Anti-forcing numbers of perfect matchings of graphs

We define the anti-forcing number of a perfect matching $M$ of a graph $G$ as the minimal number of edges of $G$ whose deletion results in a subgraph with a unique perfect matching $M$, denoted by $af(G,M)$. The anti-forcing number of a graph proposed by Vuki\v{c}evi'{c} and Trinajsti'c in Kekul'e structures of molecular graphs is in fact the minimum anti-forcing number of perfect matchings. For plane bipartite graph $G$ with a perfect matching $M$, we obtain a minimax result: $af(G,M)$ equals the maximal number of $M$-alternating cycles of $G$ where any two either are disjoint or intersect only at edges in $M$. For a hexagonal system $H$, we show that the maximum anti-forcing number of $H$ equals the Fries number of $H$. As a consequence, we have that the Fries number of $H$ is between the Clar number of $H$ and twice. Further, some extremal graphs are discussed.