Conjunctive normal forms where every clause has length at most two are called 2-CNFs. We study minimally unsatisfiable 2-CNFs (2-MUs), that is, unsatisfiable 2-CNFs where removing any clause destroys unsatisfiability, and obtain their full classification up to isomorphism. The main tool is the implication digraph: we show that for 2-MUs these digraphs are "weak double cycles" (WDCs), big cycles of small cycles with possible overlaps. Combining logical and graph-theoretical methods, we prove that WDCs have at most one skew-symmetry (a self-inverse fixed-point-free anti-automorphism reversing the direction of arcs). It follows that the isomorphisms between 2-MUs are exactly the isomorphisms between their implication digraphs, reducing the classification of 2-MUs to the classification of a well-structured class of digraphs. We obtain a variety of applications for 2-MUs of deficiency k (the difference between the number of clauses and the number of variables): the smoothing of skew-symmetric WDCs corresponds exactly to the canonical normal form obtained by 1-singular Davis-Putnam reduction, and the resulting homeomorphism types are in one-to-one correspondence with binary bracelets of length k. The automorphism group of any 2-MU of deficiency k is a subgroup of the dihedral group with 4k elements. The isomorphism problem for 2-MUs is decidable in quadratic time, and the number of isomorphism types of 2-MUs for fixed k is Theta(n^(3k-1)). The article is addressed to both the logic and the graph theory communities, with complete proofs provided throughout.

翻译：每个子句长度至多为二的合取范式称为2-CNF。本文研究极小不可满足2-CNF（2-MU），即通过移除任一子句即可消除不可满足性的不可满足2-CNF，并得到其同构意义下的完全分类。主要工具是蕴含有向图：我们证明，对于2-MU，这些有向图是“弱双环”（WDCs），即由可能重叠的小环构成的大环。结合逻辑与图论方法，我们证明WDC至多存在一个斜对称（一种逆转弧方向、无不动点的自逆反自同构）。由此可知，2-MU之间的同构恰好是其蕴含有向图之间的同构，从而将2-MU的分类问题简化为一类结构优良的有向图的分类。针对亏度k（子句数与变量数之差）的2-MU，我们得到多种应用：斜对称WDC的光滑化恰好对应于通过1-奇异Davis-Putnam归约得到的规范范式，且所得同胚类型与长度为k的二元手环一一对应。任意亏度k的2-MU的自同构群是含4k个元素的二面体群的子群。2-MU的同构问题可在二次时间内判定，且固定k的2-MU同构类型数量为Θ(n^(3k-1))。本文面向逻辑与图论两个领域的研究者，全程提供完整证明。