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 a f ( G , M ) . The anti-forcing number of a graph proposed by Vukicevic and Trinajstic in Kekule 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: a f ( 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.

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