Geodesic
Weak k-reconstruction of Cartesian products
Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 2, pp. 273-285.

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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 
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Imrich, Wilfried; Zmazek, Blaz; Zerovnik, Janez. Weak k-reconstruction of Cartesian products. Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 2, pp. 273-285. http://geodesic.mathdoc.fr/item/DMGT_2003_23_2_a5/

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