Abstract
We define certain generalisations of hypergraph hypomorphisms, which we call k-morphisms, \((k,n-k)\)-hypomorphisms, partial \((k,n-k)\)-hypomorphisms. They are special bijections between collections of k-subsets of vertex sets of hypergraphs. We show that these mappings lead to alternative representations of the automorphism groups of r-uniform hypergraphs and vertex stabilisers of graphs. We also use them to show that almost every r-uniform hypergraph is reconstructible and \((k,n-k)\)-reconstructible. As a consequence we also obtain the result that almost every r-uniform hypergraph is asymmetric.
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References
Bollobás, B.: Almost every graph has reconstruction number 3. J. Graph Theory 14, 1–4 (1990)
Chinn, P.Z.: A graph with \(p\) order subgraphs is reconstructible. In: Capobianco M. et al. (eds.) Recent Trends in Graph Theory, vol. 186 of Lecture Notes in Mathematics, pp. 71-73. Springer-Verlag (1971)
Czimmermann, P.: The \((k, n-k)\)-reconstruction of graphs. Studies of the University of Žilina 19,1–8 (2005)
Czimmermannová, O.: An introduction to the \((k, n-k)\)-reconstruction of hypergraphs. J. Inf. Control Manag. Syst. 7(1), 21–24 (2009)
Erdös, P., Rényi, A.: Asymmetric graphs. Acta Math. Acad. Sci. Hungar. 11, 295–315 (1963)
Hashiguchi, K.: The Double Reconstruction Conjectures about Colored Hypergraphs and Colored Directed Graphs, in Latin 92, vol. 583 of Lecture Notes in Computer Science. pp. 246–261. Springer-Verlag, ISBN 3-540-55284-7 (1992)
Hashiguchi, K.: The Double Reconstruction Conjecture about Finite Colored Hypergraphs. J. Comb. Theory Ser. B 54(1), 64–76 (1992)
Kocay, W.L.: A family of non-reconstructible hypergraphs. J. Comb. Theory Ser. B 42(1), 46–63 (1987)
Korshunov, A.D.: Number of nonisomorphic graphs in an \(n\)-point graph. Math. Notes Acad. USSR 9, 155–160 (1971)
Lauri, J., Scapellato, R.: Topics in graph automorphisms and reconstruction. Cambridge University Press, Cambridge (2003)
Müller, V.: Probabilistic reconstruction from subgraphs. Commen. Math. Univ. Carolinae 17, 709–719 (1976)
Ramachandran, S.: On a New Digraph Reconstruction Conjecture. J. Comb. Theory Ser. B 31(2), 143–149 (1981)
Stockmeyer, P.: The falsity of the reconstruction conjecture for tournaments. J. Graph Theory 1, 19–25 (1977)
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Czimmermann, P. Generalisations of Hypomorphisms and Reconstruction of Hypergraphs. Graphs and Combinatorics 32, 887–901 (2016). https://doi.org/10.1007/s00373-015-1639-x
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DOI: https://doi.org/10.1007/s00373-015-1639-x