Geodesic
Decomposition tree and indecomposable coverings
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 1, pp. 37-44.

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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.
Keywords: interval, indecomposable, k-covering, decomposition tree 
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Breiner, Andrew; Deogun, Jitender; Ille, Pierre. Decomposition tree and indecomposable coverings. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 1, pp. 37-44. http://geodesic.mathdoc.fr/item/DMGT_2011_31_1_a2/

[1] R. McConnell and F. de Montgolfier, Linear-time modular decomposition of directed graphs, Discrete Appl. Math. 145 (2005) 198-209, doi: 10.1016/j.dam.2004.02.017.

[2] A. Cournier and M. Habib, An efficient algorithm to recognize prime undirected graphs, in: Graph-theoretic Concepts in Computer Science, Lecture Notes in Computer Science 657, E.W. Mayr, (Editor), (Springer, Berlin, 1993) 212-224.

[3] A. Ehrenfeucht and G. Rozenberg, Theory of 2-structures. I. Clans, basic subclasses, and morphisms, Theoret. Comput. Sci. 70 (1990) 277-303, doi: 10.1016/0304-3975(90)90129-6.

[4] T. Gallai, Transitiv orientierbare Graphen, Acta Math. Acad. Sci. Hungar. 18 (1967) 25-66, doi: 10.1007/BF02020961.

[5] M. Habib, Substitution des structures combinatoires, théorie et algorithmes, Ph.D. Thesis, Université Pierre et Marie Curie, Paris VI, 1981.

[6] P. Ille, Indecomposable graphs, Discrete Math. 173 (1997) 71-78, doi: 10.1016/S0012-365X(96)00097-0.

[7] D. Kelly, Comparability graphs, in: Graphs and Orders, I. Rival, (Editor), Reidel (Drodrecht, 1985) 3-40.

[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.

[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.

[10] J. Spinrad, P₄-trees and substitution decomposition, Discrete Appl. Math. 39 (1992) 263-291, doi: 10.1016/0166-218X(92)90180-I.