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ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 20(1) (2000) 143-154
DOI: 10.7151/dmgt.1114

Unique Factorization Theorem

Peter Mihók

Mathematical Institute of Slovak Academy of Sciences
Grešákova 6, 040 01 Košice, Slovakia
Faculty of Economics, Technical University
B. Nšemcovej 32, 040 01 Košice, Slovakia



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 P1,P2, …,Pn be properties of graphs. A graph G is (P1,P2,...,Pn)-partitionable (G has property P1 ºP2 º… ºPn) if the vertex set V(G) of G can be partitioned into n sets V1,V2,..., Vn such that the subgraph G[Vi] of G induced by Vi belongs to Pi; i = 1,2,...,n. A property ℜ is said to be reducible if there exist properties P1 and P2 such that ℜ = P1 ºP2; otherwise the property ℜ 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 (P1,P2, ...,Pn)-partitionable graphs for any irreducible properties P1,P2, ...,Pn.

Keywords: induced-hereditary, additive property of graphs, reducible property of graphs, unique factorization, uniquely partitionable graphs, generating sets


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Received 22 March 2000
Revised 4 May 2000