Categorifying connected domination via graph \'uberhomology
| Autor: | Caputi, Luigi, Celoria, Daniele, Collari, Carlo |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Journal of Pure and Applied Algebra (2023) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.jpaa.2023.107381 |
| Popis: | \"Uberhomology is a recently defined homology theory for simplicial complexes, which yields subtle information on graphs. We prove that bold homology, a certain specialisation of \"uberhomology, is related to dominating sets in graphs. To this end, we interpret \"uberhomology as a poset homology, and investigate its functoriality properties. We then show that the Euler characteristic of the bold homology of a graph coincides with an evaluation of its connected domination polynomial. Even more, the bold chain complex retracts onto a complex generated by connected dominating sets. We conclude with several computations of this homology on families of graphs; these include a vanishing result for trees, and a characterisation result for complete graphs. Comment: 20 pages, 11 figures and 1 table. Comments are welcome! |
| Databáze: | arXiv |
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