on the anti-forcing number of graph powers

Let $g=(v,e)$ be a simple connected graph. a perfect matching (or kekul’e structure in chemical literature) of $g$ is a set of disjoint edges which covers all vertices of $g$. the anti-forcing number of $g$ is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by $af(g)$.     for every $minmathbb{n}$, the $m$th power of $g$, denoted by $g^m$, is a graph with the same vertex set as $g$ such that two vertices are adjacent in $g^m$ if and only if their distance is at most $m$ in $g$. in this paper, we study the anti-forcing number of the powers of some graphs.