Let G = (V,A) be a directed graph. With any subset X of V is associated the directed subgraph G[X] = (X,A ∩ (X×X)) of G induced by X. A subset X of V is an interval of G provided that for a,b ∈ X and x ∈ V∖X, (a,x) ∈ A if and only if (b,x) ∈ A, and similarly for (x,a) and (x,b). For example ∅, V, and {x}, where x ∈ V, are intervals of G which are the trivial intervals. A directed graph is indecomposable if all its intervals are trivial. Given an integer k > 0, a directed graph G = (V,A) is called an indecomposable k-covering provided that for every subset X of V with |X| ≤ k, there exists a subset Y of V such that X ⊆ Y, G[Y] is indecomposable with |Y| ≥ 3. In this paper, the indecomposable k-covering directed graphs are characterized for any k > 0.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1528,
author = {Andrew Breiner and Jitender Deogun and Pierre Ille},
title = {Decomposition tree and indecomposable coverings},
journal = {Discussiones Mathematicae Graph Theory},
volume = {31},
year = {2011},
pages = {37-44},
zbl = {1284.05112},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1528}
}
Andrew Breiner; Jitender Deogun; Pierre Ille. Decomposition tree and indecomposable coverings. Discussiones Mathematicae Graph Theory, Tome 31 (2011) pp. 37-44. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1528/
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