In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure, is emphasized. Many related topics, including questions on the complexity of related problems, are investigated.

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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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