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：PL 几何类别 $P$, 意指 $\hbox{plgcat} (P) $\hbox{plgcat} (P), 为 Lusternik- Schnirelmann 类别提供了一个天然的上方框, 它被定义为 $P$ 的PL 折叠 Subpolyhedra 最低数量。 在范围2 中, PL 几何类别最多为~ 3 。 因此, 很容易用 $\hbox{plcat} (P) = 1$。 Boghini 提供了2$- pollyhedra 的部分定性。 我们补充了他的结果, 显示它很难确定 $\hbox{plc> (P)\ legclat 类别是否是部分的描述, 至少在算法意义上, 我们的削减是基于2- $P$(P$) 和 attleqkbox 显示可变的 $2 和 a suble suble suble a suble suble sult suffik} pal\\\\ suffik} a sufflible $\\\\\\\\\\\\\\\\\\\\\\ lagkbox a pal_ suffilty\\\\\\\\ $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la la $= a a a a a la la la la la la la la la