[LazyPup]

It depends upon which national model code your local code is patterned after.

Under the International Residential Code all structures are required to have on "Main Vent" which must run undiminished in size from the building "main drain" through the roof. After that is a achieved you may terminate auxiliary vents through the roof, in the attic space, through the wall under some conditions or you may use an unlimited number of AAV's (air admittance valves).

Under the Uniform Plumbing code "ALL Vents" must terminate through the roof and you are limited to a maximum of one AAV per structure with the expressed written consent of the local AHJ (authority having jurisdiction)

The preferred method of venting an island sink is by means of an "island loop vent".

I am attaching an illustration that shows how the loop vents are created.