Perfect codes on graphs

The set of codewords for a standard error-correcting code can be viewed as a subset of the vertices of a hypercube. Two vertices are adjacent in a hypercube exactly when their Hamming distance is 1. Such a code is a perfect 1-error-correcting code if no two codewords are adjacent, and if every non-codeword is adjacent to exactly one codeword. Since a perfect code can be described using only vertices and adjacency, the definition applies to general graphs rather than only to hypercubes. How does one decide if a graph can support a perfect 1-error-correcting code? The obvious way to show that such a code exists is to display the code. On the other hand, it seems difficult to show that a graph does not support such a code. We show that this intuition is right by showing that to determine if a graph has a perfect 1-error-correcting code is an NP-complete problem.