Unique factorization theorem
Discussiones Mathematicae Graph Theory (2000)
- Volume: 20, Issue: 1, page 143-154
- ISSN: 2083-5892
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topPeter Mihók. "Unique factorization theorem." Discussiones Mathematicae Graph Theory 20.1 (2000): 143-154. <http://eudml.org/doc/270654>.
@article{PeterMihók2000,
abstract = {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 Vi 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.},
author = {Peter Mihók},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {induced-hereditary; additive property of graphs; reducible property of graphs; unique factorization; uniquely partitionable graphs; generating sets; additive property; reducible property; partitionable graphs; partition; induced-hereditary property; factors},
language = {eng},
number = {1},
pages = {143-154},
title = {Unique factorization theorem},
url = {http://eudml.org/doc/270654},
volume = {20},
year = {2000},
}
TY - JOUR
AU - Peter Mihók
TI - Unique factorization theorem
JO - Discussiones Mathematicae Graph Theory
PY - 2000
VL - 20
IS - 1
SP - 143
EP - 154
AB - 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 Vi 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.
LA - eng
KW - induced-hereditary; additive property of graphs; reducible property of graphs; unique factorization; uniquely partitionable graphs; generating sets; additive property; reducible property; partitionable graphs; partition; induced-hereditary property; factors
UR - http://eudml.org/doc/270654
ER -
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Citations in EuDML Documents
top- Gabriel Semanišin, On generating sets of induced-hereditary properties
- Alastair Farrugia, R. Bruce Richter, Unique factorisation of additive induced-hereditary properties
- Jozef Bucko, Peter Mihók, On infinite uniquely partitionable graphs and graph properties of finite character
- Izak Broere, Jozef Bucko, Peter Mihók, Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties
- Jozef Bucko, Peter Mihók, On uniquely partitionable relational structures and object systems
- Peter Mihók, Gabriel Semanišin, Unique factorization theorem for object-systems
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