Width two posets are reconstructible
type: Item
instance of: scholarly article
title: Width two posets are reconstructible
zbMATH Open document ID: 0874.06002
DOI: 10.1016/0012-365X(95)00295-8
author: Dieter Kratsch
author: Jean-Xavier Rampon
published in: Discrete Mathematics
publication date: 1997-11-05T00:00:00Z
review text: Given a poset \(P\) with vertex set \(V(P)\), the collection of subposets \(D(P)= \{V(P)-X\mid X\in V(P)\}\) with the order inherited from \(P\) is the deck of \(P\). For posets ``the reconstruction problem'' is to determine whether \(D(P)= D(Q)\) implies \(P=Q\) (up to isomorphism). The reconstruction problem can be viewed in more general settings but is probably most interesting for classes of finite posets. Thus counterexamples exist for Ulam's unrestricted version. The authors of this very useful addition look at the class of width two posets and in their constructive proof are able to take advantage of known results (classes where the reconstruction conjecture holds) to construct a proof which also uses earlier technical results by these same authors. It is observed that at width 4 there may exist counterexamples since ``weak counterexamples'' do exist and also that the methods used here are width 2 specific. Thus a width \(n\to \text{width } (n-1)\) reduction method may be unavailable and the search for some other ``general parameter'' by means of which a ``most general method'' for handling this problem may be constructed is still wide open. Nevertheless, the class of width 2 posets includes many new examples not handled in other ways.
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 05C60
zbMATH DE Number: 1019042
zbMATH Keywords: deck of a poset
zbMATH Keywords: reconstruction
zbMATH Keywords: width two posets
zbMATH Keywords: width 4
reviewed by: J. Neggers
MaRDI profile type: Publication
cites work: Q3136258
cites work: Graph reconstruction—a survey
cites work: Q3975132
cites work: On the Structure of Finite T0 + T5 Spaces
cites work: Q4881848
cites work: A counterexample about poset reconstruction
cites work: Towards the reconstruction of posets
cites work: The falsity of the reconstruction conjecture for tournaments
cites work: Q4004146
description: scientific article; zbMATH DE number 1019042