Unique factorization theorem

• Autorzy: Peter Mihók
• Języki publikacji: EN

Abstrakty

EN

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 $G[V_i]$ of G induced by V_i belongs to $𝓟_i$; 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.

Słowa kluczowe

EN

induced-hereditary; additive property of graphs; reducible property of graphs; unique factorization; uniquely partitionable graphs; generating sets

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2000
• Tom: 20
• Numer: 1
• Strony: 143-154

Daty

wydano

2000

otrzymano

2000-03-22

poprawiono

2000-05-04

Twórcy

autor

Peter Mihók

• Mathematical Institute of Slovak Academy of Sciences, Grešákova 6, 040 01 Košice, Slovakia
• Faculty of Economics, Technical University Košice, Slovakia

Bibliografia

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• [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)33:1<44::AID-JGT5>3.0.CO;2-O
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Identyfikatory

DOI

10.7151/dmgt.1114