NPhardness 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 LusternikSchnirelmann 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 NPhard 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 2dimensional polyhedra $P$ admitting a shellable subdivision satisfy $\hbox{plgcat}(P) \leq 2$ and a (nontrivial) modification of the reduction of Goaoc, Paták, Patáková, Tancer and Wagner showing that shellability of $2$complexes is NPhard.
 Publication:

arXiv eprints
 Pub Date:
 April 2022
 DOI:
 10.48550/arXiv.2204.13981
 arXiv:
 arXiv:2204.13981
 Bibcode:
 2022arXiv220413981S
 Keywords:

 Computer Science  Computational Geometry;
 Mathematics  Geometric Topology
 EPrint:
 Version 2: 15 pages, 5 figures