Discrete Morse-theoretic analysis of graph-associated complexes

Construct natural discrete Morse matchings for the independence complex, perfect matching complex, cut complex, and total cut complex of Möbius ladder graphs M_{2n}, and determine whether these matchings yield a unified description of the homotopy types of the four complexes.

Background

The paper obtains homotopy-type descriptions for the independence and perfect matching complexes of M_{2n} using graph reductions, links, deletions, and known results about paths and cycles. It does not provide a discrete Morse-theoretic framework unifying these computations.

The unresolved direction is to construct Morse matchings that respect the combinatorial symmetries of Möbius ladder graphs and to assess whether the resulting critical-cell structure explains the homotopy types of the independence, perfect matching, cut, and total cut complexes in a common way.