Uniquely partitionable planar graphs with respect to properties having a forbidden tree

Let P1, P2 be graph properties. A vertex (P1,P2)-partition of a graph G is a partition {V1, V2} of V (G) such that for i = 1, 2 the induced subgraph G[Vi] has the property Pi. A property R = P1◦P2 is defined to be the set of all graphs having a vertex (P1,P2)-partition. A graph G ∈ P1◦P2 is said to be uniquely (P1,P2)-partitionable if G has exactly one vertex (P1,P2)-partition. In this note, we show the existence of uniquely partitionable planar graphs with respect to hereditary additive properties having a forbidden tree.