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 Vukičevi\'{c} and Trinajstić in Kekulé 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.