Tournaments with Feedback Path Powers

A minimum feedback arc set of a digraph is a smallest sized set of arcs whose reversal makes the resulting digraph acyclic. Given an acyclic digraph \( D \), we seek a smallest sized tournament \( T \) having \( A(D) \) as a minimum feedback arc set. The reversing number of a digraph \( D \) equals \( |V(T)| – |V(D)| \). We investigate the reversing number of the \( k \)th power of directed Hamiltonian path \( P_n^k \), when \( k \) is fixed and \( n \) tends to infinity. We show that even for small values of \( k \), where \( |A(P_n^k)| \) is much closer to \( |A(P_n)| \) than \( |A(T_n)| \), the opposite relationship holds for the reversing number.