Minimal reducible bounds for induced-hereditary properties

Abstract

Let ( M a ,⊆) and ( L a ,⊆) be the lattices of additive induced-hereditary properties of graphs and additive hereditary properties of graphs, respectively. A property R∈ M a ( ∈ L a ) is called a minimal reducible bound for a property P∈ M a ( ∈ L a ) if in the interval ( P, R) of the lattice M a ( L a ) there are only irreducible properties. The set of all minimal reducible bounds of a property P∈ M a in the lattice M a we denote by B M( P) . Analogously, the set of all minimal reducible bounds of a property P∈ L a in L a is denoted by B L( P) . We establish a method to determine minimal reducible bounds for additive degenerate induced-hereditary (hereditary) properties of graphs. We show that this method can be successfully used to determine already known minimal reducible bounds for k-degenerate graphs and outerplanar graphs in the lattice L a . Moreover, in terms of this method we describe the sets of minimal reducible bounds for partial k-trees and the graphs with restricted order of components in L a and k-degenerate graphs in M a .