In my area the plumbing lines are required to have vents, so the line is vented with no vent required at the sink itself. In my reno this required putting holes in a lot of walls to properly tie into the vent stack which goes to the roof, but now we're all up to code. Not the most technical answer, I know, but we hired a plumber to do the job. My island will have what everyone around here calls a 'cheater vent' which I gather is not up to code, and is a small vent branching off from the drain under the sink. Only way I could have an island sink without some major work due to it's location.

that's helpful regarding venting _after_ water flows through the Ptrap.

Before the P trap there is often an air gap.

Here is a link that might be useful: purpose of airgap in dishwasher hose plumbing