On the minimal reducible bound for outerplanar and planar graphs

Let L be the set of all additive and hereditary properties of graphs. For P1, P2 ∈ L we define the reducible property R = P1 P2 as follows: G ∈ P1P2 if there is a bipartition (V1, V2) of V(G) such that 〈V1〉 ∈ P1 and 〈V2〉 ∈ P2. For a property P ∈ L, a reducible property R is called a minimal reducible bound for P if P ⊆ R and for each reducible property R′, R′ ⊂ R → P ⊉ R′. It is proved that the class of all outerplanar graphs has exactly two minimal reducible bounds in L. Some related problems for planar graphs are discussed.