Near automorphisms of the complement or the square of a cycle

Let $G$ be a graph with vertex set $V(G)$, $f$ a permutation of $V(G)$. Define ${\delta }_f(x,y)=|d(x,y)-d(f(x),f(y))|$ and ${\delta }_f(G)=\sum {\delta }_f(x,y)$, where the sum is taken over all unordered pairs $x$, $y$ of distinct vertices of $G$. Let $\pi (G)$ denote the smallest positive value of ${\delta }_f(G)$ among all permutations $f$ of $V(G)$. A permutation $f$ with ${\delta }_f(G)=\pi (G)$ is called a near automorphism of $G$. In this paper, the near automorphisms of the complement or the square of a cycle are characterized. Moreover, $\pi (\overline{C_n})$ and $\pi \left(C_n^2\right)$ are determined.