Geodesic
Unique factorization theorem
Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 1, pp. 143-154.

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A property of graphs is any class of graphs closed under isomorphism. A property of graphs is induced-hereditary and additive if it is closed under taking induced subgraphs and disjoint unions of graphs, respectively. Let ₁,₂, ...,ₙ be properties of graphs. A graph G is (₁,₂,...,ₙ)-partitionable (G has property ₁ º₂ º... ºₙ) if the vertex set V(G) of G can be partitioned into n sets V₁,V₂,..., Vₙ such that the subgraph G[V_i] of G induced by V_i belongs to _i; i = 1,2,...,n. A property is said to be reducible if there exist properties ₁ and ₂ such that = ₁ º₂; otherwise the property is irreducible. We prove that every additive and induced-hereditary property is uniquely factorizable into irreducible factors. Moreover the unique factorization implies the existence of uniquely (₁,₂, ...,ₙ)-partitionable graphs for any irreducible properties ₁,₂, ...,ₙ.
Keywords: induced-hereditary, additive property of graphs, reducible property of graphs, unique factorization, uniquely partitionable graphs, generating sets 
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Mihók, Peter. Unique factorization theorem. Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 1, pp. 143-154. http://geodesic.mathdoc.fr/item/DMGT_2000_20_1_a11/

[1] D. Achlioptas, J.I. Brown, D.G. Corneil and M.S.O. Molloy, The existence of uniquely -G colourable graphs, Discrete Math. 179 (1998) 1-11, doi: 10.1016/S0012-365X(97)00022-8.

[2] A. Berger, Reducible properties have infinitely many minimal forbidden subgraphs, manuscript.

[3] B. Bollobás and A.G. Thomason, Hereditary and monotone properties of graphs, in: R.L. Graham and J. Nesetril, eds., The mathematics of Paul Erdős, II, Algorithms and Combinatorics vol. 14 (Springer-Verlag, 1997), 70-78.

[4] M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, Survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997) 5-50, doi: 10.7151/dmgt.1037.

[5] I. Broere, J. Bucko, Divisibility in additive hereditary graph properties and uniquely partitionable graphs, Tatra Mountains Math. Publications 18 (1999) 79-87.

[6] E.J. Cockayne, Color clasess for r-graphs, Canad. Math. Bull. 15 (3) (1972) 349-354, doi: 10.4153/CMB-1972-063-2.

[7] R.L. Graham, M. Grötschel and L. Lovász, Handbook of combinatorics (Elsevier Science B.V., Amsterdam, 1995).

[8] T.R. Jensen and B. Toft, Graph colouring problems (Wiley-Interscience Publications, New York, 1995).

[9] J. Kratochvíl, P. Mihók, Hom-properties are uniquely factorizable into irreducible factors, Discrete Math. 213 (2000) 189-194, doi: 10.1016/S0012-365X(99)00179-X.

[10] P. Mihók, Additive hereditary properties and uniquely partitionable graphs, in: M. Borowiecki and Z. Skupień, eds., Graphs, hypergraphs and matroids (Zielona Góra, 1985), 49-58.

[11] P. Mihók and R. Vasky, On the factorization of reducible properties of graphs into irreducible factors, Discuss. Math. Graph Theory 15 (1995) 195-203, doi: 10.7151/dmgt.1017.

[12] P. Mihók, Reducible properties and uniquely partitionable graphs, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 49 (1999) 213-218.

[13] P. Mihók, G. Semanišin and R. Vasky, Additive and Hereditary Properties of Graphs are Uniquely Factorizable into Irreducible Factors, J. Graph Theory 33 (2000) 44-53, doi: 10.1002/(SICI)1097-0118(200001)33:144::AID-JGT5>3.0.CO;2-O

[14] G. Semanišin, On generating sets of induced-hereditary properties, manuscript.