provide ventilations without compressions. This is also called "rescue breathing." Adults: give 1 breath every 5 to 6 seconds.read more >>