Summary: A property of graphs is any isomorphism closed class of simple graphs. For given properties of graphs \({\mathcal P}_1,{\mathcal P}_2,\dots, {\mathcal P}_n\) a vertex \(({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n)\)-partition of a graph \(G\) is a partition \(\{V_1,V_2,\dots, V_n\}\) of \(V(G)\) such that for each \(i= 1,2,\dots, n\) the induced subgraph \(G[V_i]\) has property \({\mathcal P}_i\). The class of all graphs having a vertex \(({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n)\)-partition is denoted by \({\mathcal P}_1\circ{\mathcal P}_2\circ\cdots\circ{\mathcal P}_n\). A property \({\mathcal R}\) is said to be reducible with respect to a lattice of properties of graphs \(\mathbb{L}\) if there are \(n\geq 2\) properties \({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n\in \mathbb{L}\) such that \({\mathcal R}={\mathcal P}_1\circ{\mathcal P}_2\circ\cdots\circ{\mathcal P}_n\); otherwise \({\mathcal R}\) is irreducible in \(\mathbb{L}\). We study the structure of different lattices of properties of graphs and we prove that in these lattices every reducible property of graphs has a finite factorization into irreducible properties.