  Let $G$ be a simple graph with $2n$ vertices and a perfect matching. The
forcing number $f(G,M)$ of a perfect matching $M$ of $G$ is the smallest
cardinality of a subset of $M$ that is contained in no other perfect matching
of $G$. Among all perfect matchings $M$ of $G$, the minimum and maximum values
of $f(G,M)$ are called the minimum and maximum forcing numbers of $G$, denoted
by $f(G)$ and $F(G)$, respectively. Then $f(G)\leq F(G)\leq n-1$. Che and Chen
(2011) proposed an open problem: how to characterize the graphs $G$ with
$f(G)=n-1$. Later they showed that for a bipartite graph $G$, $f(G)=n-1$ if and
only if $G$ is a complete bipartite graph $K_{n,n}$. In this paper, we
completely solve the problem of Che and Chen, and show that $f(G)=n-1$ if and
only if $G$ is a complete multipartite graph or a graph obtained from complete
bipartite graph $K_{n,n}$ by adding arbitrary edges in the same partite set.
For all graphs $G$ with $F(G)=n-1$, we prove that the forcing spectrum of each
such graph $G$ forms an integer interval by matching 2-switches and the minimum
forcing numbers of all such graphs $G$ form an integer interval from
$\lfloor\frac{n}{2}\rfloor$ to $n-1$.
