A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913654783344640 |
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| author | Mondal, Anupam Pramanik, Pritam Chandra |
| author_facet | Mondal, Anupam Pramanik, Pritam Chandra |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_08086 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs Mondal, Anupam Pramanik, Pritam Chandra Combinatorics Algebraic Topology 57Q70 (Primary), 05E45, 05C40 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. |
| title | A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs |
| topic | Combinatorics Algebraic Topology 57Q70 (Primary), 05E45, 05C40 |
| url | https://arxiv.org/abs/2305.08086 |