# The decomposability of additive hereditary properties of graphs

• Volume: 20, Issue: 2, page 281-291
• ISSN: 2083-5892

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## Abstract

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An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If ₁,...,ₙ are properties of graphs, then a (₁,...,ₙ)-decomposition of a graph G is a partition E₁,...,Eₙ of E(G) such that $G\left[{E}_{i}\right]$, the subgraph of G induced by ${E}_{i}$, is in ${}_{i}$, for i = 1,...,n. We define ₁ ⊕...⊕ ₙ as the property G ∈ : G has a (₁,...,ₙ)-decomposition. A property is said to be decomposable if there exist non-trivial hereditary properties ₁ and ₂ such that = ₁⊕ ₂. We study the decomposability of the well-known properties of graphs ₖ, ₖ, ₖ, ₖ, ₖ, ₖ and ${}^{p}$.

## How to cite

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Izak Broere, and Michael J. Dorfling. "The decomposability of additive hereditary properties of graphs." Discussiones Mathematicae Graph Theory 20.2 (2000): 281-291. <http://eudml.org/doc/270686>.

@article{IzakBroere2000,
abstract = {An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If ₁,...,ₙ are properties of graphs, then a (₁,...,ₙ)-decomposition of a graph G is a partition E₁,...,Eₙ of E(G) such that $G[E_i]$, the subgraph of G induced by $E_i$, is in $_i$, for i = 1,...,n. We define ₁ ⊕...⊕ ₙ as the property G ∈ : G has a (₁,...,ₙ)-decomposition. A property is said to be decomposable if there exist non-trivial hereditary properties ₁ and ₂ such that = ₁⊕ ₂. We study the decomposability of the well-known properties of graphs ₖ, ₖ, ₖ, ₖ, ₖ, ₖ and $^\{p\}$.},
author = {Izak Broere, Michael J. Dorfling},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {property of graphs; additive; hereditary; decomposable property of graphs; additive hereditary property; decomposability},
language = {eng},
number = {2},
pages = {281-291},
title = {The decomposability of additive hereditary properties of graphs},
url = {http://eudml.org/doc/270686},
volume = {20},
year = {2000},
}

TY - JOUR
AU - Izak Broere
AU - Michael J. Dorfling
TI - The decomposability of additive hereditary properties of graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2000
VL - 20
IS - 2
SP - 281
EP - 291
AB - An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If ₁,...,ₙ are properties of graphs, then a (₁,...,ₙ)-decomposition of a graph G is a partition E₁,...,Eₙ of E(G) such that $G[E_i]$, the subgraph of G induced by $E_i$, is in $_i$, for i = 1,...,n. We define ₁ ⊕...⊕ ₙ as the property G ∈ : G has a (₁,...,ₙ)-decomposition. A property is said to be decomposable if there exist non-trivial hereditary properties ₁ and ₂ such that = ₁⊕ ₂. We study the decomposability of the well-known properties of graphs ₖ, ₖ, ₖ, ₖ, ₖ, ₖ and $^{p}$.
LA - eng
KW - property of graphs; additive; hereditary; decomposable property of graphs; additive hereditary property; decomposability
UR - http://eudml.org/doc/270686
ER -

## References

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1. [1] M. Borowiecki and M. Hałuszczak, Decompositions of some classes of graphs, Report No. IM-3-99, Institute of Mathematics, Technical University of Zielona Góra, 1999. Zbl0905.05061
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3. [3] S.A. Burr, M.S. Jacobson, P. Mihók and G. Semanišin, Generalized Ramsey theory and decomposable properties of graphs, Discuss. Math. Graph Theory 19 (1999) 199-217, doi: 10.7151/dmgt.1095. Zbl0958.05094
4. [4] M. Hałuszczak and P. Vateha, On the completeness of decomposable properties of graphs, Discuss. Math. Graph Theory 19 (1999) 229-236, doi: 10.7151/dmgt.1097.
5. [5] 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
6. [6] J. Nesetril and V. Rödl, Simple proof of the existence of restricted Ramsey graphs by means of a partite construction, Combinatorica 1 (1981) 199-202, doi: 10.1007/BF02579274. Zbl0491.05044

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