Some tight bounds on the minimum and maximum forcing numbers of graphs

Let $G$ be a simple graph with $2n$ vertices and a perfect matching. We denote by $f(G)$ and $F(G)$ the minimum and maximum forcing number of $G$, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique perfect matching is $n^2$. We know that $G$ has a unique perfect matching if and only if $f(G)=0$. Along this line, we generalize the classical result to all graphs $G$ with $f(G)=k$ for $0\leq k\leq n-1$, and characterize corresponding extremal graphs as well. Hence we get a non-trivial lower bound of $f(G)$ in terms of the order and size. For bipartite graphs, we gain corresponding stronger results. Further, we obtain a new upper bound of $F(G)$. For bipartite graphs $G$, Che and Chen (2013) obtained that $f(G)=n-1$ if and only if $G$ is complete bipartite graph $K_{n,n}$. We completely characterize all bipartite graphs $G$ with $f(G)= n-2$.