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