On platypus graphs and the Steiner-Deogun property

A platypus is a non-hamiltonian graph in which every vertex-deleted subgraph is traceable. We prove a series of results on platypus graphs. For instance, although there are planar platypuses and bipartite platypuses, it is not known whether there is a planar bipartite platypus. Motivated by this question, we show that every tree is an induced subgraph of some planar platypus. On the other hand, there exists an infinite family of planar graphs each member of which is not an induced subgraph of any planar platypus. Throughout the article we point out connections between platypus graphs and graphs having the Steiner–Deogun property, as defined by Kratsch, Lehel, and Müller.