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Let G be a simple graph with a perfect matching. Deng and Zhang showed thatthe 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 ofG and investigate the extremal graphs. If G has a perfect matching Mwhose anti-forcing number attains this upper bound, then we say G is anextremal graph and M is a nice perfect matching. We obtain an equivalentcondition for the nice perfect matchings of G and establish a one-to-onecorrespondence between the nice perfect matchings and the edge-involutions ofG, which are the automorphisms Ξ± of order two such that v andΞ±(v) are adjacent for every vertex v. We demonstrate that all extremalgraphs can be constructed from K2β by implementing two expansion operations,and G is extremal if and only if one factor in a Cartesian decomposition ofG is extremal. As examples, we have that all perfect matchings of thecomplete 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
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