Necessary Conditions for stability in z domain?
The one and only condition for BIBO stability of a 1D discrete-time system, in the z-domain, is that its transfer functions's ROC (region of convergence) should include the unit circle : $|z| =1$. Therefore, it's a necessary and sufficient condition for BIBO stability of a 1D SISO system.

There are no other conditions (to my knowledge). 

EDIT [based on comments] :
By definition of ROC (region of convergence) there cannot be any poles inside the ROC region. This's related not with stability but with convergence. And also, causality is not related with stability too. 

Properties of ROC is a different question to be answered.