Categorifying connected domination via graph \"uberhomology

TL;DR

This paper explores a new homology theory called uberhomology, linking it to connected domination in graphs, and demonstrates its properties and computations for various graph families.

Contribution

It introduces the connection between uberhomology and connected domination, providing new insights and computational methods for graph homology.

Findings

Euler characteristic matches connected domination polynomial evaluation

Bold homology retracts onto connected dominating sets

Vanishing of homology for trees and characterization for complete graphs

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.