The recognition of the class of indecomposable digraphs under low hemimorphy

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Given a digraph G = (V, A), the subdigraph of G induced by a subset X of V is denoted by G [X]. With each digraph G = (V, A) is associated its dual G{star operator} = (V, A{star operator}) defined as follows: for any x, y ∈ V, (x, y) ∈ A{star operator} if (y, x) ∈ A. Two digraphs G and H are hemimorphic if G is isomorphic to H or to H{star operator}. Given k > 0, the digraphs G = (V, A) and H = (V, B) are k-hemimorphic if for every X ⊆ V, with | X | ≤ k, G [X] and H [X] are hemimorphic. A class C of digraphs is k-recognizable if every digraph k-hemimorphic to a digraph of C belongs to C. In another vein, given a digraph G = (V, A), 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, 0{combining long solidus overlay}, {x}, where x ∈ V, and V are intervals called trivial. A digraph is indecomposable if all its intervals are trivial. We characterize the indecomposable digraphs which are 3-hemimorphic to a non-indecomposable digraph. It follows that the class of indecomposable digraphs is 4-recognizable. © 2008 Elsevier B.V. All rights reserved.




Boussaïri, A., & Ille, P. (2009). The recognition of the class of indecomposable digraphs under low hemimorphy. Discrete Mathematics, 309(10), 3404–3407.

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