Near automorphisms of complement or square of a path

Let G be a connected graph with vertex set V(G), f a permutation of V(G). Define $$\delta _f (x,y)=|d(x,y)-d(f(x),f(y))|$$ δ f ( x , y ) = | d ( x , y ) - d ( f ( x ) , f ( y ) ) | and $$\delta _f (G)= \sum \delta _f (x,y)$$ δ f ( G ) = ∑ δ f ( x , y ) , where the sum is taken over all unordered pairs x, y of distinct vertices of G. Let $$\pi (G)$$ π ( G ) denote the smallest positive value of $$\delta _f (G)$$ δ f ( G ) among all permutations of V(G). A permutation f with $$\delta _f (G) =\pi (G)$$ δ f ( G ) = π ( G ) is called a near automorphism of G and $$\pi (G)$$ π ( 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, $$\pi (\overline{P_n})$$ π ( P n ¯ ) and $$\pi (P_n^2)$$ π ( P n 2 ) are determined. As a result, one can find how much the near automorphisms of $$\overline{P_n}$$ P n ¯ and $$P_n^2$$ P n 2 differ from those of $$P_n$$ P n .