| Abstract: | A perfectdominatingset S of a graph Γ is a set of vertices of Γ such that every vertex of Γ is either in S or is adjacent to exactly one vertex of S. We show that a perfect dominating set of the n-cube Qn induces a subgraph of Qn whose components are isomorphic to hypercubes. We conjecture that each of these hypercubes has the same dimension. We then prove that if Qr is a component of the subgraph induced by S, then n ? r ? 1 or 3 (mod 6). A number of examples are given and connections with Steiner Systems and codes are noted. |
