# Maximal graphs with respect to hereditary properties

• Volume: 17, Issue: 1, page 51-66
• ISSN: 2083-5892

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## Abstract

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A property of graphs is a non-empty set of graphs. A property P is called hereditary if every subgraph of any graph with property P also has property P. Let P₁, ...,Pₙ be properties of graphs. We say that a graph G has property P₁∘...∘Pₙ if the vertex set of G can be partitioned into n sets V₁, ...,Vₙ such that the subgraph of G induced by Vi has property ${P}_{i}$; i = 1,..., n. A hereditary property R is said to be reducible if there exist two hereditary properties P₁ and P₂ such that R = P₁∘P₂. If P is a hereditary property, then a graph G is called P- maximal if G has property P but G+e does not have property P for every e ∈ E([G̅]). We present some general results on maximal graphs and also investigate P-maximal graphs for various specific choices of P, including reducible hereditary properties.

## How to cite

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Izak Broere, Marietjie Frick, and Gabriel Semanišin. "Maximal graphs with respect to hereditary properties." Discussiones Mathematicae Graph Theory 17.1 (1997): 51-66. <http://eudml.org/doc/270401>.

@article{IzakBroere1997,
abstract = {A property of graphs is a non-empty set of graphs. A property P is called hereditary if every subgraph of any graph with property P also has property P. Let P₁, ...,Pₙ be properties of graphs. We say that a graph G has property P₁∘...∘Pₙ if the vertex set of G can be partitioned into n sets V₁, ...,Vₙ such that the subgraph of G induced by Vi has property $P_i$; i = 1,..., n. A hereditary property R is said to be reducible if there exist two hereditary properties P₁ and P₂ such that R = P₁∘P₂. If P is a hereditary property, then a graph G is called P- maximal if G has property P but G+e does not have property P for every e ∈ E([G̅]). We present some general results on maximal graphs and also investigate P-maximal graphs for various specific choices of P, including reducible hereditary properties.},
author = {Izak Broere, Marietjie Frick, Gabriel Semanišin},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {hereditary property of graphs; maximal graphs; vertex partition; hereditary property; reducible property; -degenerate graph},
language = {eng},
number = {1},
pages = {51-66},
title = {Maximal graphs with respect to hereditary properties},
url = {http://eudml.org/doc/270401},
volume = {17},
year = {1997},
}

TY - JOUR
AU - Izak Broere
AU - Marietjie Frick
AU - Gabriel Semanišin
TI - Maximal graphs with respect to hereditary properties
JO - Discussiones Mathematicae Graph Theory
PY - 1997
VL - 17
IS - 1
SP - 51
EP - 66
AB - A property of graphs is a non-empty set of graphs. A property P is called hereditary if every subgraph of any graph with property P also has property P. Let P₁, ...,Pₙ be properties of graphs. We say that a graph G has property P₁∘...∘Pₙ if the vertex set of G can be partitioned into n sets V₁, ...,Vₙ such that the subgraph of G induced by Vi has property $P_i$; i = 1,..., n. A hereditary property R is said to be reducible if there exist two hereditary properties P₁ and P₂ such that R = P₁∘P₂. If P is a hereditary property, then a graph G is called P- maximal if G has property P but G+e does not have property P for every e ∈ E([G̅]). We present some general results on maximal graphs and also investigate P-maximal graphs for various specific choices of P, including reducible hereditary properties.
LA - eng
KW - hereditary property of graphs; maximal graphs; vertex partition; hereditary property; reducible property; -degenerate graph
UR - http://eudml.org/doc/270401
ER -

## References

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