Geodesic
On infinite uniquely partitionable graphs and graph properties of finite character
Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 2, pp. 241-251

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A graph property is any nonempty isomorphism-closed class of simple (finite or infinite) graphs. A graph property is of finite character if a graph G has a property if and only if every finite induced subgraph of G has a property . Let ₁,₂,...,ₙ be graph properties of finite character, a graph G is said to be (uniquely) (₁, ₂, ...,ₙ)-partitionable if there is an (exactly one) partition V₁, V₂, ..., Vₙ of V(G) such that G[V_i] ∈ _i for i = 1,2,...,n. Let us denote by ℜ = ₁ ∘ ₂ ∘ ... ∘ ₙ the class of all (₁,₂,...,ₙ)-partitionable graphs. A property ℜ = ₁ ∘ ₂ ∘ ... ∘ ₙ, n ≥ 2 is said to be reducible. We prove that any reducible additive graph property ℜ of finite character has a uniquely (₁, ₂, ...,ₙ)-partitionable countable generating graph. We also prove that for a reducible additive hereditary graph property ℜ of finite character there exists a weakly universal countable graph if and only if each property _i has a weakly universal graph.
Keywords: graph property of finite character, reducibility, uniquely partitionable graphs, weakly universal graph 
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Bucko, Jozef; Mihók, Peter. On infinite uniquely partitionable graphs and graph properties of finite character. Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 2, pp. 241-251. http://geodesic.mathdoc.fr/item/DMGT_2009_29_2_a2/