Cut-primitive directed graphs versus clan-primitive directed graphs
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Youssef Boudabbous
Abstract
Given a directed graph G = (V,A), a subset X of V is a clan 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 instance, ∅, V and {x}, where x ∈ V, are clans of G, called trivial. A directed graph is clan-primitive if all its clans are trivial. Given a directed graph G = (V,A), a subset X of V is a cut of G if X and V \ X are clans of G. For example, ∅ and V are cuts of G, called trivial. A directed graph is cut-primitive if all its cuts are trivial. Ehrenfeucht and Rozenberg [Theoret. Comput. Sci. 70: 343–358, 1990] proved: Given a clan-primitive directed graph G = (V,A), if X is a subset of V such that |X| ≥ 3, |V \ X| ≥ 2 and G[X] is clan-primitive, then there are x ≠ y ∈ V \ X such that G[X ∪ {x, y}] is clan-primitive. We show: Given a clan-primitive directed graph G = (V,A), if X is a proper subset of V such that |X| ≥ 3 and G[X] is cut-primitive, then there are x, y ∈ V \ X such that G[X ∪ {x, y}] is cut-primitive and {x}, {y} are maximal proper clans of G[X ∪ {x, y}].
© de Gruyter 2010
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Articles in the same Issue
- An analog of Morgan's theorem for the Kontorovich–Lebedev transform
- Positivity of the Jacobi–Cherednik intertwining operator and its dual
- Absolute continuity of the representing measures of the Dunkl intertwining operator and of its dual and applications
- Cut-primitive directed graphs versus clan-primitive directed graphs
- On a class of variational-hemivariational inequalities involving set valued mappings
- Extremal problems in the class of analytic functions associated with Wright's function
- Averages on annuli of Euclidean space
- On the basis graph of a matroid of arbitrary cardinality
- Locally compact groups which are just not compact
