Let G be a simple graph with a perfect matching. Deng and Zhang showed that
the maximum anti-forcing number of G is no more than the cyclomatic number.
In this paper, we get a novel upper bound on the maximum anti-forcing number of
G and investigate the extremal graphs. If G has a perfect matching M
whose anti-forcing number attains this upper bound, then we say G is an
extremal graph and M is a nice perfect matching. We obtain an equivalent
condition for the nice perfect matchings of G and establish a one-to-one
correspondence between the nice perfect matchings and the edge-involutions of
G, which are the automorphisms Ξ± of order two such that v and
Ξ±(v) are adjacent for every vertex v. We demonstrate that all extremal
graphs can be constructed from K2β by implementing two expansion operations,
and G is extremal if and only if one factor in a Cartesian decomposition of
G is extremal. As examples, we have that all perfect matchings of the
complete graph K2nβ and the complete bipartite graph Kn,nβ are nice.
Also we show that the hypercube Qnβ, the folded hypercube FQnβ (nβ₯4)
and the enhanced hypercube Qn,kβ (0β€kβ€nβ4) have exactly n,
n+1 and n+1 nice perfect matchings respectively.Comment: 15 pages, 7 figure