Geodesic
Uniquely partitionable planar graphs with respect to properties having a forbidden tree
Discussiones Mathematicae. Graph Theory, Tome 19 (1999) no. 1, pp. 71-78

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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 G[V_i] has the property _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.
Keywords: uniquely partitionable planar graphs, forbidden graphs 
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     title = {Uniquely partitionable planar graphs with respect to properties having a forbidden tree},
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Bucko, Jozef; Ivančo, Jaroslav. Uniquely partitionable planar graphs with respect to properties having a forbidden tree. Discussiones Mathematicae. Graph Theory, Tome 19 (1999) no. 1, pp. 71-78. http://geodesic.mathdoc.fr/item/DMGT_1999_19_1_a5/