The edit distance between two graphs on the same labeled vertex set is defined to be the size of the symmetric difference of the edge sets. The edit distance between a graph, $G$, and a graph property, $\mathcal{H}$, is the minimum edit distance between $G$ and a graph in $\mathcal{H}$. The edit distance function of a graph property $\mathcal{H}$ is a function of $p\in [0,1]$ that measures, in the limit, the maximum normalized edit distance between a graph of density $p$ and $\mathcal{H}$.

In this thesis, we address the edit distance function for the property of having no induced copy of $C_h^t$, the $t^{\mbox{th}}$ power of the cycle of length $h$. For $h\geq 2t(t+1)+1$ and $h$ not divisible by $t+1$, we determine the function for all values of $p$. For $h\geq 2t(t+1)+1$ and $h$ divisible by $t+1$, the function is obtained for all but small values of $p$. We also obtain some results for smaller values of $h$, present alternative proofs of some important previous results using simple optimization techniques and discuss possible extension of the theory to hypergraphs.