Extremal anti-forcing numbers of perfect matchings of graphs

The anti-forcing number of a perfect matching M of a graph G is the minimal number of edges not in M whose removal to make M as a unique perfect matching of the resulting graph. The set of anti-forcing numbers of all perfect matchings of G is the anti-forcing spectrum of G. In this paper, we characterize the plane elementary bipartite graph whose minimum anti-forcing number is one. We show that the maximum anti-forcing number of a graph is at most its cyclomatic number. In particular, we characterize the graphs with the maximum anti-forcing number achieving the upper bound, such extremal graphs are a class of plane bipartite graphs. Finally, we determine the anti-forcing spectrum of an even polygonal chain in linear time.