  For Boolean satisfiability problems, the structure of the solution space is
characterized by the solution graph, where the vertices are the solutions, and
two solutions are connected iff they differ in exactly one variable. Motivated
by research on heuristics and the satisfiability threshold, Gopalan et al. in
2006 studied connectivity properties of the solution graph and related
complexity issues for constraint satisfaction problems in Schaefer's framework.
They found dichotomies for the diameter of connected components and for the
complexity of the st-connectivity question, and conjectured a trichotomy for
the connectivity question that we recently were able to prove.
  While Gopalan et al. considered CNF(S)-formulas with constants, we here look
at two important variants: CNF(S)-formulas without constants, and partially
quantified formulas. For the diameter and the st-connectivity question, we
prove dichotomies analogous to those of Gopalan et al. in these settings. While
we cannot give a complete classification for the connectivity problem yet, we
identify fragments where it is in P, where it is coNP-complete, and where it is
PSPACE-complete, in analogy to Gopalan et al.'s trichotomy.
