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Discussiones Mathematicae Graph Theory 23(2) (2003) 273-285
DOI: https://doi.org/10.7151/dmgt.1202

WEAK -RECONSTRUCTION OF CARTESIAN PRODUCTS

Wilfried Imrich Montanuniversität Leoben Institut für Mathematik und Angewandte Geometrie Franz-Josef Straß e 18, A-8700 Leoben, Austria e-mail: imrich@unileoben.ac.at

Blaz Zmazek and Janez Zerovnik University of Maribor Faculty of Mechanical Engineering Smetanova 17, 2000 Maribor, Slovenia and IMFM, Jadranska 19, Ljubljana e-mail: Blaz.Zmazek@uni-mb.si e-mail: Janez.Zerovnik@uni-lj.si

Abstract

By Ulam's conjecture every finite graph G can be reconstructed from its deck of vertex deleted subgraphs. The conjecture is still open, but many special cases have been settled. In particular, one can reconstruct Cartesian products.

We consider the case of k-vertex deleted subgraphs of Cartesian products, and prove that one can decide whether a graph H is a k-vertex deleted subgraph of a Cartesian product G with at least k+1 prime factors on at least k+1 vertices each, and that H uniquely determines G.

This extends previous work of the authors and Sims. The paper also contains a counterexample to a conjecture of MacAvaney.

Keywords: reconstruction problem, Cartesian product, composite graphs.

2000 Mathematics Subject Classification: 05C.

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Received 26 September 2001
Revised 12 April 2002