A 3-SAT problem is called positive and planar if all the literals are positive and the clause-variable incidence graph (i.e., SAT graph) is planar. The NAE 3-SAT and 1-in-3-SAT are two variants of 3-SAT that remain NP-complete even when they are positive. The positive 1-in-3-SAT problem remains NP-complete under planarity constraint, but planar NAE 3-SAT is solvable in $O(n^{1.5}\log n)$ time. In this paper we prove that a positive planar NAE 3-SAT is always satisfiable when the underlying SAT graph is 3-connected, and a satisfiable assignment can be obtained in linear time. We also show that without 3-connectivity constraint, existence of a linear-time algorithm for positive planar NAE 3-SAT problem is unlikely as it would imply a linear-time algorithm for finding a spanning 2-matching in a planar subcubic graph. We then prove that positive planar 1-in-3-SAT remains NP-complete under the 3-connectivity constraint, even when each variable appears in at most 4 clauses. However, we show that the 3-connected planar 1-in-3-SAT is always satisfiable when each variable appears in an even number of clauses.

中文翻译：

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3-连通性约束下的正平面可满足性问题

如果所有文字都是正的并且子句变量关联图（即 SAT 图）是平面的，则 3-SAT 问题被称为正的和平面的。NAE 3-SAT 和 1-in-3-SAT 是 3-SAT 的两种变体，即使它们是正数，它们也保持 NP 完全性。正 1-in-3-SAT 问题在平面约束下仍然是 NP-完全问题，但平面 NAE 3-SAT 可以在 $O(n^{1.5}\log n)$ 时间内求解。在本文中，我们证明了当底层 SAT 图为 3-连通时，正平面 NAE 3-SAT 总是可满足的，并且可以在线性时间内获得可满足的分配。我们还表明，如果没有 3 连通性约束，则不太可能存在用于正平面 NAE 3-SAT 问题的线性时间算法，因为它暗示了用于在平面亚三次图中寻找跨越 2 匹配的线性时间算法。然后我们证明正平面 1-in-3-SAT 在 3-connectivity 约束下仍然是 NP-complete，即使每个变量最多出现在 4 个子句中。然而，我们表明当每个变量出现在偶数个子句中时，3-连通平面 1-in-3-SAT 总是可满足的。