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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Elsevier
2025-06-01
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| Series: | Examples and Counterexamples |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666657X25000011 |
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| author | Anupam Mondal Pritam Chandra Pramanik |
| author_facet | Anupam Mondal Pritam Chandra Pramanik |
| author_sort | Anupam Mondal |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-e623f654fb3e41649f8c475f5e59db04 |
| institution | Kabale University |
| issn | 2666-657X |
| language | English |
| publishDate | 2025-06-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Examples and Counterexamples |
| spelling | doaj-art-e623f654fb3e41649f8c475f5e59db042025-01-19T06:26:45ZengElsevierExamples and Counterexamples2666-657X2025-06-017100174A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphsAnupam Mondal0Pritam Chandra Pramanik1Institute for Advancing Intelligence (IAI), TCG CREST, Kolkata – 700091, West Bengal, India; Academy of Scientific & Innovative Research (AcSIR), Ghaziabad – 201002, Uttar Pradesh, India; Corresponding author at: Institute for Advancing Intelligence (IAI), TCG CREST, Kolkata – 700091, West Bengal, India.Institute for Advancing Intelligence (IAI), TCG CREST, Kolkata – 700091, West Bengal, India; Department of Mathematics, National Institute of Technology (NIT) Durgapur, Durgapur – 713209, West Bengal, IndiaRobin 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.http://www.sciencedirect.com/science/article/pii/S2666657X25000011Disconnected graphSimplicial complexDiscrete Morse theoryGradient vector fieldHomotopy |
| spellingShingle | Anupam Mondal Pritam Chandra Pramanik A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs Examples and Counterexamples Disconnected graph Simplicial complex Discrete Morse theory Gradient vector field Homotopy |
| title | A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs |
| title_full | A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs |
| title_fullStr | A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs |
| title_full_unstemmed | A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs |
| title_short | A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs |
| title_sort | note on an application of discrete morse theoretic techniques on the complex of disconnected graphs |
| topic | Disconnected graph Simplicial complex Discrete Morse theory Gradient vector field Homotopy |
| url | http://www.sciencedirect.com/science/article/pii/S2666657X25000011 |
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