Catalytic Converters (Brands and Questions)

[PHOENIX]

I am getting ready to buy a few more of my exhuast components and trying to figure some pricing.

Catalytic Converter Requirements for my exhuast plans:
3" In and 3" out (no exception)
Quantity Required = 1

First my question:
What does the air tube on a catalytic converter do?
Does it allow the flowing exhuast gases to suck fresh air into the cat to help it work (air helps heat, cats work when hot)?
Is there an advantage to getting one with or without an air tube?

I believe my current catalytic converter does not have an air tube, but I have not checked in awhile.
Eitherway it doesn't matter what I have know.
The question is about the air tube; what is it for and is it an advantage?

Brands:
I would like to get something nice like a Random Technologies catalytic converter because they are very high flow and would work well with the high flow muffler.
But they are not cheap at $199.99 each.

Any comments (good, bad, etc.) about these units (no specific order):
(1) Dynomax Super Converter ($130-$140 each)
(2) Magnaflow Performance Hi-Flow Converter ($110 each)
(3) Catco ($66 each)

Thanks!