Geodesic
On Vertices Enforcing a Hamiltonian Cycle
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 71-89

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A nonempty vertex set X ⊆ V (G) of a hamiltonian graph G is called an H-force set of G if every X-cycle of G (i.e. a cycle of G containing all vertices of X) is hamiltonian. The H-force number h(G) of a graph G is defined to be the smallest cardinality of an H-force set of G. In the paper the study of this parameter is introduced and its value or a lower bound for outerplanar graphs, planar graphs, k-connected graphs and prisms over graphs is determined.
Keywords: cycle, hamiltonian, 1-hamiltonian 
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Fabrici, Igor; Hexel, Erhard; Jendrol’, Stanislav. On Vertices Enforcing a Hamiltonian Cycle. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 71-89. http://geodesic.mathdoc.fr/item/DMGT_2013_33_1_a6/