1-perfect codes in Sierpinski graphs

Sierpinski graphs S(n,k) generalise the Tower of Hanoi graphs-the graph S(n,3) is isomorphic to the graph H-n 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, k) possess unique 1-perfect codes, thus extending a previously known result for H-n. An efficient decoding algorithm is also presented. The present approach, in particular the proposed (de)coding, is intrinsically different from the approach to H-n.