Anyone the vampire targets must succeed on a Will save or fall instantly under the vampire’s influence as though by a dominate person spell (caster level 12th). (3.5 SRD; Vampire)
After reading the SRD (my only source for 3.5 material) regarding compulsion and charm supernatural abilities, and comparing that to the Dominate Person spell, it is unclear to me whether or not a vampire's Dominate(Su) ability (if the PC fails the Will save) will continue after the vampire dies/is slain, or, if the dominated state persists until the next time a refresh is needed (which by the spell description would be the next day).
Once control is established, the range at which it can be exercised is unlimited, as long as you and the subject are on the same plane. You need not see the subject to control it.
If you don’t spend at least 1 round concentrating on the spell each day, the subject receives a new saving throw to throw off the domination. (3.5 SRD; Dominate Person)
While I am predisposed to believe that it is dispelled or removed when the vampire is destroyed (stake in the heart, sun melts, etc) the rules text I have found leaves that unclear. After reviewing the SRD entries for the spell, for vampires, for the Dominate(Su), and for Charmed/Compulsion the 'what happens when the vampire is destroyed' leaves me unsure. While being reduced to 0 HP incapacitates the vampire, the SRD does not have the incapacitated condition. (Given the ability to change into gaseous form, and the ability to start regaining HP, I'd be cautious about assuming that "incapacitated" would break the compulsion on the PC. Nothing in the Helpless condition helps me either).
Was this covered in Sage Advice, FAQ, or in one of the books dedicated to vampires?