Minimal reducible bounds for induced-hereditary properties

Let (M ;⊆) and (L ;⊆) be the lattices of additive induced-hereditary properties of graphs and additive hereditary properties of graphs, respectively. A property R∈Ma (∈ L) is called a minimal reducible bound for a property P∈Ma (∈ L) if in the interval (P;R) of the lattice M (L) there are only irreducible properties. The set of all minimal reducible bounds of a property P∈Ma in the lattice M we denote by BM (P). Analogously, the set of all minimal reducible bounds of a property P∈ L in L is denoted by BL(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. 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 and k-degenerate graphs in M. c © 2004 Elsevier B.V. All rights reserved.