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1-perfect codes in Sierpiński graphs

Published online by Cambridge University Press:  17 April 2009

Sandi Klavžar ,
Uroš Milutinović  and
Ciril Petr
Sandi Klavžar
Affiliation:
Department of Mathematics, PeF, University of Maribor, Koroška 160, 2000 Maribor, Slovenia e-mail: [email protected], [email protected]
Uroš Milutinović
Affiliation:
Department of Mathematics, PeF, University of Maribor, Koroška 160, 2000 Maribor, Slovenia e-mail: [email protected], [email protected]
Ciril Petr
Affiliation:
Iskratel Telecommunications Systems Ltd., Tržaška 37a, 2000 Maribor, Slovenia e-mail: [email protected]
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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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 66 , Issue 3 , December 2002 , pp. 369 - 384
Copyright
Copyright © Australian Mathematical Society 2002

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