Geodesic
Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties
Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 1, pp. 31-37

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Let ₁,₂,...,ₙ be graph properties, a graph G is said to be uniquely (₁,₂, ...,ₙ)-partitionable if there is exactly one (unordered) partition V₁,V₂,...,Vₙ of V(G) such that G[V_i] ∈ _i for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (₁,₂,...,ₙ)-partitionable graphs exist if and only if _i and _j are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ 1,2,...,n.
Keywords: induced-hereditary properties, reducibility, divisibility, uniquely partitionable graphs. 
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Broere, Izak; Bucko, Jozef; Mihók, Peter. Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties. Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 1, pp. 31-37. http://geodesic.mathdoc.fr/item/DMGT_2002_22_1_a3/