The PL geometric category of a polyhedron $P$, denoted $\hbox{plgcat}(P)$, provides a natural upper bound for the Lusternik--Schnirelmann category and it is defined as the minimum number of PL collapsible subpolyhedra of $P$ that cover $P$. In dimension 2 the PL geometric category is at most~3. It is easy to characterize/recognize $2$-polyhedra $P$ with $\hbox{plgcat}(P) = 1$. Borghini provided a partial characterization of $2$-polyhedra with $\hbox{plgcat}(P) = 2$. We complement his result by showing that it is NP-hard to decide whether $\hbox{plgcat}(P)\leq 2$. Therefore, we should not expect much more than a partial characterization, at least in algorithmic sense. Our reduction is based on the observation that 2-dimensional polyhedra $P$ admitting a shellable subdivision satisfy $\hbox{plgcat}(P) \leq 2$ and a (nontrivial) modification of the reduction of Goaoc, Pat\'{a}k, Pat\'{a}kov\'{a}, Tancer and Wagner showing that shellability of $2$-complexes is NP-hard.

翻译：多面体$P$的PL几何范畴（记作$\hbox{plgcat}(P)$）为Lusternik–Schnirelmann范畴提供了自然的上界，其定义为覆盖$P$所需的最小PL可缩子多面体数量。在二维情形下，PL几何范畴至多为3。$\hbox{plgcat}(P) = 1$的二维多面体$P$易于刻画与识别。Borghini给出了$\hbox{plgcat}(P) = 2$的二维多面体的部分刻画。我们通过证明判定$\hbox{plgcat}(P)\leq 2$是NP困难的，从而补充了他的结论。因此，至少在算法意义上，我们不应预期比部分刻画更深入的结果。我们的归约基于以下观察：允许可壳子划分的二维多面体$P$满足$\hbox{plgcat}(P) \leq 2$，并结合对Goaoc、Paták、Patáková、Tancer与Wagner关于二维复形可壳性NP困难性归约的（非平凡）改进。