HH-force sets of locally semicomplete digraphs

Let DD be a hamiltonian digraph. A nonempty vertex set X⊆V(D)X⊆V(D) is called an H-force set   of DD if every XX-cycle of DD (i.e. a cycle of DD containing all vertices of XX) is hamiltonian. The H-force number  h(D)h(D) of a digraph DD is defined to be the smallest cardinality of an HH-force set of DD. In this paper, the minimal HH-force sets of locally semicomplete digraphs are characterized and the HH-force number is given.