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 Gwhose 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 Vukičević 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 Gwhere any two either are disjoint or intersect only at edges inM . 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. © 2015 Elsevier B.V. All rights reserved.