NP-hardness of computing PL geometric category in dimension 2

Abstract: 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.