Step 1: Map the three cerebral arteries to the surfaces of the hemisphere. Medial surface goes mainly to the anterior cerebral, the occipital and inferior temporal regions to the posterior cerebral, and the broad outer face to a third vessel.
Step 2: Identify that third vessel - the middle cerebral artery, the continuation and largest branch of the internal carotid, travelling in the lateral sulcus.
Step 3: Its cortical branches fan over almost the entire lateral surface, feeding the frontal, parietal, and temporal opercular cortex, which is why a middle cerebral stroke classically affects the face and arm more than the leg.
Step 4: The PICA is a cerebellar/medullary vessel and is irrelevant to the cerebrum. So the dominant artery of the lateral hemispheric surface is the middle cerebral artery.
\[\boxed{\text{Middle cerebral artery}}\]