Classification of minimally unsatisfiable 2-CNFs

We consider minimally unsatisfiable 2-CNFs (short 2-MUs). Characterisations of 2-MUs in the literature have been restricted to the nonsingular case (where every variable occurs positively and negatively at least twice), and those with a unit-clause. We provide the full classification of 2-MUs F. The main tool is the implication digraph, and we show that the implication digraph of F is a "weak double cycle" (WDC), a big cycle of small cycles (with possible overlaps). Combining logical and graph-theoretical methods, we prove that WDCs have at most one skew-symmetry, and thus we obtain that the isomorphisms between 2-MUs F, F' are exactly the isomorphisms between their implication digraphs. We obtain a variety of applications. For fixed deficiency k, the difference of the number of clauses of F and the number n of variables of F, the automorphism group of F is a subgroup of the Dihedral group with 4k elements. The isomorphism problem restricted to 2-MUs F is decidable in linear time for fixed k. The number of isomorphism types of 2-MUs for fixed k is Theta(n^3k-1). The smoothing (removal of linear vertices) of skew-symmetric WDCs corresponds exactly to the canonical normal form of F obtained by 1-singular DP-reduction, a restricted form of DP-reduction (or "variable elimination") only reducing variables of degree 2. The isomorphism types of these normal forms, i.e., the homeomorphism types of skew-symmetric WDCs, are in one-to-one correspondence with binary bracelets (or "turnover necklaces") of length k.