# Decomposition tree and indecomposable coverings

• Volume: 31, Issue: 1, page 37-44
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Andrew Breiner, Jitender Deogun, and Pierre Ille. "Decomposition tree and indecomposable coverings." Discussiones Mathematicae Graph Theory 31.1 (2011): 37-44. <http://eudml.org/doc/270976>.

@article{AndrewBreiner2011,
abstract = {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.},
author = {Andrew Breiner, Jitender Deogun, Pierre Ille},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {interval; indecomposable; k-covering; decomposition tree; -covering},
language = {eng},
number = {1},
pages = {37-44},
title = {Decomposition tree and indecomposable coverings},
url = {http://eudml.org/doc/270976},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Andrew Breiner
AU - Jitender Deogun
AU - Pierre Ille
TI - Decomposition tree and indecomposable coverings
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 1
SP - 37
EP - 44
AB - 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.
LA - eng
KW - interval; indecomposable; k-covering; decomposition tree; -covering
UR - http://eudml.org/doc/270976
ER -

## References

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7. [7] D. Kelly, Comparability graphs, in: Graphs and Orders, I. Rival, (Editor), Reidel (Drodrecht, 1985) 3-40.
8. [8] F. Maffray and M. Preissmann, A translation of Tibor Gallai's paper: transitiv orientierbare Graphen, in: Perfect Graphs, J.J. Ramirez-Alfonsin and B.A. Reed, (Editors) (Wiley, New York, 2001) 25-66. Zbl0989.05050
9. [9] J. Schmerl and W. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures, Discrete Math. 113 (1993) 191-205, doi: 10.1016/0012-365X(93)90516-V. Zbl0776.06002
10. [10] J. Spinrad, P₄-trees and substitution decomposition, Discrete Appl. Math. 39 (1992) 263-291, doi: 10.1016/0166-218X(92)90180-I. Zbl0758.68036

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