Geodesic
Minimal forbidden subgraphs of reducible graph properties
Discussiones Mathematicae. Graph Theory, Tome 21 (2001) no. 1, pp. 111-117

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A property of graphs is any class of graphs closed under isomorphism. Let ₁,₂,...,ₙ be properties of graphs. A graph G is (₁,₂,...,ₙ)-partitionable if the vertex set V(G) can be partitioned into n sets, V₁,V₂,..., Vₙ, such that for each i = 1,2,...,n, the graph G[V_i] ∈ _i. We write ₁∘₂∘...∘ₙ for the property of all graphs which have a (₁,₂,...,ₙ)-partition. An additive induced-hereditary property is called reducible if there exist additive induced-hereditary properties ₁ and ₂ such that = ₁∘₂. Otherwise is called irreducible. An additive induced-hereditary property can be defined by its minimal forbidden induced subgraphs: those graphs which are not in but which satisfy that every proper induced subgraph is in . We show that every reducible additive induced-hereditary property has infinitely many minimal forbidden induced subgraphs. This result is also seen to be true for reducible additive hereditary properties.
Keywords: reducible graph properties, forbidden subgraphs, induced subgraphs 
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Berger, Amelie. Minimal forbidden subgraphs of reducible graph properties. Discussiones Mathematicae. Graph Theory, Tome 21 (2001) no. 1, pp. 111-117. http://geodesic.mathdoc.fr/item/DMGT_2001_21_1_a7/