  Robin Forman's highly influential 2002 paper titled A User's Guide to
Discrete Morse Theory presents an overview of the subject in a very readable
manner. As a proof of concept, the author determines the topology (homotopy
type) of the abstract simplicial complex of disconnected graphs of order $n$
(which was previously done by Victor Vassiliev using classical topological
methods) using discrete Morse theoretic techniques, which are purely
combinatorial in nature. The techniques involve the construction (and
verification) of a discrete gradient vector field on the complex. However, the
verification part relies on a claim that doesn't seem to hold. In this note, we
provide a couple of counterexamples against this specific claim and also an
alternative proof of the claim that the constructed discrete vector field is
indeed a gradient vector field.
