  In 1981, Duffus, Gould, and Jacobson showed that every connected graph either
has a Hamiltonian path, or contains a claw ($K_{1,3}$) or a net (a fixed
six-vertex graph) as an induced subgraph. This implies that subject to being
connected, these two are the only minimal (under taking induced subgraphs)
graphs with no Hamiltonian path.
  Brousek (1998) characterized the minimal graphs that are $2$-connected,
non-Hamiltonian and do not contain the claw as an induced subgraph. We
characterize the minimal graphs that are $2$-connected and non-Hamiltonian for
two classes of graphs: (1) split graphs, (2) triangle-free graphs. We remark
that testing for Hamiltonicity is NP-hard in both of these classes.
