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

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