Discussiones Mathematicae Graph Theory 31(1) (2011) 37-44
doi: 10.7151/dmgt.1528

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DECOMPOSITION TREE AND INDECOMPOSABLE COVERINGS

Andrew Breiner

Department of Mathematics and Computer Science
Nebraska Wesleyan University
5000 St. Paul Avenue, Lincoln, NE 68504, USA
e-mail: abreiner@nebrwesleyan.edu

Jitender Deogun

Department of Computer Science and Engineering
University of Nebraska - Lincoln
Lincoln, NE 68588-0115, USA
e-mail: deogun@cse.unl.edu

Pierre Ille

C.N.R.S. - UMR 6206
Institut de Mathématiques de Luminy
163, Avenue de Luminy - Case 907
13288 Marseille Cedex 9, France
e-mail: ille@iml.univ-mrs.fr

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.

Keywords: interval, indecomposable, k-covering, decomposition tree.

2010 Mathematics Subject Classification: 05C20, 05C75.

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Received 11 June 2008
Revised 29 March 2010
Accepted 6 April 2010