Shellings of simplicial complexes have long been a useful tool in topological and algebraic combinatorics. Shellings of a complex expose a large amount of information in a helpful way, but are not easy to construct, often requiring deep information about the structure of the complex. It is natural to ask whether shellings may be efficiently found computationally. In a recent paper, Goaoc, Paták, Patáková, Tancer, and Wagner gave a negative answer to this question (assuming \(\mathsf {P}\ne \mathsf {NP}\)), showing that the problem of deciding whether a simplicial complex is shellable is \(\mathsf {NP}\)-complete. In this paper, we give simplified constructions of various gadgets used in the \(\mathsf {NP}\)-completeness proof of these authors. Using these gadgets combined with relative shellability and other ideas, we also exhibit a simpler proof of the \(\mathsf {NP}\)-completeness of the shellability decision problem. Our method systematically uses relative shellings to build up large shellable complexes with desired properties.

长期以来，简单络合物的脱壳一直是拓扑和代数组合论中的有用工具。复合物的外壳以有用的方式公开了大量信息，但不容易构建，通常需要有关复合物结构的深入信息。很自然地问，是否可以通过计算有效地发现炮弹。在最近的一篇论文中，Goaoc，Paták，Patáková，Tancer和Wagner对这个问题给出了否定的答案（假设\（\ mathsf {P} \ ne \ mathsf {NP} \）），这表明了决定是否可简化的简单复数是\（\ mathsf {NP} \）-完全。在本文中，我们给出了\（\ mathsf {NP} \）中使用的各种小工具的简化构造。这些作者的完整性证明。通过将这些小工具与相对可移植性和其他思想相结合，我们还展示了可（\ mathsf {NP} \） -可移植性决策问题的完整性的更简单证明。我们的方法系统地使用相对脱壳来构建具有所需特性的大型可脱壳复合物。