  A platypus graph is a non-hamiltonian graph for which every vertex-deleted
subgraph is traceable. They are closely related to families of graphs
satisfying interesting conditions regarding longest paths and longest cycles,
for instance hypohamiltonian, leaf-stable, and maximally non-hamiltonian
graphs.
  In this paper, we first investigate cubic platypus graphs, covering all
orders for which such graphs exist: in the general and polyhedral case as well
as for snarks. We then present (not necessarily cubic) platypus graphs of girth
up to 16---whereas no hypohamiltonian graphs of girth greater than 7 are
known---and study their maximum degree, generalising two theorems of Chartrand,
Gould, and Kapoor. Using computational methods, we determine the complete list
of all non-isomorphic platypus graphs for various orders and girths. Finally,
we address two questions raised by the third author in [J. Graph Theory
\textbf{86} (2017) 223--243].
