Near automorphisms of complement or square of a path

Abstract

Let G be a connected 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 of V(G). A permutation f with $\delta _f (G) =\pi (G)$ is called a near automorphism of G and $\pi (G)$ is called the value of near automorphisms of G. In this paper, the near automorphisms of the complement of a path and the near automorphisms of the square of a path are characterized, respectively. Moreover, $\pi (\overline{P_n})$ and $\pi (P_n^2)$ are determined. As a result, one can find how much the near automorphisms of $\overline{P_n}$ and $P_n^2$ differ from those of $P_n$.