Reconstruction of Posets with the Same Comparability Graph

Ille, Pierre; Rampon, Jean-Xavier

doi:10.1006/jctb.1998.1859

Cited by 9 publications

(4 citation statements)

References 9 publications

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“…The inverse operation of the quotient is the lexicographical sum [22] defined as follows. Let P be a poset and let Q be a family of posets indexed by V (P ), write…”

Section: Posetsmentioning

confidence: 99%

On the power graph of a finite group

Feng¹,

Ma²,

Wang³

2014

Preprint

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The power graph P G of a finit group G is the graph with the vertex set G, where two elements are adjacent if one is a power of the other. We first show that P G has an transitive orientation, so it is a perfect graph and its core is a complete graph. Then we use the poset on all cyclic subgroups (under usual inclusion) to characterise the structure of P G . Finally, the closed formula for the metric dimension of P G is established. As an application, we compute the metric dimension of the power graph of a cyclic group.

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“…The inverse operation of the quotient is the lexicographical sum [22] defined as follows. Let P be a poset and let Q be a family of posets indexed by V (P ), write…”

Section: Posetsmentioning

confidence: 99%

On the power graph of a finite group

Feng¹,

Ma²,

Wang³

2014

Preprint

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“…Theorem 7 [9] Two f inite orders P and P on the same set V with n = |V| ≥ 4 elements are isomorphic provided that for every subset X of V such that |X| ∈ {2, n − 1}, the suborders P(X) and P (X) are isomorphic.…”

Section: Corollarymentioning

confidence: 99%

Prime Orders All of Whose Prime Suborders Are Selfdual

Boudabbous

Zaguia

2010

Order

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Let P be an order on a set V. A subset A of V is autonomous in P if every element of V not in A is either less than or greater than or incomparable to all elements of A. The empty set, the singletons from V and the set V are autonomous sets and are called trivial. Call an order prime if all its autonomous sets are trivial. We give the complete list of all finite prime orders all of whose prime suborders are selfdual. Mathematics Subject Classifications (2000) 06A06 · 06A07Throughout, P = (V, ≤) denotes a f inite ordered set. When this does not cause confusion, we shall not distinguish between the order and the ordered set, using the same letter, say P, to denote both. The suborder of P induced by a subset X of V is denoted P(X). A subset A of V is autonomous (or an interval or a module) in P if for all v ∈ V \ A and for all a, a ∈ A,The empty set, the singletons from V and the whole set V are autonomous sets and are said to be trivial. P is prime if all its autonomous sets are trivial. With this

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“…Nontrivial reconstructible classes are tree-like orders [6], interval orders [11] and width two orders [12]. Three other recent results are the recognizability of decomposable orders with at least 12 elements shown by Ille [8], the recognizability of orders defined by a wide class of forbidden substructures shown by Basso-Gerbelli and Ille [1], and the fÀ1; 2g-reconstruction of orders shown by Ille and Rampon [9]. Note that the two former results have been obtained in the more general context of binary relations.…”

Section: Introductionmentioning

confidence: 98%

Reconstruction of Finite Truncated Semi-Modular Lattices

Rampon¹,

Schröder²

2002

Journal of Combinatorial Theory, Series B

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In this paper, we prove reconstruction results for truncated lattices. The main results are that truncated lattices that contain a 4-crown and truncated semi-modular lattices are reconstructible. Reconstruction of the truncated lattices not covered by this work appears challenging. Indeed, the remaining truncated lattices possess very little lattice-typical structure. This seems to indicate that further progress on the reconstruction of truncated lattices is closely correlated with progress on reconstructing ordered sets in general. # 2002 Elsevier Science (USA) MSC: 06A07.

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Reconstruction of Posets with the Same Comparability Graph

Abstract: Given two finite posets P and P$ with the same comparability graph, we show that if |V(P)| 4 and if for all x # V(P), P&x & P$&x, then P &P$. This result leads us to characterize the finite posets P such that for all x # V(P), P&x & P*&x. Academic Press

Cited by 9 publications

References 9 publications

On the power graph of a finite group

On the power graph of a finite group

Prime Orders All of Whose Prime Suborders Are Selfdual

Reconstruction of Finite Truncated Semi-Modular Lattices

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