A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs
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 wa...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Elsevier
2025-06-01
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| Series: | Examples and Counterexamples |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666657X25000011 |
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| Summary: | 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. |
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| ISSN: | 2666-657X |