Forbidden subgraphs and the existence of paths and cycles passing through specified vertices

In [H. Broersma, H. Li, J. Li, F. Tian, H.J. Veldman, Cycles through subsets with large degree sums, Discrete Math. 171 (1997) 43-54], Duffus et al. showed that every connected graph G which contains no induced subgraph isomorphic to a claw or a net is traceable. They also showed that if a 2-connected graph G satisfies the above conditions, then G is hamiltonian. In this paper, modifying the conditions of Duffus et al.'s theorems, we give forbidden structures for a specified set of vertices which assures the existence of paths and cycles passing through these vertices.