1. A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs
- Author
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Anupam Mondal and Pritam Chandra Pramanik
- Subjects
Disconnected graph ,Simplicial complex ,Discrete Morse theory ,Gradient vector field ,Homotopy ,Mathematics ,QA1-939 - Abstract
Robin Forman’s highly influential 2002 paper 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 does not seem to hold. In this note, we provide a couple of counterexamples against this specific claim. We also provide an alternative proof of the bigger claim that the constructed discrete vector field is indeed a gradient vector field. Our proof technique relies on a key observation which is not specific to the problem at hand, and thus is applicable while verifying a constructed discrete vector field is a gradient one in general.
- Published
- 2025
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