Minimal reducible bounds for planar graphs

AbstractFor properties of graphs P1 and P2 a vertex (P1,P2)-partition of a graph G is a partition (V1,V2) of V(G) such that each subgraph G[Vi] induced by Vi has property Pi,i=1,2. The class of all vertex (P1,P2)-partitionable graphs is denoted by P1∘P2. An additive hereditary property R is reducible if there exist additive hereditary properties P1 and P2 such that R=P1∘P2, otherwise it is irreducible. For a given property P a reducible property R is called a minimal reducible bound for P if P⊆R and there is no reducible property R′ satisfying P⊆R′⊂R. In this paper we give a survey of known reducible bounds and we prove some new minimal reducible bounds for important classes of planar graphs. The connection between our results and Barnette's conjecture is also presented