Criteria of the existence of uniquely partionable graphs with respect to additive induced-hereditary properties

Let P1,P2, . . . ,Pn be graph properties, a graph G is said to be uniquely (P1,P2, . . . ,Pn)-partitionable if there is exactly one (unordered) partition {V1, V2, . . . , Vn} of V (G) such that G[Vi] ∈ Pi for i = 1, 2, . . . , n. We prove that for additive and induced-hereditary properties uniquely (P1,P2, . . . ,Pn)-partitionable graphs exist if and 32 I. Broere, J. Bucko and P. Mihók only if Pi and Pj are either coprime or equal irreducible properties of graphs for every i 6= j, i, j ∈ {1, 2, . . . , n}.