  The anti-forcing number of a perfect matching $M$ of a graph $G$ is the
minimum number of edges of $G$ whose deletion results in a subgraph with a
unique perfect matching $M$, denoted by $af(G,M)$. When $G$ is a plane
bipartite graph, Lei et al. established a minimax result: For any perfect
matching $M$ of $G$, $af(G,M)$ equals the maximum number of $M$-alternating
cycles of $G$ where any two either are disjoint or intersect only at edges in
$M$; For a hexagonal system, the maximum anti-forcing number equals the fries
number. In this paper we show that for every perfect matching $M$ of a
hexagonal system $H$ with the maximum anti-forcing number or minus one,
$af(H,M)$ equals the number of $M$-alternating hexagons of $H$. Further we show
that a hexagonal system $H$ has a triphenylene as nice subgraph if and only
$af(H,M)$ always equals the number of $M$-alternating hexagons of $H$ for every
perfect matching $M$ of $H$.
