Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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L. Caputi - D. Celoria - C. Collari

Categorifying connected domination via graph ├╝berhomology

created by collari on 04 Jan 2022
modified on 21 Apr 2023


Published Paper

Inserted: 4 jan 2022
Last Updated: 21 apr 2023

Journal: Journal of Pure and Applied Algebra
Year: 2022
Doi: 10.1016/j.jpaa.2023.107381

ArXiv: 2201.00721 PDF


\"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.

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