What does a vent stack look like?
There will always be a vent stack in a home if you have plumbing in it.  they usually poke out the roof to vent and go all the way to the basement and under the basement floor.  All the waste water dumps into these pipes and exits the house to the main waste pipe outside your home.  Depending on how many bathrooms you have the number of vent stacks can differ but most homes with one bathroom and a kitchen have two vent stacks.  The bathroom vent stack will be the larger one (3-4 inches) and the kitchen is generally around a 2 inch pipe I believe.  Usually when you tie into a vent stack you cut out a chunk and insert a "y" fitting of some sort.  I am no plumber by any means but I have observed this being done. Also, when you tie into a vent stack there is a code in my state where you have to have the vent for the tie-in go above the fixture above it (i.e. above where the kitchen sink drain dumps into the stack if you tie into that stack)  I think the name of that would be a branch vent ( see my simple diagram below). Good luck.