On the complexity of unfrozen problems

We consider questions such as what is the complexity of recognizing instances of (monotonic) NP-complete problems in which no variable is fixed (or frozen) by the set of solutions. Since this unfrozenness is also a monotonic property in NP, this leads to an inductive sequence of properties for each monotone NP-complete property. In some cases the sequence remains NP-complete, while in others it at some point enters P. Determining the boundaries is particularly challenging. We also consider the related questions of recognizing maximal properties. This study was motivated by results from statistical mechanics being applied to phase transitions of NP-complete problems, which show a correlation of hard instances with a sudden increase in frozen variables.