Abstract:A hereditary property of graphs is any class of graphs closed under isomorphism and subgraphs. Let P 1 , P 2 , . . . , P n be hereditary properties of graphs. We say that a graph G has property P 1 • P 2 • · · · • P n if the vertex set of G can be partitioned into n sets V 1 , V 2 , . . . , V n such that the subgraph of G induced by V i belongs to P i ; i = 1, 2, . . . , n. A hereditary property is said to be reducible if there exist hereditary properties P 1 and P 2 such that R = P 1 • P 2 ; otherwise it is irreducible. We prove that the factorization of a reducible hereditary property into irreducible factors is unique whenever the property is additive, i.e., it is closed under the disjoint union of graphs.

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Let P 1 , . . . , P n be properties of graphs. A (P 1 , . . . , P n )-partition of a graph G is a partition of the vertex set V (G) into subsets V 1 , . . . , V n such that the subgraph G[V i ] induced by V i has property P i ; i = 1, . . . , n. A graph G is said to be uniquely (P 1 , . . . , P n )-partitionable if G has exactly one (P 1 , . . . , P n )-partition. A property P is called hereditary if every subgraph of every graph with property P also has property P. If every graph that is a disjoint union of two graphs that have property P also has property P, then we say that P is additive. A property P is called degenerate if there exists a bipartite graph that does not have property P. In this paper, we prove that if P 1 , . . . , P n are degenerate, additive, hereditary properties of graphs, then there exists a uniquely (P 1 , . . . , P n )-partitionable graph.

A hereditary property R of graphs is said to be reducible if there exist hereditary properties P 1 , P 2 such that G ∈ R if and only if the set of vertices of G can be partitioned into V (G) = V 1 ∪ V 2 so that V 1 ∈ P 1 and V 2 ∈ P 2. The problem of the factorization of reducible properties into irreducible factors is investigated.

A property of graphs is any isomorphism closed class of simple graphs. For given properties of graphs P 1 , P 2 ,. .. , P n a vertex (P 1 , P 2 ,. .. , P n)-partition of a graph G is a partition {V 1 , V 2 ,. .. , V n } of V (G) such that for each i = 1, 2,. .. , n the induced subgraph G[V i ] has property P i. The class of all graphs having a vertex (P 1 , P 2 ,. .. , P n)partition is denoted by P 1 •P 2 • • • • •P n. A property R is said to be reducible with respect to a lattice of properties of graphs L if there are n ≥ 2 properties P 1 , P 2 ,. .. , P n ∈ L such that R =P 1 •P 2 • • • • •P n ; otherwise R is irreducible in 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.