NP-hardness of computing PL geometric category in dimension 2
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The PL geometric category of a polyhedron P, denoted 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 plgcat(P) = 1. Borghini provided a partial characterization of 2-polyhedra with plgcat(P) = 2. We complement his result by showing that it is NP-hard to decide whether plgcat(P)≤ 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 plgcat(P) ≤ 2 and a (nontrivial) modification of the reduction of Goaoc, Paták, Patáková, Tancer and Wagner showing that shellability of 2-complexes is NP-hard.
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