Originally Posted by TG Twinkie
The galley sink does not have a vented bowl like the bath sink, thus the reason for the galley sink vent.
I disagree with this statement. Not all bathroom sinks have a separate hole in them. The ones we've have in RV's never have. Plus, the extra hole near the top of the sink is an overflow hole, not a vent. I think that's the vent you're talking about about anyway. If not, my apologies.
Regardless, each fixture needs to be vented somehow after the trap. If not vented through the roof, than an air-admittance valve needs to be installed. These are typically installed in the cabinet under the kitchen sink and bathroom sink if there is no through the roof vent for those drain lines. The purpose of the vent or air-admittance valve in the drain line is to ensure good water flow from the sink to the gray tank. The longer the drain run to the tank, the more need for the vent/air-admittance valve.
When we did the venting on Little Girl, we tied the two new gray water tanks and black tank together through a single roof vent. I added an air-admittance valve under the bathroom sink. The tub is on the same drain line as the bathroom sink, so it's covered by the same air-admittance valve. I will add another one under the kitchen sink.