  Let $G$ be a Hamiltonian graph with $n$ vertices. A nonempty vertex set
$X\subseteq V(G)$ is called a Hamiltonian cycle enforcing set (in short, an
$H$-force set) of $G$ if every $X$-cycle of $G$ (i.e., a cycle of $G$
containing all vertices of $X$) is a Hamiltonian cycle. For the graph $G$,
$h(G)$ is the smallest cardinality of an $H$-force set of $G$ and call it the
$H$-force number of $G$. Ore's theorem states that the graph $G$ is Hamiltonian
if $d(u)+d(v)\geq n$ for every pair of nonadjacent vertices $u,v$ of $G$. In
this paper, we study the $H$-force sets of the graphs satisfying the condition
of Ore's theorem, show that the $H$-force number of these graphs is possibly
$n$, or $n-2$, or $\frac{n}{2}$ and give a classification of these graphs due
to the $H$-force number.
