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1-perfect codes in Sierpiński graphs

  • Sandi Klavžar (a1), Uroš Milutinović (a1) and Ciril Petr (a2)
Abstract

Sierpiński graphs S (n, κ) generalise the Tower of Hanoi graphs—the graph S (n, 3) is isomorphic to the graph Hn of the Tower of Hanoi with n disks. A 1-perfect code (or an efficient dominating set) in a graph G is a vertex subset of G with the property that the closed neighbourhoods of its elements form a partition of V (G). It is proved that the graphs S (n, κ) possess unique 1-perfect codes, thus extending a previously known result for Hn. An efficient decoding algorithm is also presented. The present approach, in particular the proposed (de)coding, is intrinsically different from the approach to Hn.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[4] N. Biggs , ‘Perfect codes in graphs’, J. Combin. Theory Ser. B 15 (1973), 289296.

[8] P. Hammond , ‘On the nonexistence of perfect and nearly perfect codes’, Discrete Math. 39 (1982), 105109.

[11] S. Klavžar and U. Milutinović , ‘Graphs S (n, kappa;) and a variant of the Tower of Hanoi problem’, Czechoslovak Math. J. 47 (1997), 95104.

[13] J. Kratochvíl , ‘Perfect codes over graphs’, J. Combin. Theory Ser. B 40 (1986), 224228.

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[17] J. Kratochvíl and M. Křivánek , ‘On the computational complexity of codes in graphs’, in Mathematical Foundations of Computer Science, Carlsbad, 1988, Lecture Notes in Comput. Sci. 324 (Springer-Verlag, Berlin, 1988), pp. 396404.

[21] S.L. Lipscomb and J.C. Perry , ‘Lipscomb's L (A) space fractalized in Hilbert's l2(A) space’, Proc. Amer. Math. Soc. 115 (1992), 11571165.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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