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arXiv · 2305.08086

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

Anupam Mondal·Pritam Chandra Pramanik
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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 doesn't 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.

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Anupam Mondal, Pritam Chandra Pramanik. 2023-05-14. A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs. https://doi.org/10.1016/j.exco.2025.100174

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