# Weak k-reconstruction of Cartesian products

Wilfried Imrich; Blaz Zmazek; Janez Zerovnik

Discussiones Mathematicae Graph Theory (2003)

- Volume: 23, Issue: 2, page 273-285
- ISSN: 2083-5892

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topWilfried Imrich, Blaz Zmazek, and Janez Zerovnik. "Weak k-reconstruction of Cartesian products." Discussiones Mathematicae Graph Theory 23.2 (2003): 273-285. <http://eudml.org/doc/270511>.

@article{WilfriedImrich2003,

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.},

author = {Wilfried Imrich, Blaz Zmazek, Janez Zerovnik},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {reconstruction problem; Cartesian product; composite graphs; Cartesian products},

language = {eng},

number = {2},

pages = {273-285},

title = {Weak k-reconstruction of Cartesian products},

url = {http://eudml.org/doc/270511},

volume = {23},

year = {2003},

}

TY - JOUR

AU - Wilfried Imrich

AU - Blaz Zmazek

AU - Janez Zerovnik

TI - Weak k-reconstruction of Cartesian products

JO - Discussiones Mathematicae Graph Theory

PY - 2003

VL - 23

IS - 2

SP - 273

EP - 285

AB - 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.

LA - eng

KW - reconstruction problem; Cartesian product; composite graphs; Cartesian products

UR - http://eudml.org/doc/270511

ER -

## References

top- [1] W. Dörfler, Some results on the reconstruction of graphs, Colloq. Math. Soc. János Bolyai, 10, Keszthely, Hungary (1973) 361-383.
- [2] T. Feder, Product graph representations, J. Graph Theory 16 (1992) 467-488, doi: 10.1002/jgt.3190160508. Zbl0766.05092
- [3J. Feigenbaum and R. Haddad, On factorable extensions and subgraphs of prime graphs, SIAM J. Discrete Math. 2 (1989) 197-218. Zbl0736.05061
- [4] J. Fisher, A counterexample to the countable version of a conjecture of Ulam, J. Combin. Theory 7 (1969) 364-365, doi: 10.1016/S0021-9800(69)80063-3. Zbl0187.21304
- [5] J. Hagauer and J. Zerovnik, An algorithm for the weak reconstruction of Cartesian-product graphs, J. Combin. Information & System Sciences 24 (1999) 87-103. Zbl1219.05094
- [6W. Imrich, Embedding graphs into Cartesian products, Graph Theory and Applications: East and West, Ann. New York Acad. Sci. 576 (1989) 266-274.
- [7] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (John Wiley & Sons, New York, 2000).
- [8] W. Imrich and J. Zerovnik, Factoring Cartesian product graphs, J. Graph Theory 18 (1994) 557-567. Zbl0811.05054
- [9] W. Imrich and J. Zerovnik, On the weak reconstruction of Cartesian-product graphs, Discrete Math. 150 (1996) 167-178, doi: 10.1016/0012-365X(95)00185-Y. Zbl0858.05076
- [10] S. Klavžar, personal communication.
- [11] K.L. MacAvaney, A conjecture on two-vertex deleted subgraphs of Cartesian products, Lecture Notes in Math. 829 (1980) 172-185, doi: 10.1007/BFb0088911.
- [12] G. Sabidussi, Graph multiplication, Math. Z. 72 (1960) 446-457, doi: 10.1007/BF01162967. Zbl0093.37603
- [13] J. Sims, Stability of the cartesian product of graphs (M. Sc. thesis, University of Melbourne, 1976).
- [14] J. Sims and D.A. Holton, Stability of cartesian products, J. Combin. Theory (B) 25 (1978) 258-282, doi: 10.1016/0095-8956(78)90002-3. Zbl0405.05052
- [15] S.M. Ulam, A Collection of Mathematical Problems, (Wiley, New York, 1960) p. 29. Zbl0086.24101

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