Categorifying connected domination via graph überhomology
arXiv:2201.00721 · doi:10.1016/j.jpaa.2023.107381
Abstract
Überhomology is a recently defined homology theory for simplicial complexes, which yields subtle information on graphs. We prove that bold homology, a certain specialisation of überhomology, is related to dominating sets in graphs. To this end, we interpret überhomology 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.
20 pages, 11 figures and 1 table. Comments are welcome!