# Factorizations of properties of graphs

• Volume: 19, Issue: 2, page 167-174
• ISSN: 2083-5892

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## Abstract

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A property of graphs is any isomorphism closed class of simple graphs. For given properties of graphs ₁,₂,...,ₙ a vertex (₁, ₂, ...,ₙ)-partition of a graph G is a partition V₁,V₂,...,Vₙ of V(G) such that for each i = 1,2,...,n the induced subgraph $G\left[{V}_{i}\right]$ has property ${}_{i}$. The class of all graphs having a vertex (₁, ₂, ...,ₙ)-partition is denoted by ₁∘₂∘...∘ₙ. A property is said to be reducible with respect to a lattice of properties of graphs if there are n ≥ 2 properties ₁,₂,...,ₙ ∈ such that = ₁∘₂∘...∘ₙ; otherwise is irreducible in . We study the structure of different lattices of properties of graphs and we prove that in these lattices every reducible property of graphs has a finite factorization into irreducible properties.

## How to cite

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Izak Broere, et al. "Factorizations of properties of graphs." Discussiones Mathematicae Graph Theory 19.2 (1999): 167-174. <http://eudml.org/doc/270424>.

@article{IzakBroere1999,
abstract = {A property of graphs is any isomorphism closed class of simple graphs. For given properties of graphs ₁,₂,...,ₙ a vertex (₁, ₂, ...,ₙ)-partition of a graph G is a partition V₁,V₂,...,Vₙ of V(G) such that for each i = 1,2,...,n the induced subgraph $G[V_i]$ has property $_i$. The class of all graphs having a vertex (₁, ₂, ...,ₙ)-partition is denoted by ₁∘₂∘...∘ₙ. A property is said to be reducible with respect to a lattice of properties of graphs if there are n ≥ 2 properties ₁,₂,...,ₙ ∈ such that = ₁∘₂∘...∘ₙ; otherwise is irreducible in . We study the structure of different lattices of properties of graphs and we prove that in these lattices every reducible property of graphs has a finite factorization into irreducible properties.},
author = {Izak Broere, Samuel John Teboho Moagi, Peter Mihók, Roman Vasky},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {factorization; property of graphs; irreducible property; reducible property; lattice of properties of graphs; partition; irreducible properties},
language = {eng},
number = {2},
pages = {167-174},
title = {Factorizations of properties of graphs},
url = {http://eudml.org/doc/270424},
volume = {19},
year = {1999},
}

TY - JOUR
AU - Izak Broere
AU - Samuel John Teboho Moagi
AU - Peter Mihók
TI - Factorizations of properties of graphs
JO - Discussiones Mathematicae Graph Theory
PY - 1999
VL - 19
IS - 2
SP - 167
EP - 174
AB - A property of graphs is any isomorphism closed class of simple graphs. For given properties of graphs ₁,₂,...,ₙ a vertex (₁, ₂, ...,ₙ)-partition of a graph G is a partition V₁,V₂,...,Vₙ of V(G) such that for each i = 1,2,...,n the induced subgraph $G[V_i]$ has property $_i$. The class of all graphs having a vertex (₁, ₂, ...,ₙ)-partition is denoted by ₁∘₂∘...∘ₙ. A property is said to be reducible with respect to a lattice of properties of graphs if there are n ≥ 2 properties ₁,₂,...,ₙ ∈ such that = ₁∘₂∘...∘ₙ; otherwise is irreducible in . We study the structure of different lattices of properties of graphs and we prove that in these lattices every reducible property of graphs has a finite factorization into irreducible properties.
LA - eng
KW - factorization; property of graphs; irreducible property; reducible property; lattice of properties of graphs; partition; irreducible properties
UR - http://eudml.org/doc/270424
ER -

## References

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1. [1] 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. Zbl0902.05026
2. [2] M. Borowiecki and P. Mihók, Hereditary properties of graphs, in: V.R. Kulli, ed., Advances in Graph Theory (Vishwa International Publication, Gulbarga, 1991), 42-69.
3. [3] A. Haviar and R. Nedela, On varieties of graphs, Discuss. Math. Graph Theory 18 (1998) 209-223, doi: 10.7151/dmgt.1077. Zbl0926.05033
4. [4] R.L. Graham, M. Grötschel and L. Lovász, Handbook of combinatorics (Elsevier Science B.V., Amsterdam, 1995). Zbl0833.05001
5. [5] T.R. Jensen and B. Toft, Graph colouring problems (Wiley-Interscience Publications, New York, 1995). Zbl0971.05046
6. [6] 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. Zbl0623.05043
7. [7] P. Mihók, G. Semanišin, Reducible Properties of Graphs, Discuss. Math. Graph Theory 15 (1995) 11-18, doi: 10.7151/dmgt.1002. Zbl0829.05057
8. [8] 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 Zbl0942.05056

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