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Robin Forman's highly influential 2002 paper titled A User's Guide toDiscrete Morse Theory presents an overview of the subject in a very readablemanner. As a proof of concept, the author determines the topology (homotopytype) of the abstract simplicial complex of disconnected graphs of order $n$(which was previously done by Victor Vassiliev using classical topologicalmethods) using discrete Morse theoretic techniques, which are purelycombinatorial in nature. The techniques involve the construction (andverification) of a discrete gradient vector field on the complex. However, theverification part relies on a claim that doesn't seem to hold. In this note, weprovide a couple of counterexamples against this specific claim and also analternative proof of the claim that the constructed discrete vector field isindeed a gradient vector field.

A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs