In this thesis we introduce coding, codes and perfect codes, their generalization to graphs, and a connection between 1-perfect codes and dominating sets in graphs. We describe hypercubes and prove some of their characteristics. Dual-cubes, which are similar to hypercubes, are introduced. We show the characteristics that are inherited from hypercubes and some that are important for generating 1-perfect codes. We prove that the dual-cube $DQ_m$ consists of $2^{m+1}$ induced hypercubes. Hypercubes contain Hamming codes which are 1-perfect codes, and this is taken as a basis of creating 1-perfect codes in dual-cubes. Hamming codes are further transformed with algorithms that eventually lead to 1-perfect codes. With this we show that the dual-cube $DQ_m$ admits a 1-perfect code if and only if $m=2^k-2$ for $k\ge 2$. This result is used for proving tight bounds on the domination number of dual-cube with an arbitrary parameter.
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