An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove that the prime ideals of the lattice of all additive induced-hereditary properties are divided into two groups, determined either by a set of excluded join-irreducible properties or determined by a set of excluded properties with infinite join-decomposability number. We provide non-trivial examples of each type.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1189,
author = {Amelie J. Berger and Peter Mih\'ok},
title = {Prime ideals in the lattice of additive induced-hereditary graph properties},
journal = {Discussiones Mathematicae Graph Theory},
volume = {23},
year = {2003},
pages = {117-127},
zbl = {1037.05044},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1189}
}
Amelie J. Berger; Peter Mihók. Prime ideals in the lattice of additive induced-hereditary graph properties. Discussiones Mathematicae Graph Theory, Tome 23 (2003) pp. 117-127. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1189/
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