A property of graphs is any nonempty class of graphs closed under isomorphism. A property of graphs is called induced-hereditary and additive if it is closed under taking induced subgraphs and disjoint unions of graphs, respectively. Let \({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n\) be properties of graphs. A \(({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n)\)-partition of a graph \(G\) is a partition \((V_1,V_2,\dots, V_n)\) of the vertex set of \(G\) such that the subgraph of \(G\) induced by \(V_i\) belongs to \({\mathcal P}_i\); \(i= 1,\dots, n\). In the case \({\mathcal P}_1={\mathcal P}_2=\cdots={\mathcal P}_n\) such a partition is called a \(({\mathcal P},n)\)-partition. The class of \(({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n)\)-partitionable graphs is denoted by \({\mathcal P}_1\circ{\mathcal P}_2\circ\cdots\circ{\mathcal P}_n\). If \(G\) has exactly one (unordered) \(({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n)\)-partition, then \(G\) is uniquely \(({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n)\)-partitionable. An induced-hereditary property \({\mathcal R}\) is reducible if there exist induced-hereditary properties, \({\mathcal P}_1\), \({\mathcal P}_2\), such that \({\mathcal R}={\mathcal P}_1\circ{\mathcal P}_2\); otherwise \({\mathcal R}\) is irreducible. The author proves the following very significant result: Every additive, induced-hereditary property is uniquely factorizable into irreducible factors. One corollary of this result is that if \({\mathcal P}\) is an additive, induced-hereditary property then, for \(n\geq 2\) there exist uniquely \(({\mathcal P},n)\)-partitionable graphs if and only if \({\mathcal P}\) is irreducible. Another interesting corollary is that if \({\mathcal R}={\mathcal P}_1\circ{\mathcal P}_2\circ\cdots\circ{\mathcal P}_n\) is the unique factorization of a reducible additive, induced-hereditary property \({\mathcal R}\) into irreducible factors, then every graph \(G\in{\mathcal R}\) is an induced subgraph of some uniquely \(({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n)\)-partitionable graph.