Minimal reducible bounds in the lattice of additive hereditary graph properties

AbstractAn additive hereditary property of graphs is any class of graphs closed under subgraphs, disjoint unions and isomorphisms. These properties can be ordered under set inclusion to form a lattice. In this lattice, we show that every irreducible property has at least one minimal reducible bound, and that if an irreducible property is contained in a reducible property, there exists a minimal reducible bound for the irreducible property between them. We give an example of a property with uncountably many minimal reducible bounds. In addition we show that if a reducible property strictly contains another property, then the reducible property is a minimal reducible bound for some property between them