### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 37-44 |

Number of pages | 8 |

Journal | Discussiones Mathematicae - Graph Theory |

Volume | 31 |

Issue number | 1 |

DOIs | |

State | Published - 2011 |

### Fingerprint

### Keywords

- decomposition tree
- indecomposable
- interval
- k-covering

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discussiones Mathematicae - Graph Theory*,

*31*(1), 37-44. https://doi.org/10.7151/dmgt.1528

**Decomposition tree and indecomposable coverings.** / Breiner, Andrew; Deogun, Jitender; Ille, Pierre.

Research output: Contribution to journal › Article

*Discussiones Mathematicae - Graph Theory*, vol. 31, no. 1, pp. 37-44. https://doi.org/10.7151/dmgt.1528

}

TY - JOUR

T1 - Decomposition tree and indecomposable coverings

AU - Breiner, Andrew

AU - Deogun, Jitender

AU - Ille, Pierre

PY - 2011

Y1 - 2011

N2 - 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.

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.

KW - decomposition tree

KW - indecomposable

KW - interval

KW - k-covering

UR - http://www.scopus.com/inward/record.url?scp=78751504992&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78751504992&partnerID=8YFLogxK

U2 - 10.7151/dmgt.1528

DO - 10.7151/dmgt.1528

M3 - Article

AN - SCOPUS:78751504992

VL - 31

SP - 37

EP - 44

JO - Discussiones Mathematicae - Graph Theory

JF - Discussiones Mathematicae - Graph Theory

SN - 1234-3099

IS - 1

ER -