The minimal non-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:mo>⩽</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-reconstructible relations

Boudabbous, Youssef; Lopez, Gérard

doi:10.1016/j.disc.2004.08.016

Cited by 14 publications

(2 citation statements)

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“…2) Use respectively 2) of Theorem 1.5 and Theorem 1. Let T = (V, E) and T ′ = (V, E ′ ) be 2 tournaments and let k be a nonnegative integer; T and T ′ are khypomorphic [8,27] (resp. k -hypomorphic up to duality) if for every k -element subset K of V , the induced subtournaments T ′ ↾K and T ↾K are isomorphic (resp.…”

Section: Theorem 44mentioning

confidence: 99%

On a generalization of Kelly''s combinatorial lemma

Amira¹,

Dammak²,

Kaddour³

2014

Turk J Math

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Section: Theorem 44mentioning

confidence: 99%

On a generalization of Kelly''s combinatorial lemma

Amira¹,

Dammak²,

Kaddour³

2014

Turk J Math

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“…On the other hand, in [7] Boudabbous and C. Delhommé characterized the (≤ k)-reconstructible binary relations (finite or not), for each k ≥ 4. For the (≤ 3)-reconstruction, Boudabbous and Lopez [8] characterized the finite binary relations that are (≤ 3)-reconstructible. Hagendorf [4] proved that every finite poset with at least 4 vertices is (≤ 3)-reconstructible.…”

Section: Introductionmentioning

confidence: 99%

Description of the \((\leq 3)\)-hypomorphic Posets and Bichains. Application to the \((\leq 3)\)-reconstruction

Brahim,

Sayar

2024

Ars Comb.

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Two binary structures \(\mathfrak{R}\) and \(\mathfrak{R’}\) on the same vertex set \(V\) are \((\leq k)\)-hypomorphic for a positive integer \(k\) if, for every set \(K\) of at most \(k\) vertices, the two binary structures induced by \(\mathfrak{R}\) and \(\mathfrak{R’}\) on \(K\) are isomorphic. A binary structure \(\mathfrak{R}\) is \((\leq k)\)-reconstructible if every binary structure \(\mathfrak{R’}\) that is \((\leq k)\)-hypomorphic to \(\mathfrak{R}\) is isomorphic to \(\mathfrak{R}\). In this paper, we describe the pairs of \((\leq 3)\)-hypomorphic posets and the pairs of \((\leq 3)\)-hypomorphic bichains. As a consequence, we characterize the \((\leq 3)\)-reconstructible posets and the \((\leq 3)\)-reconstructible bichains. This answers a question suggested by Y. Boudabbous and C. Delhommé during a personal communication.

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Reconstruction from the deck of ‐vertex induced subgraphs

Spinoza

West

2018

Journal of Graph Theory

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The k‐ deck of a graph is its multiset of subgraphs induced by k vertices; we study what can be deduced about a graph from its k‐deck. We strengthen a result of Manvel by proving for ℓ ∈ double-struckN that when n is large enough ( n > 2 ℓ ( ℓ + 1 ) 2 suffices), the ( n − ℓ )‐deck determines whether an n‐vertex graph is connected ( n ≥ 25 suffices when ℓ = 3, and n ≤ 2 l cannot suffice). The reconstructibility ρ ( G ) of a graph G with n vertices is the largest ℓ such that G is determined by its ( n − ℓ )‐deck. We generalize a result of Bollobás by showing ρ ( G ) ≥ ( 1 − o ( 1 ) ) n ∕ 2 for almost all graphs. As an upper bound on min ρ ( G ), we have ρ ( C n ) = ⌈ n ∕ 2 ⌉ = ρ ( P n ) + 1. More generally, we compute ρ ( G ) whenever normalΔ ( G ) = 2, which involves extending a result of Stanley. Finally, we show that a complete r‐partite graph is reconstructible from its ( r + 1 )‐deck.

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The minimal non-(⩽k)-reconstructible relations

Cited by 14 publications

References 6 publications

On a generalization of Kelly''s combinatorial lemma

On a generalization of Kelly''s combinatorial lemma

Description of the \((\leq 3)\)-hypomorphic Posets and Bichains. Application to the \((\leq 3)\)-reconstruction

Reconstruction from the deck of ‐vertex induced subgraphs

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