We give a short, self-contained, and easily verifiable proof that determining the outerthickness of a general graph is NP-hard. This resolves a long-standing open problem on the computational complexity of outerthickness. Moreover, our hardness result applies to a more general covering problem $P_F$, defined as follows. Fix a proper graph class $F$ whose membership is decidable. Given an undirected simple graph $G$ and an integer $k$, the task is to cover the edge set $E(G)$ by at most $k$ subsets $E_1,\ldots,E_k$ such that each subgraph $(V(G),E_i)$ belongs to $F$. Note that if $F$ is monotone (in particular, when $F$ is the class of all outerplanar graphs), any such cover can be converted into an edge partition by deleting overlaps; hence, in this case, covering and partitioning are equivalent. Our result shows that for every proper graph class $F$ whose membership is decidable and that satisfies all of the following conditions: (a) $F$ is closed under topological minors, (b) $F$ is closed under $1$-sums, and (c) $F$ contains a cycle of length $3$, the problem $P_F$ is NP-hard for every fixed integer $k\ge 3$. In particular: For $F$ equal to the class of all outerplanar graphs, our result settles the long-standing open problem on the complexity of determining outerthickness. For $F$ equal to the class of all planar graphs, our result complements Mansfield's NP-hardness result for the thickness, which applies only to the case $k=2$. It is also worth noting that each of the three conditions above is necessary. If $F$ is the class of all eulerian graphs, then cond. (a) fails. If $F$ is the class of all pseudoforests, then cond. (b) fails. If $F$ is the class of all forests, then cond. (c) fails. For each of these three classes $F$, the problem $P_F$ is solvable in polynomial time for every fixed integer $k\ge 3$, showing that none of the three conditions can be dropped.

翻译：我们给出一个简短、自包含且易于验证的证明，表明确定一般图的外厚度是NP难的。这解决了一个长期悬而未决的关于外厚度计算复杂性的开放问题。此外，我们的困难性结果适用于一个更一般的覆盖问题$P_F$，其定义如下。固定一个成员资格可判定的真图类$F$。给定一个无向简单图$G$和一个整数$k$，任务是用最多$k$个子集$E_1,\ldots,E_k$覆盖边集$E(G)$，使得每个子图$(V(G),E_i)$属于$F$。注意，如果$F$是单调的（特别是当$F$是所有外平面图的类时），任何这样的覆盖都可以通过删除重叠转换为边划分；因此，在这种情况下，覆盖和划分是等价的。我们的结果表明，对于每个成员资格可判定且满足以下所有条件的真图类$F$：(a) $F$在拓扑子图下封闭，(b) $F$在$1$-和下封闭，以及(c) $F$包含一个长度为$3$的环，问题$P_F$对于每个固定的整数$k\ge 3$是NP难的。特别地：当$F$等于所有外平面图的类时，我们的结果解决了关于确定外厚度复杂性的长期开放问题。当$F$等于所有平面图的类时，我们的结果补充了Mansfield关于厚度的NP困难性结果，该结果仅适用于$k=2$的情况。还值得注意的是，上述三个条件中的每一个都是必要的。如果$F$是所有欧拉图的类，则条件(a)不成立。如果$F$是所有伪森林的类，则条件(b)不成立。如果$F$是所有森林的类，则条件(c)不成立。对于这三个类$F$中的每一个，问题$P_F$对于每个固定的整数$k\ge 3$都可以在多项式时间内求解，这表明三个条件中的任何一个都不能被省略。