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arXiv · 2201.00721

Categorifying connected domination via graph \"uberhomology

Luigi Caputi·Daniele Celoria·Carlo Collari
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Abstract

\"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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BibTeXRIS

Luigi Caputi, Daniele Celoria, Carlo Collari. 2022-01-03. Categorifying connected domination via graph \"uberhomology. https://doi.org/10.1016/j.jpaa.2023.107381

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