What are the bare minimum parts needed to operate a magnetron?
I believe the way microwave oven magnetrons are wired is that there are two coils on the transformer more or less in series, one to supply the HV and one to supply filament.  The filament doubles as the cathode and it has some series inductors to keep the RF from leaking out the power terminals.  The HV return from the anode could just be the metal case as it is less than an amp.  An inverter-driven microwave likely has the same setup, however the transformer will be smaller as the switching frequency will be much higher (60 Hz vs several kHz).  The inverter should have essentially the same connection to the magnetron as the older style single transformer solution.  Also, you won't be able to run the inverter transformer by itself, you'll also need the rest of the inverter drive and control electronics.  I'm not sure if it will work without the front panel/user interface; it may be required to turn the inverter on and possibly select the power level in some way.  

However, turning on a magnetron outside of a case is very dangerous.  Not only could you kill yourself with the high voltage, the high power microwaves could give you serious burns as well as possibly causing interference and damage to electronic deivces.  The magnetron could also overheat due to reflected power if it is not properly coupled to a waveguide.  

Edit: Looks like the inverter microwaves operate on exactly the same principle as the older ones, just at a much higher frequency.  You'll either need to keep the front panel or figure out what signal it sends to the inverter to turn it on.  



Image from http://www.electronicspoint.com/threads/microwave-inverter.234684/