On the Structure of Solution-Graphs for Boolean Formulas

In this work we extend the study of solution graphs and prove that for boolean formulas in a class called CPSS, all connected components are partial cubes of small dimension, a statement which was proved only for some cases in [16]. In contrast, we show that general Schaefer formulas are powerful enough to encode graphs of exponential isometric dimension and graphs which are not even partial cubes. Our techniques shed light on the detailed structure of st-connectivity for Schaefer and connectivity for CPSS formulas, problems which were already known to be solvable in polynomial time. We refine this classification and show that the problems in these cases are equivalent to the satisfiability problem of related formulas by giving mutual reductions between (st-)connectivity and satisfiability. An immediate consequence is that st-connectivity in (undirected) solution graphs of Horn-formulas is P-complete while for 2SAT formulas st-connectivity is NL-complete.