Summary: Let \(G\) be a Hamiltonian graph. 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) (called the H-force number of \(G)\) is the smallest cardinality of an H-force set of \(G\). Ore’s theorem states that an \(n\)-vertex graph \(G\) is Hamiltonian if \(d(u)+d(v)\ge n\) for every pair of nonadjacent vertices \(u,v\) of \(G\). In this article, 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.