La demi-isomorphie et les tournois fortement connexes finis

Bouaziz, Moncef; Boudabbous, Youssef

doi:10.1016/s1631-073x(02)02439-1

Cited by 8 publications

(6 citation statements)

References 7 publications

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“…Second, we obtain the following corollary, which is an easy consequence of Lemma 3.22. Notice that, in the case of tournaments, the above corollary was obtained by M. Bouaziz and Y. Boudabbous in [5].…”

Section: Balanced Lemma Of Boussaïri and Applicationsmentioning

confidence: 62%

Two {4,n-3}-isomorphic n-vertex digraphs are hereditarily isomorphic

Boudabbous

2019

Filomat

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Let D and D be two digraphs with the same vertex set V, and let F be a set of positive integers. The digraphs D and D are hereditarily isomorphic whenever the (induced) subdigraphs D[X] and D [X] are isomorphic for each nonempty vertex subset X. They are F-isomorphic if the subdigraphs D[X] and D [X] are isomorphic for each vertex subset X with | X |∈ F. In this paper, we prove that if D and D are two {4, n − 3}-isomorphic n-vertex digraphs, where n ≥ 9, then D and D are hereditarily isomorphic. As a corollary, we obtain that given integers k and n with 4 ≤ k ≤ n − 6, if D and D are two {n − k}-isomorphic n-vertex digraphs, then D and D are hereditarily isomorphic. To the memory of my dear master Gérard LOPEZ who taught me and gave me the passion of reconstruction. With all my gratitude and admiration. 1. Introduction All digraphs mentioned here are finite, and have no loops and no multiple edges. Thus a digraph (or directed graph) D consists of a nonempty and finite set V(D) of vertices with a collection E(D) of ordered pairs of distinct vertices, called the set of edges of D. Such a digraph is denoted by (V(D), E(D)). We recall the basic notions of the reconstruction problem in the theory of relations what we apply to the case of digraphs. Consider two digraphs D and D on the same vertex set V with |V| = n ≥ 1, and let k be a positive integer. The digraphs D and D are hereditarily isomorphic if the subdigraphs D[X] and D [X], induced on X, are isomorphic for each nonempty subset X of V. They are k-isomorphic whenever for every k-element vertex subset X, the subdigraphs D[X] and D [X] are isomorphic. They are (≤ k)-isomorphic if they are k-isomorphic for every positive integer k with k ≤ k. The digraphs D and D are (−k)-isomorphic whenever either k ≥ n or D and D are (n − k)-isomorphic with k < n. Let F be a set of non zero integers. The digraphs D and D are F-isomorphic whenever D and D are p-isomorphic for every p ∈ F. The digraph D is F-reconstructible if every digraph F-isomorphic to D is 2010 Mathematics Subject Classification. 05C20;05C60

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Section: Balanced Lemma Of Boussaïri and Applicationsmentioning

confidence: 62%

Two {4,n-3}-isomorphic n-vertex digraphs are hereditarily isomorphic

Boudabbous

2019

Filomat

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“…A tournament is indecomposable if all its intervals are trivial; otherwise, it is decomposable. For example, the tournament C 3 = ({1, 2, 3}, {(1, 2), (2,3), (3, 1)}) is indecomposable, whereas, the tournaments C 4 = ({1, 2, 3, 4}, {(1, 2), (2,3), (3,4), (4,1), (3,1), (2, 4)}), δ + = ({1, 2, 3, 4}, {(1, 2), (2,3), (3,1), (1,4), (2,4), (3,4)}) and δ − = (δ + ) * are decomposable.…”

Section: Preliminaries On Tournamentsmentioning

confidence: 99%

“…Lemma 30 [3] Let T = (V, A) and T ′ = (V ′ , A ′ ) be two isomorphic tournaments, f be an isomorphism from T onto T ′ , i ∈ V and R i (resp. R ′ i ) be a tournament defined on a vertex set I i (resp.…”

Section: Given a Non Empty Subset A Of V And Let γmentioning

confidence: 99%

The {-3}-reconstruction and the {-3}-self duality of tournaments

Achour,

Boudabbous,

Boussairi

2012

Preprint

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Let T = (V, A) be a (finite) tournament and k be a non negative integer. For every subset X of V is associated the subtournament T [X] = (X, A ∩ (X × X)) of T , induced by X. The dual tournament of T , denoted by T * , is the tournament obtained from T by reversing all its arcs. The tournament T is self dual if it is isomorphic to its dual. T is {−k}-self dual if for each set X of k vertices, T [V \ X] is self dual. T is strongly self dual if each of its induced subtournaments is self dual. A subset I of V is an interval of T if for a, b ∈ I and for x ∈ V \ I, (a, x) ∈ A if and only if (b, x) ∈ A. For instance, ∅, V and {x}, where x ∈ V , are intervals of T called trivial intervals. T is indecomposable if all its intervals are trivial; otherwise, it is decomposableSuppose that T is decomposable and | V |≥ 9. In this paper, we begin by proving the equivalence between the {−3}-self duality and the strong self duality of T . Then we characterize each tournament {−3}-hypomorphic to T . As a consequence of this characterization, we prove that if there is no intervalFinally, we conclude by reducing the {−3}-reconstruction problem to the indecomposable case (between a tournament and its dual). In particular, we find and improve, in a less complicated way, the results of [6] found by Y. Boudabbous and A. Boussaïri.

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“…Say then that a tournament is strongly connected when it admits at most a single strongly connected component. For example, the tournament C 3 = ({1, 2, 3}, { (1,2), (2,3), (3, 1)}) is strongly connected. However, the two tournaments D 1 = ({1, 2, 3, 4}, { (1,2), (2,3), (3,1), (1,4), (2,4), (3,4)}) and D 2 = D * 1 , are non-strongly connected and admit two strongly connected components: {1, 2, 3} and {4}.…”

Section: Strong Connectivity Indecomposability Interval Partition mentioning

confidence: 99%

HEREDITARY HEMIMORPHY OF {-Κ}-Hemimorphic TOURNAMENTS FOR ≥ 5

Bouaziz¹,

Boudabbous²,

Amri³

2011

Journal of the Korean Mathematical Society

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Abstract. Let T = (V, A) be a tournament. With every subset X of V is associated the subtournament T [X] = (X, A ∩ (X × X)) of T , induced by X. The dual of T , denoted by T * , is the tournament obtained from T by reversing all its arcs. Given a tournament T ′ = (V, A ′ ) and a nonnegative integer k, T and T ′ are {−k}-hemimorphic provided that for allare isomorphic. The tournaments T and T ′ are said to be hereditarily hemimorphic if for all subset X of V , the subtournaments T [X] and T ′ [X] are hemimorphic. The purpose of this paper is to establish the hereditary hemimorphy of the {−k}-hemimorphic tournaments on at least k + 7 vertices, for every k ≥ 5.

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La demi-isomorphie et les tournois fortement connexes finis

Cited by 8 publications

References 7 publications

Two {4,n-3}-isomorphic n-vertex digraphs are hereditarily isomorphic

Two {4,n-3}-isomorphic n-vertex digraphs are hereditarily isomorphic

The {-3}-reconstruction and the {-3}-self duality of tournaments

HEREDITARY HEMIMORPHY OF {-Κ}-Hemimorphic TOURNAMENTS FOR ≥ 5

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